Subgroups — C₁₀₉×#22648 #42140642

Order 27904.

1116 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1940	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#1851	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1850	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#1797	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1767	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	#435×C₁₀₉	—	2	8	no	—
20	#423×C₁₀₉	—	2	4	no	—
21	C₂×C₄⋊C₈×C₁₀₉	—	2	4	no	—
22	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
32	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
43	#336×C₁₀₉	C₂²	1	2	yes	—
44	#336×C₁₀₉	C₂²	1	2	yes	—
45	#336×C₁₀₉	C₂²	1	2	yes	—
46	C₂²×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₁₀₉	—	2	4	no	—
54	C₂×M₁₆×C₁₀₉	—	2	4	no	—
55	C₂×M₁₆×C₁₀₉	—	4	2	no	—
56	C₂²×C₈×C₁₀₉	—	2	4	no	—
57	C₂²×C₈×C₁₀₉	—	2	4	no	—
58	C₂²×C₈×C₁₀₉	—	4	2	no	—
59	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
62	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
63	C₄×D₄×C₁₀₉	—	2	4	no	—
64	C₄×D₄×C₁₀₉	—	2	4	no	—
65	C₄×D₄×C₁₀₉	D₄	1	4	yes	—
66	C₄×D₄×C₁₀₉	Q₈	1	4	yes	—
67	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₁₀₉	—	2	2	no	—
77	C₂×C₄²×C₁₀₉	—	2	2	no	—
78	C₂×C₄²×C₁₀₉	—	2	2	no	—
79	C₂×C₄²×C₁₀₉	—	2	2	no	—
80	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₁₀₉	—	2	4	no	—
94	C₄⋊C₈×C₁₀₉	—	2	4	no	—
95	C₄⋊C₈×C₁₀₉	—	2	2	no	—
96	C₄⋊C₈×C₁₀₉	—	2	2	no	—
97	C₄⋊C₈×C₁₀₉	—	2	4	no	—
98	C₄⋊C₈×C₁₀₉	—	2	2	no	—
99	C₄⋊C₈×C₁₀₉	—	4	2	no	—
100	C₄⋊C₈×C₁₀₉	D₄	1	4	yes	complemented
101	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
102	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
103	#97×C₁₀₉	—	2	4	no	—
104	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
105	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
106	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
107	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
108	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
109	#95×C₁₀₉	C₂×C₄	1	1	yes	characteristic
110	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
111	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×D₄×C₁₀₉	—	2	2	no	—
113	C₂×D₄×C₁₀₉	—	2	2	no	—
114	C₂×D₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
115	C₂²×C₄×C₁₀₉	—	2	2	no	—
116	C₂²×C₄×C₁₀₉	—	2	2	no	—
117	C₂²×C₄×C₁₀₉	—	2	1	no	—
118	C₂²×C₄×C₁₀₉	—	2	1	no	—
119	C₂²×C₄×C₁₀₉	—	2	1	no	—
120	C₂²×C₄×C₁₀₉	—	2	1	no	—
121	C₂²×C₄×C₁₀₉	—	2	1	no	—
122	C₂²×C₄×C₁₀₉	—	2	1	no	—
123	C₂²×C₄×C₁₀₉	—	2	1	no	—
124	C₂²×C₄×C₁₀₉	—	2	1	no	—
125	C₂²×C₄×C₁₀₉	—	2	2	no	—
126	C₂²×C₄×C₁₀₉	—	2	2	no	—
127	C₂²×C₄×C₁₀₉	—	2	2	no	—
128	C₂²×C₄×C₁₀₉	—	2	2	no	—
129	C₂²×C₄×C₁₀₉	—	2	2	no	—
130	C₂²×C₄×C₁₀₉	—	2	2	no	—
131	C₂²×C₄×C₁₀₉	—	2	2	no	—
132	C₂²×C₄×C₁₀₉	—	4	1	no	—
133	C₂²×C₄×C₁₀₉	—	4	1	no	—
134	C₂²×C₄×C₁₀₉	M₁₆	1	4	yes	complemented
135	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
150	C₂²×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
151	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₁₀₉	—	4	8	no	—
153	C₂×C₈×C₁₀₉	—	2	2	no	—
154	C₂×C₈×C₁₀₉	—	2	2	no	—
155	C₂×C₈×C₁₀₉	—	2	2	no	—
156	C₂×C₈×C₁₀₉	—	2	2	no	—
157	C₂×C₈×C₁₀₉	—	2	2	no	—
158	C₂×C₈×C₁₀₉	—	2	2	no	—
159	C₂×C₈×C₁₀₉	—	2	2	no	—
160	C₂×C₈×C₁₀₉	—	2	2	no	—
161	C₂×C₈×C₁₀₉	—	2	2	no	—
162	C₂×C₈×C₁₀₉	—	2	2	no	—
163	C₂×C₈×C₁₀₉	—	2	2	no	—
164	C₂×C₈×C₁₀₉	—	4	8	no	—
165	C₂×C₈×C₁₀₉	C₂×D₄	1	4	yes	—
166	C₄⋊C₄×C₁₀₉	—	2	2	no	—
167	C₄⋊C₄×C₁₀₉	—	2	2	no	—
168	C₄⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
