Subgroups — C₁₀₉×#22522 #42140516

Order 27904.

1110 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#446×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1992	C₂	1	2	yes	maximal, complemented
5	C₁₀₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1989	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1802	C₂	1	1	yes	maximal, characteristic
9	C₁₀₉×#1802	C₂	1	2	yes	maximal
10	C₁₀₉×#1762	C₂	1	2	yes	maximal
11	C₄×#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic
12	#554×C₁₀₉	—	2	4	no	—
13	#554×C₁₀₉	C₄	1	8	yes	complemented
14	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×#125×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
18	C₂×C₄×Q₈×C₁₀₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
23	#446×C₁₀₉	—	2	4	no	—
24	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
28	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×#97×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	#389×C₁₀₉	—	2	1	no	—
32	#389×C₁₀₉	—	2	2	no	—
33	#384×C₁₀₉	C₂²	1	1	yes	characteristic
34	#384×C₁₀₉	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
37	#340×C₁₀₉	C₂²	1	2	yes	—
38	#340×C₁₀₉	C₂²	1	2	yes	—
39	#340×C₁₀₉	C₂²	1	1	yes	characteristic
40	#340×C₁₀₉	C₂²	1	2	yes	—
41	#340×C₁₀₉	C₂²	1	2	yes	—
42	#340×C₁₀₉	C₂²	1	2	yes	—
43	#340×C₁₀₉	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₁₀₉	—	2	1	no	—
45	C₂²×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₁₀₉	—	2	1	no	—
48	C₂³×C₄×C₁₀₉	—	2	2	no	—
49	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
53	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
54	#125×C₁₀₉	—	2	4	no	—
55	#125×C₁₀₉	C₂×C₄	1	8	yes	—
56	C₂²⋊Q₈×C₁₀₉	—	2	8	no	—
57	C₂²⋊Q₈×C₁₀₉	—	2	16	no	—
58	C₄×Q₈×C₁₀₉	—	4	4	no	—
59	C₄×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
60	C₄×Q₈×C₁₀₉	D₄	1	4	yes	—
61	C₄×D₄×C₁₀₉	—	2	2	no	—
62	C₄×D₄×C₁₀₉	—	2	4	no	—
63	C₄×D₄×C₁₀₉	C₂×C₄	1	8	yes	—
64	C₄×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₁₀₉	—	2	8	no	—
69	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
70	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
73	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₄²×C₁₀₉	—	2	2	no	—
83	C₂×C₄²×C₁₀₉	—	2	1	no	—
84	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
85	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₁₀₉	—	2	2	no	—
87	C₄⋊C₈×C₁₀₉	—	2	4	no	—
88	#102×C₁₀₉	—	2	2	no	—
89	#102×C₁₀₉	—	2	4	no	—
90	#102×C₁₀₉	Q₈	1	4	yes	complemented
91	#97×C₁₀₉	—	2	4	no	—
92	C₄×C₈×C₁₀₉	—	2	2	no	—
93	#95×C₁₀₉	—	2	1	no	—
94	#95×C₁₀₉	—	2	2	no	—
95	C₂⁴×C₁₀₉	—	2	1	no	—
96	C₂×Q₈×C₁₀₉	—	2	4	no	—
97	C₂×Q₈×C₁₀₉	—	2	2	no	—
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	—	8	1	no	—
100	C₂×Q₈×C₁₀₉	—	8	1	no	—
101	C₂×Q₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
102	C₂×D₄×C₁₀₉	—	2	1	no	—
103	C₂×D₄×C₁₀₉	—	2	1	no	—
104	C₂×D₄×C₁₀₉	—	2	1	no	—
105	C₂×D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
107	C₂×D₄×C₁₀₉	#41	1	2	yes	—
108	C₂²×C₄×C₁₀₉	—	2	1	no	—
109	C₂²×C₄×C₁₀₉	—	2	1	no	—
110	C₂²×C₄×C₁₀₉	—	2	2	no	—
111	C₂²×C₄×C₁₀₉	—	2	4	no	—
112	C₂²×C₄×C₁₀₉	—	2	8	no	—
113	C₂²×C₄×C₁₀₉	—	2	2	no	—
114	C₂²×C₄×C₁₀₉	—	2	1	no	—
115	C₂²×C₄×C₁₀₉	—	4	2	no	—
116	C₂²×C₄×C₁₀₉	—	4	2	no	—
117	C₂²×C₄×C₁₀₉	—	8	2	no	—
118	C₂²×C₄×C₁₀₉	—	8	2	no	—
119	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₁₀₉	—	2	1	no	—
125	C₂×C₈×C₁₀₉	—	2	1	no	—
126	C₂×C₈×C₁₀₉	—	2	1	no	—
127	C₂×C₈×C₁₀₉	—	2	2	no	—
128	C₂×C₈×C₁₀₉	—	2	1	no	—
129	C₂×C₈×C₁₀₉	—	2	2	no	—
130	C₂×C₈×C₁₀₉	—	2	1	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
