Subgroups — C₁₀₉×#22487 #42140481

Order 27904.

1258 conjugacy classes of subgroups.

The subgroup lattice has 522 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×C₈×D₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#1987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#1892	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1872	C₂	1	2	yes	maximal, complemented
8	C₁₀₉×#1824	C₂	1	2	yes	maximal, complemented
9	C₁₀₉×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#340×C₁₀₉	C₂	1	2	yes	maximal, complemented
12	C₂³×C₈×C₁₀₉	C₂²	1	2	yes	—
13	#554×C₁₀₉	—	2	4	no	—
14	#554×C₁₀₉	C₄	1	8	yes	complemented
15	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×#125×C₁₀₉	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
18	C₂×C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₀₉	—	2	1	no	—
21	C₂×C₄×D₄×C₁₀₉	—	2	1	no	—
22	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₈×D₄×C₁₀₉	—	2	4	no	—
26	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×#103×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
32	#389×C₁₀₉	—	2	1	no	—
33	#385×C₁₀₉	—	2	2	no	—
34	#384×C₁₀₉	C₂²	1	1	yes	characteristic
35	#380×C₁₀₉	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₀₉	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₀₉	—	2	2	no	—
48	C₂³×C₄×C₁₀₉	—	2	2	no	—
49	C₂³×C₄×C₁₀₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₀₉	—	2	4	no	—
51	C₂²×C₈×C₁₀₉	—	2	4	no	—
52	C₂²×C₈×C₁₀₉	—	4	2	no	—
53	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
54	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
57	C₄⋊Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
58	#125×C₁₀₉	—	2	8	no	—
59	C₂²⋊Q₈×C₁₀₉	—	2	4	no	—
60	C₂²⋊Q₈×C₁₀₉	—	2	16	no	—
61	C₂²⋊Q₈×C₁₀₉	C₂×C₄	1	8	yes	complemented
62	C₄×Q₈×C₁₀₉	—	2	2	no	—
63	C₄×Q₈×C₁₀₉	C₂×C₄	1	4	yes	complemented
64	C₄×D₄×C₁₀₉	—	2	8	no	—
65	C₄×D₄×C₁₀₉	—	4	4	no	—
66	C₄×D₄×C₁₀₉	C₂×C₄	1	4	yes	—
67	C₄×D₄×C₁₀₉	D₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄⋊C₄×C₁₀₉	—	2	4	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	—	2	4	no	—
74	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₁₀₉	—	4	2	no	—
78	C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	complemented
79	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
88	C₂×C₄²×C₁₀₉	—	2	1	no	—
89	C₂×C₄²×C₁₀₉	—	2	1	no	—
90	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₁₀₉	—	2	2	no	—
92	#103×C₁₀₉	—	2	2	no	—
93	#103×C₁₀₉	—	2	2	no	—
94	#103×C₁₀₉	—	4	2	no	—
95	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
96	C₄×C₈×C₁₀₉	—	2	2	no	—
97	#95×C₁₀₉	—	2	1	no	—
98	#95×C₁₀₉	—	2	2	no	—
99	C₂⁴×C₁₀₉	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₁₀₉	—	2	4	no	—
101	C₂×Q₈×C₁₀₉	—	2	1	no	—
102	C₂×Q₈×C₁₀₉	—	2	1	no	—
103	C₂×Q₈×C₁₀₉	—	2	1	no	—
104	C₂×Q₈×C₁₀₉	—	2	1	no	—
105	C₂×Q₈×C₁₀₉	—	2	1	no	—
106	C₂×Q₈×C₁₀₉	—	8	1	no	—
107	C₂×Q₈×C₁₀₉	—	8	1	no	—
108	C₂×Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
109	C₂×D₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
110	C₂×D₄×C₁₀₉	SD₁₆	1	4	yes	—
111	C₂×D₄×C₁₀₉	Q₁₆	1	4	yes	—
112	C₂²×C₄×C₁₀₉	—	2	2	no	—