169	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
170	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
171	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
172	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
173	C₂²⋊C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
174	C₂²⋊C₄×C₁₀₉	M₁₆	1	4	yes	complemented
175	C₄²×C₁₀₉	—	2	2	no	—
176	C₄²×C₁₀₉	—	2	2	no	—
177	C₄²×C₁₀₉	—	2	2	no	—
178	C₄²×C₁₀₉	—	2	2	no	—
179	C₄²×C₁₀₉	—	2	2	no	—
180	C₄²×C₁₀₉	—	2	2	no	—
181	C₄²×C₁₀₉	—	2	2	no	—
182	C₄²×C₁₀₉	—	2	1	no	—
183	C₄²×C₁₀₉	—	2	1	no	—
184	C₄²×C₁₀₉	—	2	1	no	—
185	C₄²×C₁₀₉	—	2	1	no	—
186	C₄²×C₁₀₉	—	2	1	no	—
187	C₄²×C₁₀₉	—	2	1	no	—
188	C₄²×C₁₀₉	—	4	1	no	—
189	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
190	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
191	C₄²×C₁₀₉	C₂×Q₈	1	2	yes	—
192	C₄²×C₁₀₉	#41	1	2	yes	—
193	C₂³×C₁₀₉	—	2	1	no	—
194	C₂³×C₁₀₉	—	2	1	no	—
195	C₂³×C₁₀₉	—	2	1	no	—
196	C₂³×C₁₀₉	—	2	1	no	—
197	C₂³×C₁₀₉	—	2	1	no	—
198	C₂³×C₁₀₉	—	2	1	no	—
199	C₂³×C₁₀₉	—	2	1	no	—
200	C₂³×C₁₀₉	—	2	1	no	—
201	C₂³×C₁₀₉	—	2	1	no	—
202	C₂³×C₁₀₉	—	2	1	no	—
203	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
204	C₂³×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
205	C₂³×C₁₀₉	C₂×M₁₆	1	2	yes	—
206	C₂³×C₁₀₉	C₂×M₁₆	1	2	yes	—
207	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
208	D₄×C₁₀₉	—	2	8	no	—
209	C₂×C₄×C₁₀₉	—	2	1	no	—
210	C₂×C₄×C₁₀₉	—	2	1	no	—
211	C₂×C₄×C₁₀₉	—	2	1	no	—
212	C₂×C₄×C₁₀₉	—	2	1	no	—
213	C₂×C₄×C₁₀₉	—	2	1	no	—
214	C₂×C₄×C₁₀₉	—	2	1	no	—
215	C₂×C₄×C₁₀₉	—	2	1	no	—
216	C₂×C₄×C₁₀₉	—	2	1	no	—
217	C₂×C₄×C₁₀₉	—	2	1	no	—
218	C₂×C₄×C₁₀₉	—	2	1	no	—
219	C₂×C₄×C₁₀₉	—	2	1	no	—
220	C₂×C₄×C₁₀₉	—	2	1	no	—
221	C₂×C₄×C₁₀₉	—	2	1	no	—
222	C₂×C₄×C₁₀₉	—	2	1	no	—
223	C₂×C₄×C₁₀₉	—	2	1	no	—
224	C₂×C₄×C₁₀₉	—	2	1	no	—
225	C₂×C₄×C₁₀₉	—	2	2	no	—
226	C₂×C₄×C₁₀₉	—	2	2	no	—
227	C₂×C₄×C₁₀₉	—	2	2	no	—
228	C₂×C₄×C₁₀₉	—	2	2	no	—
229	C₂×C₄×C₁₀₉	—	2	2	no	—
230	C₂×C₄×C₁₀₉	—	2	2	no	—
231	C₂×C₄×C₁₀₉	—	2	2	no	—
232	C₂×C₄×C₁₀₉	—	2	2	no	—
233	C₂×C₄×C₁₀₉	—	2	8	no	—
234	C₂×C₄×C₁₀₉	—	2	1	no	—
235	C₂×C₄×C₁₀₉	—	2	1	no	—
236	C₂×C₄×C₁₀₉	—	2	1	no	—
237	C₂×C₄×C₁₀₉	—	2	1	no	—
238	C₂×C₄×C₁₀₉	—	2	1	no	—
239	C₂×C₄×C₁₀₉	—	2	1	no	—
240	C₂×C₄×C₁₀₉	—	2	1	no	—
241	C₂×C₄×C₁₀₉	—	2	1	no	—
242	C₂×C₄×C₁₀₉	—	2	1	no	—
243	C₂×C₄×C₁₀₉	—	4	4	no	—
244	C₂×C₄×C₁₀₉	—	4	4	no	—
245	C₂×C₄×C₁₀₉	—	4	4	no	—
246	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
247	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
248	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
249	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
250	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
251	C₂×C₄×C₁₀₉	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
253	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
254	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
255	C₂×C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
256	C₂×C₄×C₁₀₉	#131	1	2	yes	—
257	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
260	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
262	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
263	C₈×C₁₀₉	—	2	4	no	—
264	C₈×C₁₀₉	—	4	2	no	—
265	C₈×C₁₀₉	—	4	2	no	—
266	C₈×C₁₀₉	—	4	2	no	—
267	C₂²×C₁₀₉	—	2	2	no	—
268	C₂²×C₁₀₉	—	2	2	no	—
269	C₂²×C₁₀₉	—	2	2	no	—
270	C₂²×C₁₀₉	—	2	2	no	—
271	C₂²×C₁₀₉	—	2	2	no	—
272	C₂²×C₁₀₉	—	2	2	no	—
273	C₂²×C₁₀₉	—	2	2	no	—
274	C₂²×C₁₀₉	—	2	2	no	—
275	C₂²×C₁₀₉	—	2	2	no	—
276	C₂²×C₁₀₉	—	4	1	no	—
277	C₂²×C₁₀₉	—	4	1	no	—
278	C₂²×C₁₀₉	—	4	1	no	—