133	C₄⋊C₄×C₁₀₉	—	2	8	no	—
134	C₄⋊C₄×C₁₀₉	—	2	4	no	—
135	C₄⋊C₄×C₁₀₉	—	2	8	no	—
136	C₄⋊C₄×C₁₀₉	—	2	1	no	—
137	C₄⋊C₄×C₁₀₉	—	2	2	no	—
138	C₄⋊C₄×C₁₀₉	—	2	1	no	—
139	C₄⋊C₄×C₁₀₉	—	2	1	no	—
140	C₄⋊C₄×C₁₀₉	—	2	2	no	—
141	C₄⋊C₄×C₁₀₉	—	2	1	no	—
142	C₄⋊C₄×C₁₀₉	—	2	2	no	—
143	C₄⋊C₄×C₁₀₉	—	4	4	no	—
144	C₄⋊C₄×C₁₀₉	—	4	2	no	—
145	C₄⋊C₄×C₁₀₉	—	8	2	no	—
146	C₄⋊C₄×C₁₀₉	—	8	1	no	—
147	C₄⋊C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
148	C₄⋊C₄×C₁₀₉	C₂²⋊C₄	1	8	yes	—
149	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
150	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
151	C₄⋊C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
152	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
153	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
154	C₄²×C₁₀₉	—	4	4	no	—
155	C₄²×C₁₀₉	—	4	2	no	—
156	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
157	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
158	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
159	C₄²×C₁₀₉	C₂×D₄	1	4	yes	—
160	C₂³×C₁₀₉	—	2	2	no	—
161	C₂³×C₁₀₉	—	2	2	no	—
162	C₂³×C₁₀₉	—	2	2	no	—
163	C₂³×C₁₀₉	—	4	1	no	—
164	C₂³×C₁₀₉	—	4	1	no	—
165	C₂³×C₁₀₉	—	4	1	no	—
166	C₂³×C₁₀₉	—	4	1	no	—
167	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₁₀₉	—	2	8	no	—
169	Q₈×C₁₀₉	—	8	4	no	—
170	D₄×C₁₀₉	—	2	2	no	—
171	D₄×C₁₀₉	—	2	2	no	—
172	D₄×C₁₀₉	—	2	2	no	—
173	D₄×C₁₀₉	C₄×Q₈	1	4	yes	complemented
174	C₂×C₄×C₁₀₉	—	2	2	no	—
175	C₂×C₄×C₁₀₉	—	2	2	no	—
176	C₂×C₄×C₁₀₉	—	2	2	no	—
177	C₂×C₄×C₁₀₉	—	2	1	no	—
178	C₂×C₄×C₁₀₉	—	2	2	no	—
179	C₂×C₄×C₁₀₉	—	2	1	no	—
180	C₂×C₄×C₁₀₉	—	2	2	no	—
181	C₂×C₄×C₁₀₉	—	2	4	no	—
182	C₂×C₄×C₁₀₉	—	2	2	no	—
183	C₂×C₄×C₁₀₉	—	4	1	no	—
184	C₂×C₄×C₁₀₉	—	4	1	no	—
185	C₂×C₄×C₁₀₉	—	4	1	no	—
186	C₂×C₄×C₁₀₉	—	4	2	no	—
187	C₂×C₄×C₁₀₉	—	4	1	no	—
188	C₂×C₄×C₁₀₉	—	4	2	no	—
189	C₂×C₄×C₁₀₉	—	4	8	no	—
190	C₂×C₄×C₁₀₉	—	4	2	no	—
191	C₂×C₄×C₁₀₉	—	4	1	no	—
192	C₂×C₄×C₁₀₉	—	4	2	no	—
193	C₂×C₄×C₁₀₉	—	4	1	no	—
194	C₂×C₄×C₁₀₉	—	4	1	no	—
195	C₂×C₄×C₁₀₉	—	4	2	no	—
196	C₂×C₄×C₁₀₉	—	8	8	no	—
197	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
198	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
199	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
200	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
201	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
202	C₂×C₄×C₁₀₉	#137	1	2	yes	—
203	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
206	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₁₀₉	—	2	2	no	—
210	C₈×C₁₀₉	—	4	1	no	—
211	C₈×C₁₀₉	—	4	2	no	—
212	C₂²×C₁₀₉	—	2	4	no	—
213	C₂²×C₁₀₉	—	4	2	no	—
214	C₂²×C₁₀₉	—	4	2	no	—
215	C₂²×C₁₀₉	—	4	2	no	—
216	C₂²×C₁₀₉	—	4	2	no	—
217	C₂²×C₁₀₉	—	4	2	no	—
218	C₂²×C₁₀₉	—	4	2	no	—
219	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₁₀₉	#548	1	1	yes	characteristic
221	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
222	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₁₀₉	C₂×#137	1	1	yes	characteristic
224	C₂²×C₁₀₉	#576	1	1	yes	characteristic
225	C₂²×C₁₀₉	#577	1	1	yes	characteristic
226	C₄×C₁₀₉	—	2	4	no	—
227	C₄×C₁₀₉	—	2	4	no	—
228	C₄×C₁₀₉	—	2	2	no	—
229	C₄×C₁₀₉	—	4	2	no	—
230	C₄×C₁₀₉	—	4	2	no	—
231	C₄×C₁₀₉	—	4	4	no	—
232	C₄×C₁₀₉	—	8	2	no	—
233	C₄×C₁₀₉	—	8	1	no	—
234	C₄×C₁₀₉	#417	1	4	yes	—
235	C₄×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
236	C₄×C₁₀₉	#519	1	1	yes	characteristic
237	C₄×C₁₀₉	#548	1	1	yes	characteristic
238	C₄×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
239	#22522	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₁₀₉	—	4	4	no	—