113	C₂²×C₄×C₁₀₉	—	2	2	no	—
114	C₂²×C₄×C₁₀₉	—	2	2	no	—
115	C₂²×C₄×C₁₀₉	—	2	2	no	—
116	C₂²×C₄×C₁₀₉	—	2	1	no	—
117	C₂²×C₄×C₁₀₉	—	2	1	no	—
118	C₂²×C₄×C₁₀₉	—	2	1	no	—
119	C₂²×C₄×C₁₀₉	—	2	2	no	—
120	C₂²×C₄×C₁₀₉	—	2	8	no	—
121	C₂²×C₄×C₁₀₉	—	2	1	no	—
122	C₂²×C₄×C₁₀₉	—	2	2	no	—
123	C₂²×C₄×C₁₀₉	—	4	4	no	—
124	C₂²×C₄×C₁₀₉	—	4	4	no	—
125	C₂²×C₄×C₁₀₉	—	8	2	no	—
126	C₂²×C₄×C₁₀₉	—	8	2	no	—
127	C₂²×C₄×C₁₀₉	—	8	2	no	—
128	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
129	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
134	C₂×C₈×C₁₀₉	—	2	1	no	—
135	C₂×C₈×C₁₀₉	—	2	1	no	—
136	C₂×C₈×C₁₀₉	—	2	1	no	—
137	C₂×C₈×C₁₀₉	—	2	2	no	—
138	C₂×C₈×C₁₀₉	—	2	1	no	—
139	C₂×C₈×C₁₀₉	—	2	2	no	—
140	C₂×C₈×C₁₀₉	—	2	2	no	—
141	C₂×C₈×C₁₀₉	—	2	1	no	—
142	C₂×C₈×C₁₀₉	—	4	8	no	—
143	C₂×C₈×C₁₀₉	#41	1	2	yes	—
144	C₄⋊C₄×C₁₀₉	—	2	8	no	—
145	C₄⋊C₄×C₁₀₉	—	2	4	no	—
146	C₄⋊C₄×C₁₀₉	—	2	4	no	—
147	C₄⋊C₄×C₁₀₉	—	2	8	no	—
148	C₄⋊C₄×C₁₀₉	—	2	1	no	—
149	C₄⋊C₄×C₁₀₉	—	2	2	no	—
150	C₄⋊C₄×C₁₀₉	—	2	1	no	—
151	C₄⋊C₄×C₁₀₉	—	2	1	no	—
152	C₄⋊C₄×C₁₀₉	—	2	2	no	—
153	C₄⋊C₄×C₁₀₉	—	2	2	no	—
154	C₄⋊C₄×C₁₀₉	—	2	1	no	—
155	C₄⋊C₄×C₁₀₉	—	4	4	no	—
156	C₄⋊C₄×C₁₀₉	—	4	2	no	—
157	C₄⋊C₄×C₁₀₉	—	8	1	no	—
158	C₄⋊C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
159	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
160	C₄⋊C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
161	C₄⋊C₄×C₁₀₉	#41	1	2	yes	—
162	C₂²⋊C₄×C₁₀₉	—	2	8	no	—
163	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
164	C₂²⋊C₄×C₁₀₉	C₂²⋊C₄	1	8	yes	—
165	C₄²×C₁₀₉	—	2	4	no	—
166	C₄²×C₁₀₉	—	4	2	no	—
167	C₄²×C₁₀₉	C₂²×C₄	1	2	yes	—
168	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
169	C₂³×C₁₀₉	—	2	2	no	—
170	C₂³×C₁₀₉	—	2	2	no	—
171	C₂³×C₁₀₉	—	2	2	no	—
172	C₂³×C₁₀₉	—	2	2	no	—
173	C₂³×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
174	C₂³×C₁₀₉	C₂×SD₁₆	1	4	yes	—
175	C₂³×C₁₀₉	C₂×Q₁₆	1	4	yes	—
176	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₁₀₉	—	2	2	no	—
178	Q₈×C₁₀₉	—	2	2	no	—
179	Q₈×C₁₀₉	—	4	1	no	—
180	Q₈×C₁₀₉	—	4	1	no	—
181	Q₈×C₁₀₉	—	8	4	no	—
182	D₄×C₁₀₉	#103	1	16	yes	complemented
183	C₂×C₄×C₁₀₉	—	2	2	no	—
184	C₂×C₄×C₁₀₉	—	2	2	no	—
185	C₂×C₄×C₁₀₉	—	2	1	no	—
186	C₂×C₄×C₁₀₉	—	2	1	no	—
187	C₂×C₄×C₁₀₉	—	2	4	no	—
188	C₂×C₄×C₁₀₉	—	2	4	no	—
189	C₂×C₄×C₁₀₉	—	2	4	no	—
190	C₂×C₄×C₁₀₉	—	2	4	no	—
191	C₂×C₄×C₁₀₉	—	2	2	no	—
192	C₂×C₄×C₁₀₉	—	2	2	no	—
193	C₂×C₄×C₁₀₉	—	2	4	no	—
194	C₂×C₄×C₁₀₉	—	2	2	no	—
195	C₂×C₄×C₁₀₉	—	4	1	no	—
196	C₂×C₄×C₁₀₉	—	4	1	no	—
197	C₂×C₄×C₁₀₉	—	4	1	no	—
198	C₂×C₄×C₁₀₉	—	4	1	no	—
199	C₂×C₄×C₁₀₉	—	4	1	no	—
200	C₂×C₄×C₁₀₉	—	4	2	no	—
201	C₂×C₄×C₁₀₉	—	4	1	no	—
202	C₂×C₄×C₁₀₉	—	4	2	no	—
203	C₂×C₄×C₁₀₉	—	4	2	no	—
204	C₂×C₄×C₁₀₉	—	4	1	no	—
205	C₂×C₄×C₁₀₉	—	4	1	no	—
206	C₂×C₄×C₁₀₉	—	4	2	no	—
207	C₂×C₄×C₁₀₉	—	4	2	no	—
208	C₂×C₄×C₁₀₉	—	4	1	no	—
209	C₂×C₄×C₁₀₉	—	8	8	no	—
210	C₂×C₄×C₁₀₉	—	8	8	no	—
211	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
212	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	4	yes	—
213	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
215	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