279	C₂²×C₁₀₉	—	4	1	no	—
280	C₂²×C₁₀₉	#423	1	4	yes	complemented
281	C₂²×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
282	C₂²×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
283	C₂²×C₁₀₉	#520	1	1	yes	characteristic
284	C₂²×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
285	C₂²×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
286	C₂²×C₁₀₉	C₂×#131	1	1	yes	characteristic
287	C₂²×C₁₀₉	#568	1	1	yes	characteristic
288	C₄×C₁₀₉	—	2	2	no	—
289	C₄×C₁₀₉	—	2	2	no	—
290	C₄×C₁₀₉	—	2	2	no	—
291	C₄×C₁₀₉	—	2	2	no	—
292	C₄×C₁₀₉	—	2	2	no	—
293	C₄×C₁₀₉	—	2	2	no	—
294	C₄×C₁₀₉	—	2	2	no	—
295	C₄×C₁₀₉	—	2	2	no	—
296	C₄×C₁₀₉	—	4	1	no	—
297	C₄×C₁₀₉	—	4	1	no	—
298	C₄×C₁₀₉	—	4	1	no	—
299	C₄×C₁₀₉	#435	1	4	yes	complemented
300	C₄×C₁₀₉	D₄²	1	1	yes	characteristic
301	C₄×C₁₀₉	#547	1	1	yes	characteristic
302	C₄×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
303	C₄×C₁₀₉	#554	1	1	yes	characteristic
304	#22648	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₁₀₉	—	2	4	no	—
306	C₂×C₁₀₉	—	4	2	no	—
307	C₂×C₁₀₉	D₄×C₄⋊C₄	1	1	yes	characteristic
308	C₂×C₁₀₉	C₂×#423	1	1	yes	characteristic
309	C₂×C₁₀₉	#2906	1	1	yes	characteristic
310	C₂×C₁₀₉	C₂×#435	1	1	yes	characteristic
311	C₂×C₁₀₉	D₄×M₁₆	1	1	yes	characteristic
312	C₂×C₁₀₉	#2989	1	1	yes	characteristic
313	C₂×C₁₀₉	#2990	1	1	yes	characteristic
314	C₂×#435	C₂×C₁₀₉	1	2	yes	—
315	C₂×#423	C₂×C₁₀₉	1	1	yes	characteristic, complemented
316	C₂²×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic, complemented
317	C₄²×D₄	C₂×C₁₀₉	1	1	yes	characteristic
318	#1940	C₂×C₁₀₉	1	2	yes	complemented
319	#1851	C₂×C₁₀₉	1	2	yes	complemented
320	#1850	C₂×C₁₀₉	1	2	yes	complemented
321	#1842	C₂×C₁₀₉	1	1	yes	characteristic, complemented
322	#1797	C₂×C₁₀₉	1	1	yes	characteristic, complemented
323	#1797	C₂×C₁₀₉	1	1	yes	characteristic, complemented
324	#1767	C₂×C₁₀₉	1	1	yes	characteristic, complemented
325	C₁₀₉	#22648	1	1	yes	characteristic, direct product factor, Sylow
326	C₂²×M₁₆	C₂²×C₁₀₉	1	2	yes	—
327	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
328	C₂×C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
329	C₂×C₄×D₄	C₄×C₁₀₉	1	1	yes	characteristic
330	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
331	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
332	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
333	#435	—	2	8	no	—
334	#423	—	2	4	no	—
335	C₂×C₄⋊C₈	—	2	4	no	—
336	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
337	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
339	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
340	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
341	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
342	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
343	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
345	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
346	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
347	C₄×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
350	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
351	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
352	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
353	C₄×C₂²⋊C₄	C₄×C₁₀₉	1	1	yes	characteristic
354	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
355	C₄×C₂²⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
356	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
357	#336	C₂²×C₁₀₉	1	2	yes	—
358	#336	C₂²×C₁₀₉	1	2	yes	—
359	#336	C₂²×C₁₀₉	1	2	yes	—