241	C₂×C₁₀₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₁₀₉	C₂×#417	1	1	yes	characteristic
243	C₂×C₁₀₉	#2897	1	1	yes	characteristic
244	C₂×C₁₀₉	#3366	1	1	yes	characteristic
245	C₂×C₁₀₉	#3372	1	1	yes	characteristic
246	C₂×C₁₀₉	#3384	1	1	yes	characteristic
247	C₂×C₁₀₉	#3385	1	1	yes	characteristic
248	C₂×#554	C₂×C₁₀₉	1	1	yes	characteristic
249	C₂×#446	C₂×C₁₀₉	1	1	yes	characteristic, complemented
250	Q₈×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
251	#1992	C₂×C₁₀₉	1	2	yes	complemented
252	#1990	C₂×C₁₀₉	1	1	yes	characteristic, complemented
253	#1989	C₂×C₁₀₉	1	2	yes	complemented
254	#1987	C₂×C₁₀₉	1	1	yes	characteristic, complemented
255	#1802	C₂×C₁₀₉	1	1	yes	characteristic
256	#1802	C₂×C₁₀₉	1	2	yes	—
257	#1762	C₂×C₁₀₉	1	2	yes	—
258	C₄×#102	C₂×C₁₀₉	1	1	yes	characteristic
259	C₁₀₉	#22522	1	1	yes	characteristic, direct product factor, Sylow
260	#554	—	2	4	no	—
261	#554	C₄×C₁₀₉	1	8	yes	complemented
262	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
263	C₂×#125	C₂²×C₁₀₉	1	2	yes	—
264	C₂×C₂²⋊Q₈	—	2	1	no	—
265	C₂×C₂²⋊Q₈	—	2	2	no	—
266	C₂×C₄×Q₈	—	2	1	no	—
267	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
270	C₂²×C₄⋊C₄	—	2	1	no	—
271	#446	—	2	4	no	—
272	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
274	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
275	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
276	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
277	C₂×#97	C₂²×C₁₀₉	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
279	#389	—	2	1	no	—
280	#389	—	2	2	no	—
281	#384	C₂²×C₁₀₉	1	1	yes	characteristic
282	#384	C₂²×C₁₀₉	1	2	yes	—
283	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
284	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
285	#340	C₂²×C₁₀₉	1	2	yes	—
286	#340	C₂²×C₁₀₉	1	2	yes	—
287	#340	C₂²×C₁₀₉	1	2	yes	—
288	#340	C₂²×C₁₀₉	1	2	yes	—
289	#340	C₂²×C₁₀₉	1	1	yes	characteristic
290	#340	C₂²×C₁₀₉	1	2	yes	—
291	#340	C₂²×C₁₀₉	1	1	yes	characteristic
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	D₄×C₁₀₉	1	1	yes	characteristic
294	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
295	C₂³×C₄	—	2	1	no	—
296	C₂³×C₄	—	2	2	no	—
297	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
299	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	4	yes	—
302	#125	—	2	4	no	—
303	#125	C₂×C₄×C₁₀₉	1	8	yes	—
304	C₂²⋊Q₈	—	2	8	no	—
305	C₂²⋊Q₈	—	2	16	no	—
306	C₄×Q₈	—	4	4	no	—
307	C₄×Q₈	C₂×C₄×C₁₀₉	1	4	yes	—
308	C₄×Q₈	D₄×C₁₀₉	1	4	yes	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	4	no	—
311	C₄×D₄	C₂×C₄×C₁₀₉	1	8	yes	—
312	C₄×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	8	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
333	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	2	4	no	—
336	#102	—	2	2	no	—
337	#102	—	2	4	no	—
338	#102	Q₈×C₁₀₉	1	4	yes	complemented
339	#97	—	2	4	no	—
340	C₄×C₈	—	2	2	no	—
341	#95	—	2	1	no	—
342	#95	—	2	2	no	—
343	C₂⁴	—	2	1	no	—
344	C₂×Q₈	—	2	4	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	8	1	no	—
348	C₂×Q₈	—	8	1	no	—
349	C₂×Q₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	C₂²×C₄×C₁₀₉	1	2	yes	—
354	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
355	C₂×D₄	#41×C₁₀₉	1	2	yes	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	8	2	no	—
366	C₂²×C₄	—	8	2	no	—
367	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
371	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	8	2	no	—
394	C₄⋊C₄	—	8	1	no	—
395	C₄⋊C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