216	C₂×C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
222	C₈×C₁₀₉	—	2	2	no	—
223	C₈×C₁₀₉	—	4	2	no	—
224	C₈×C₁₀₉	—	4	1	no	—
225	C₂²×C₁₀₉	—	2	4	no	—
226	C₂²×C₁₀₉	—	2	4	no	—
227	C₂²×C₁₀₉	—	2	4	no	—
228	C₂²×C₁₀₉	—	2	4	no	—
229	C₂²×C₁₀₉	—	2	4	no	—
230	C₂²×C₁₀₉	—	2	4	no	—
231	C₂²×C₁₀₉	C₂×#103	1	8	yes	—
232	C₂²×C₁₀₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
233	C₂²×C₁₀₉	#546	1	1	yes	characteristic
234	C₂²×C₁₀₉	#547	1	1	yes	characteristic
235	C₂²×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
236	C₂²×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
237	C₂²×C₁₀₉	#576	1	1	yes	characteristic
238	C₂²×C₁₀₉	#577	1	1	yes	characteristic
239	C₄×C₁₀₉	—	2	4	no	—
240	C₄×C₁₀₉	—	2	2	no	—
241	C₄×C₁₀₉	—	4	2	no	—
242	C₄×C₁₀₉	—	4	2	no	—
243	C₄×C₁₀₉	—	4	4	no	—
244	C₄×C₁₀₉	—	4	2	no	—
245	C₄×C₁₀₉	—	8	1	no	—
246	C₄×C₁₀₉	—	8	2	no	—
247	C₄×C₁₀₉	C₂×#103	1	4	yes	—
248	C₄×C₁₀₉	C₄×#41	1	1	yes	characteristic
249	C₄×C₁₀₉	#520	1	1	yes	characteristic
250	C₄×C₁₀₉	#546	1	1	yes	characteristic
251	C₄×C₁₀₉	#547	1	1	yes	characteristic
252	#22487	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₁₀₉	—	2	8	no	—
254	C₂×C₁₀₉	#2339	1	1	yes	characteristic
255	C₂×C₁₀₉	C₂²×#103	1	1	yes	characteristic
256	C₂×C₁₀₉	#2897	1	1	yes	characteristic
257	C₂×C₁₀₉	#3295	1	1	yes	characteristic
258	C₂×C₁₀₉	#3299	1	1	yes	characteristic
259	C₂×C₁₀₉	#3335	1	1	yes	characteristic
260	C₂×C₁₀₉	#3338	1	1	yes	characteristic
261	C₂×#554	C₂×C₁₀₉	1	1	yes	characteristic
262	C₂×C₈×D₄	C₂×C₁₀₉	1	1	yes	characteristic
263	D₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic
264	#1990	C₂×C₁₀₉	1	1	yes	characteristic, complemented
265	#1987	C₂×C₁₀₉	1	1	yes	characteristic, complemented
266	#1892	C₂×C₁₀₉	1	2	yes	complemented
267	#1872	C₂×C₁₀₉	1	2	yes	complemented
268	#1824	C₂×C₁₀₉	1	2	yes	complemented
269	#1803	C₂×C₁₀₉	1	1	yes	characteristic, complemented
270	C₄×#103	C₂×C₁₀₉	1	1	yes	characteristic, complemented
271	C₂×#340	C₂×C₁₀₉	1	2	yes	complemented
272	C₁₀₉	#22487	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³×C₈	C₂²×C₁₀₉	1	2	yes	—
274	#554	—	2	4	no	—
275	#554	C₄×C₁₀₉	1	8	yes	complemented
276	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
277	C₂×#125	—	2	1	no	—
278	C₂×C₂²⋊Q₈	—	2	2	no	—
279	C₂×C₂²⋊Q₈	C₂²×C₁₀₉	1	2	yes	—
280	C₂×C₄×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
281	C₂×C₄×D₄	—	2	1	no	—
282	C₂×C₄×D₄	—	2	1	no	—
283	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
284	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
285	C₂²×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
286	C₈×D₄	—	2	4	no	—
287	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
288	C₂×#103	C₂²×C₁₀₉	1	1	yes	characteristic
289	C₂×#103	C₂²×C₁₀₉	1	1	yes	characteristic
290	C₂×#103	C₂²×C₁₀₉	1	2	yes	—
291	C₂×C₂²⋊C₈	C₂²×C₁₀₉	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
293	#389	—	2	1	no	—
294	#385	—	2	2	no	—
295	#384	C₂²×C₁₀₉	1	1	yes	characteristic
296	#380	C₂²×C₁₀₉	1	2	yes	—
297	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
298	#340	C₂²×C₁₀₉	1	2	yes	—