360	C₂²×D₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
361	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
362	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
363	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
364	C₂³×C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
365	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
367	C₂×M₁₆	—	2	4	no	—
368	C₂×M₁₆	—	2	4	no	—
369	C₂×M₁₆	—	4	2	no	—
370	C₂²×C₈	—	2	4	no	—
371	C₂²×C₈	—	2	4	no	—
372	C₂²×C₈	—	4	2	no	—
373	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
374	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
375	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
376	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	D₄×C₁₀₉	1	4	yes	—
380	C₄×D₄	Q₈×C₁₀₉	1	4	yes	—
381	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
386	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	C₂×C₄×C₁₀₉	1	1	yes	characteristic
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	—	2	2	no	—
393	C₂×C₄²	—	2	2	no	—
394	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
395	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
396	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
397	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
398	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
399	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
400	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
401	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
407	C₄⋊C₈	—	2	4	no	—
408	C₄⋊C₈	—	2	4	no	—
409	C₄⋊C₈	—	2	2	no	—
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	2	4	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	4	2	no	—
414	C₄⋊C₈	D₄×C₁₀₉	1	4	yes	complemented
415	C₂²⋊C₈	—	2	4	no	—
416	C₂²⋊C₈	—	2	4	no	—
417	#97	—	2	4	no	—
418	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
419	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
420	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
421	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
422	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
423	#95	C₂×C₄×C₁₀₉	1	1	yes	characteristic
424	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
425	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	M₁₆×C₁₀₉	1	4	yes	complemented
449	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
459	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
460	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
462	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
463	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
464	C₂²×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
466	M₁₆	—	4	8	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	4	8	no	—
479	C₂×C₈	C₂×D₄×C₁₀₉	1	4	yes	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
487	C₂²⋊C₄	C₄⋊C₄×C₁₀₉	1	4	yes	—
488	C₂²⋊C₄	M₁₆×C₁₀₉	1	4	yes	complemented
489	C₄²	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	4	1	no	—
503	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
504	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
505	C₄²	C₂×Q₈×C₁₀₉	1	2	yes	—
506	C₄²	#41×C₁₀₉	1	2	yes	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	1	no	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
518	C₂³	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
519	C₂³	C₂×M₁₆×C₁₀₉	1	2	yes	—
520	C₂³	C₂×M₁₆×C₁₀₉	1	2	yes	—
521	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