396	C₄⋊C₄	C₂²⋊C₄×C₁₀₉	1	8	yes	—
397	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
398	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
399	C₄⋊C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₂²⋊C₄	—	2	8	no	—
402	C₄²	—	4	4	no	—
403	C₄²	—	4	2	no	—
404	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
405	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
406	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
407	C₄²	C₂×D₄×C₁₀₉	1	4	yes	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	2	2	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
416	Q₈	—	2	8	no	—
417	Q₈	—	8	4	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	2	no	—
421	D₄	C₄×Q₈×C₁₀₉	1	4	yes	complemented
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	8	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	8	8	no	—
445	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
446	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
447	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
448	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
449	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
450	C₂×C₄	#137×C₁₀₉	1	2	yes	—
451	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
453	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
455	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
456	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
457	C₈	—	2	2	no	—
458	C₈	—	4	1	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
468	C₂²	#548×C₁₀₉	1	1	yes	characteristic
469	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
470	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
471	C₂²	C₂×#137×C₁₀₉	1	1	yes	characteristic
472	C₂²	#576×C₁₀₉	1	1	yes	characteristic
473	C₂²	#577×C₁₀₉	1	1	yes	characteristic
474	C₄	—	2	4	no	—
475	C₄	—	2	4	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	4	no	—
480	C₄	—	8	2	no	—
481	C₄	—	8	1	no	—
482	C₄	#417×C₁₀₉	1	4	yes	—
483	C₄	C₂×C₄×Q₈×C₁₀₉	1	1	yes	characteristic
484	C₄	#519×C₁₀₉	1	1	yes	characteristic
485	C₄	#548×C₁₀₉	1	1	yes	characteristic
486	C₄	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
487	C₂	—	4	4	no	—
488	C₂	Q₈×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
489	C₂	C₂×#417×C₁₀₉	1	1	yes	characteristic
490	C₂	C₁₀₉×#2897	1	1	yes	characteristic
491	C₂	C₁₀₉×#3366	1	1	yes	characteristic
492	C₂	C₁₀₉×#3372	1	1	yes	characteristic
493	C₂	C₁₀₉×#3384	1	1	yes	characteristic
494	C₂	C₁₀₉×#3385	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄×Q₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#446×C₁₀₉	C₂	semidirect
3. Q₈×C₄⋊C₄×C₁₀₉	C₂	semidirect
4. C₁₀₉×#1992	C₂	semidirect
5. C₁₀₉×#1990	C₂	semidirect
6. C₁₀₉×#1989	C₂	semidirect
7. C₁₀₉×#1987	C₂	semidirect
13. #554×C₁₀₉	C₄	semidirect
90. #102×C₁₀₉	Q₈	semidirect
173. D₄×C₁₀₉	C₄×Q₈	semidirect
239. #22522	C₁₀₉	direct (N × H)
249. C₂×#446	C₂×C₁₀₉	semidirect
250. Q₈×C₄⋊C₄	C₂×C₁₀₉	semidirect
251. #1992	C₂×C₁₀₉	semidirect
252. #1990	C₂×C₁₀₉	semidirect
253. #1989	C₂×C₁₀₉	semidirect
254. #1987	C₂×C₁₀₉	semidirect
259. C₁₀₉	#22522	direct (N × H)
261. #554	C₄×C₁₀₉	semidirect
338. #102	Q₈×C₁₀₉	semidirect
421. D₄	C₄×Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₁₀₉	2
C₂×#446×C₁₀₉	2
Q₈×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1992	2
C₁₀₉×#1990	2
C₁₀₉×#1989	2
C₁₀₉×#1987	2
C₁₀₉×#1802	2
C₁₀₉×#1762	2
C₄×#102×C₁₀₉	2
#22522	109

Subgroups — C109×#22522 #42140516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22522 #42140516