299	#340	C₂²×C₁₀₉	1	1	yes	characteristic
300	#340	C₂²×C₁₀₉	1	2	yes	—
301	#340	C₂²×C₁₀₉	1	2	yes	—
302	#340	C₂²×C₁₀₉	1	2	yes	—
303	#340	C₂²×C₁₀₉	1	2	yes	—
304	#340	C₂²×C₁₀₉	1	1	yes	characteristic
305	C₂²×Q₈	—	2	1	no	—
306	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
307	C₂²×D₄	D₄×C₁₀₉	1	1	yes	characteristic
308	C₂³×C₄	—	2	2	no	—
309	C₂³×C₄	—	2	2	no	—
310	C₂³×C₄	C₂³×C₁₀₉	1	2	yes	—
311	C₂²×C₈	—	2	4	no	—
312	C₂²×C₈	—	2	4	no	—
313	C₂²×C₈	—	4	2	no	—
314	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
315	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
316	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
319	#125	—	2	8	no	—
320	C₂²⋊Q₈	—	2	4	no	—
321	C₂²⋊Q₈	—	2	16	no	—
322	C₂²⋊Q₈	C₂×C₄×C₁₀₉	1	8	yes	complemented
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	C₂×C₄×C₁₀₉	1	4	yes	complemented
325	C₄×D₄	—	2	8	no	—
326	C₄×D₄	—	4	4	no	—
327	C₄×D₄	C₂×C₄×C₁₀₉	1	4	yes	—
328	C₄×D₄	D₄×C₁₀₉	1	4	yes	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	4	2	no	—
339	C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	1	8	yes	complemented
340	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
342	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
345	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	D₄×C₁₀₉	1	2	yes	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
352	C₄⋊C₈	—	2	2	no	—
353	#103	—	2	2	no	—
354	#103	—	2	2	no	—
355	#103	—	4	2	no	—
356	C₂²⋊C₈	—	2	4	no	—
357	C₄×C₈	—	2	2	no	—
358	#95	—	2	1	no	—
359	#95	—	2	2	no	—
360	C₂⁴	C₂×D₄×C₁₀₉	1	2	yes	—
361	C₂×Q₈	—	2	4	no	—
362	C₂×Q₈	—	2	1	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	8	1	no	—
368	C₂×Q₈	—	8	1	no	—
369	C₂×Q₈	C₂²×C₄×C₁₀₉	1	2	yes	—
370	C₂×D₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
371	C₂×D₄	SD₁₆×C₁₀₉	1	4	yes	—
372	C₂×D₄	Q₁₆×C₁₀₉	1	4	yes	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	8	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	4	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
390	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₁₀₉	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	4	8	no	—
404	C₂×C₈	#41×C₁₀₉	1	2	yes	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	4	4	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	—	8	1	no	—
419	C₄⋊C₄	C₂²⋊C₄×C₁₀₉	1	4	yes	—
420	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	2	yes	—
421	C₄⋊C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
422	C₄⋊C₄	#41×C₁₀₉	1	2	yes	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	4	4	no	—
425	C₂²⋊C₄	C₂²⋊C₄×C₁₀₉	1	8	yes	—
426	C₄²	—	2	4	no	—
427	C₄²	—	4	2	no	—
428	C₄²	C₂²×C₄×C₁₀₉	1	2	yes	—
429	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
435	C₂³	C₂×SD₁₆×C₁₀₉	1	4	yes	—
436	C₂³	C₂×Q₁₆×C₁₀₉	1	4	yes	—
437	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	4	1	no	—
441	Q₈	—	4	1	no	—
442	Q₈	—	8	4	no	—
443	D₄	#103×C₁₀₉	1	16	yes	complemented
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	8	8	no	—
471	C₂×C₄	—	8	8	no	—
472	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