522	D₄	—	2	8	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	8	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	4	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
561	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
562	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
563	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	2	yes	—
564	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
565	C₂×C₄	C₄×D₄×C₁₀₉	1	2	yes	—
566	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
567	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
568	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
569	C₂×C₄	C₂×M₁₆×C₁₀₉	1	2	yes	—
570	C₂×C₄	#131×C₁₀₉	1	2	yes	—
571	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
573	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
574	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
575	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
576	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
577	C₈	—	2	4	no	—
578	C₈	—	4	2	no	—
579	C₈	—	4	2	no	—
580	C₈	—	4	2	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	#423×C₁₀₉	1	4	yes	complemented
595	C₂²	C₂²×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
596	C₂²	C₂×C₄×D₄×C₁₀₉	1	1	yes	characteristic
597	C₂²	#520×C₁₀₉	1	1	yes	characteristic
598	C₂²	C₂²×M₁₆×C₁₀₉	1	1	yes	characteristic
599	C₂²	C₂²×M₁₆×C₁₀₉	1	1	yes	characteristic
600	C₂²	C₂×#131×C₁₀₉	1	1	yes	characteristic
601	C₂²	#568×C₁₀₉	1	1	yes	characteristic
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	2	no	—
609	C₄	—	2	2	no	—
610	C₄	—	4	1	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	#435×C₁₀₉	1	4	yes	complemented
614	C₄	D₄²×C₁₀₉	1	1	yes	characteristic
615	C₄	#547×C₁₀₉	1	1	yes	characteristic
616	C₄	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
617	C₄	#554×C₁₀₉	1	1	yes	characteristic
618	C₂	—	2	4	no	—
619	C₂	—	4	2	no	—
620	C₂	D₄×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
621	C₂	C₂×#423×C₁₀₉	1	1	yes	characteristic
622	C₂	C₁₀₉×#2906	1	1	yes	characteristic
623	C₂	C₂×#435×C₁₀₉	1	1	yes	characteristic
624	C₂	D₄×M₁₆×C₁₀₉	1	1	yes	characteristic
625	C₂	C₁₀₉×#2989	1	1	yes	characteristic
626	C₂	C₁₀₉×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂³×C₄
prime radical	C₂²×D₄×C₁₀₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×#423×C₁₀₉	C₂	semidirect
3. C₂²×C₄⋊C₈×C₁₀₉	C₂	semidirect
5. C₁₀₉×#1940	C₂	semidirect
6. C₁₀₉×#1851	C₂	semidirect
7. C₁₀₉×#1850	C₂	semidirect
8. C₁₀₉×#1842	C₂	semidirect
9. C₁₀₉×#1797	C₂	semidirect
10. C₁₀₉×#1797	C₂	semidirect
11. C₁₀₉×#1767	C₂	semidirect
100. C₄⋊C₈×C₁₀₉	D₄	semidirect
134. C₂²×C₄×C₁₀₉	M₁₆	semidirect
174. C₂²⋊C₄×C₁₀₉	M₁₆	semidirect
280. C₂²×C₁₀₉	#423	semidirect
299. C₄×C₁₀₉	#435	semidirect
304. #22648	C₁₀₉	direct (N × H)
315. C₂×#423	C₂×C₁₀₉	semidirect
316. C₂²×C₄⋊C₈	C₂×C₁₀₉	semidirect
318. #1940	C₂×C₁₀₉	semidirect
319. #1851	C₂×C₁₀₉	semidirect
320. #1850	C₂×C₁₀₉	semidirect
321. #1842	C₂×C₁₀₉	semidirect
322. #1797	C₂×C₁₀₉	semidirect
323. #1797	C₂×C₁₀₉	semidirect
324. #1767	C₂×C₁₀₉	semidirect
325. C₁₀₉	#22648	direct (N × H)
414. C₄⋊C₈	D₄×C₁₀₉	semidirect
448. C₂²×C₄	M₁₆×C₁₀₉	semidirect
488. C₂²⋊C₄	M₁₆×C₁₀₉	semidirect
594. C₂²	#423×C₁₀₉	semidirect
613. C₄	#435×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#435×C₁₀₉	2
C₂×#423×C₁₀₉	2
C₂²×C₄⋊C₈×C₁₀₉	2
C₄²×D₄×C₁₀₉	2
C₁₀₉×#1940	2
C₁₀₉×#1851	2
C₁₀₉×#1850	2
C₁₀₉×#1842	2
C₁₀₉×#1797	2
C₁₀₉×#1767	2
#22648	109

Subgroups — C109×#22648 #42140642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22648 #42140642