473	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	4	yes	—
474	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	2	yes	—
475	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
476	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	2	yes	—
477	C₂×C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
478	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
480	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
481	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
482	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
483	C₈	—	2	2	no	—
484	C₈	—	4	2	no	—
485	C₈	—	4	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	4	no	—
492	C₂²	C₂×#103×C₁₀₉	1	8	yes	—
493	C₂²	C₂²×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
494	C₂²	#546×C₁₀₉	1	1	yes	characteristic
495	C₂²	#547×C₁₀₉	1	1	yes	characteristic
496	C₂²	C₂²×SD₁₆×C₁₀₉	1	1	yes	characteristic
497	C₂²	C₂²×Q₁₆×C₁₀₉	1	1	yes	characteristic
498	C₂²	#576×C₁₀₉	1	1	yes	characteristic
499	C₂²	#577×C₁₀₉	1	1	yes	characteristic
500	C₄	—	2	4	no	—
501	C₄	—	2	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	4	no	—
506	C₄	—	8	2	no	—
507	C₄	—	8	1	no	—
508	C₄	C₂×#103×C₁₀₉	1	4	yes	—
509	C₄	C₄×#41×C₁₀₉	1	1	yes	characteristic
510	C₄	#520×C₁₀₉	1	1	yes	characteristic
511	C₄	#546×C₁₀₉	1	1	yes	characteristic
512	C₄	#547×C₁₀₉	1	1	yes	characteristic
513	C₂	—	2	8	no	—
514	C₂	C₁₀₉×#2339	1	1	yes	characteristic
515	C₂	C₂²×#103×C₁₀₉	1	1	yes	characteristic
516	C₂	C₁₀₉×#2897	1	1	yes	characteristic
517	C₂	C₁₀₉×#3295	1	1	yes	characteristic
518	C₂	C₁₀₉×#3299	1	1	yes	characteristic
519	C₂	C₁₀₉×#3335	1	1	yes	characteristic
520	C₂	C₁₀₉×#3338	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	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄×D₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₀₉×#1990	C₂	semidirect
5. C₁₀₉×#1987	C₂	semidirect
6. C₁₀₉×#1892	C₂	semidirect
7. C₁₀₉×#1872	C₂	semidirect
8. C₁₀₉×#1824	C₂	semidirect
9. C₁₀₉×#1803	C₂	semidirect
10. C₄×#103×C₁₀₉	C₂	semidirect
11. C₂×#340×C₁₀₉	C₂	semidirect
14. #554×C₁₀₉	C₄	semidirect
57. C₄⋊Q₈×C₁₀₉	C₂×C₄	semidirect
61. C₂²⋊Q₈×C₁₀₉	C₂×C₄	semidirect
63. C₄×Q₈×C₁₀₉	C₂×C₄	semidirect
78. C₂×C₄⋊C₄×C₁₀₉	C₂×C₄	semidirect
182. D₄×C₁₀₉	#103	semidirect
252. #22487	C₁₀₉	direct (N × H)
264. #1990	C₂×C₁₀₉	semidirect
265. #1987	C₂×C₁₀₉	semidirect
266. #1892	C₂×C₁₀₉	semidirect
267. #1872	C₂×C₁₀₉	semidirect
268. #1824	C₂×C₁₀₉	semidirect
269. #1803	C₂×C₁₀₉	semidirect
270. C₄×#103	C₂×C₁₀₉	semidirect
271. C₂×#340	C₂×C₁₀₉	semidirect
272. C₁₀₉	#22487	direct (N × H)
275. #554	C₄×C₁₀₉	semidirect
318. C₄⋊Q₈	C₂×C₄×C₁₀₉	semidirect
322. C₂²⋊Q₈	C₂×C₄×C₁₀₉	semidirect
324. C₄×Q₈	C₂×C₄×C₁₀₉	semidirect
339. C₂×C₄⋊C₄	C₂×C₄×C₁₀₉	semidirect
443. D₄	#103×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×C₈×D₄×C₁₀₉	2
D₄×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#1990	2
C₁₀₉×#1987	2
C₁₀₉×#1892	2
C₁₀₉×#1872	2
C₁₀₉×#1824	2
C₁₀₉×#1803	2
C₄×#103×C₁₀₉	2
C₂×#340×C₁₀₉	2
#22487	109

Subgroups — C109×#22487 #42140481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#22487 #42140481