Subgroups — #27179582

Order 12928.

774 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13108074	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692263	C₂	1	2	yes	maximal, complemented
3	#20265145	C₂	1	2	yes	maximal, complemented
4	#20265142	C₂	1	4	yes	maximal, complemented
5	#20265139	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×D₁₀₁	C₂	1	2	yes	maximal, complemented
8	#407×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13108074	—	2	8	no	—
10	C₂×#13108074	C₄	1	1	yes	characteristic
11	C₂×#13108074	C₄	1	1	yes	characteristic
12	C₂×#13108074	C₄	1	2	yes	—
13	C₂×#13108074	C₂²	1	2	yes	—
14	C₂×#13108074	C₂²	1	1	yes	characteristic
15	C₂×#13108074	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic
17	C₂²×D₄₀₄	C₄	1	1	yes	characteristic
18	C₂²×C₂²⋊D₁₀₁	C₄	1	2	yes	—
19	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
20	#15692263	—	2	8	no	—
21	C₂×#13108057	C₂²	1	4	yes	complemented
22	C₂²×#11717579	C₂²	1	2	yes	complemented
23	C₂×C₈×D₁₀₁	C₂²	1	4	yes	complemented
24	C₂²⋊#11717579	C₂²	1	1	yes	characteristic
25	C₂²⋊#11717579	C₂²	1	1	yes	characteristic
26	C₂²⋊#11717579	C₂²	1	2	yes	complemented
27	#15692167	C₂²	1	4	yes	complemented
28	#15692167	C₂²	1	4	yes	complemented
29	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	complemented
31	C₂²⋊C₈×C₁₀₁	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₁₀₁	C₂²	1	2	yes	complemented
33	C₂³×D₁₀₁	C₂×C₄	1	2	yes	—
34	#13108074	—	2	4	no	—
35	#13108074	—	2	8	no	—
36	#13108074	—	2	4	no	—
37	#13108074	—	2	8	no	—
38	#13108074	—	2	4	no	—
39	#13108074	D₄	1	8	yes	—
40	C₂×Dic₂₀₂	—	2	4	no	—
41	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
42	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
43	C₂×Dic₂₀₂	C₂×C₄	1	1	yes	characteristic
44	C₂×D₄₀₄	—	2	4	no	—
45	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
46	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₀₁	—	2	8	no	—
49	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
53	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₁₀₁	—	2	8	no	—
55	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
56	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
57	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
58	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
59	#13108057	—	2	8	no	—
60	C₂×#11717579	—	2	4	no	—
61	C₂×#11717579	C₂³	1	2	yes	—
62	C₂×#11717579	C₂³	1	2	yes	—
63	C₈×D₁₀₁	—	2	8	no	—
64	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₀₁	—	2	4	no	—
66	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	M₁₆×C₁₀₁	—	2	4	no	—
73	C₂×C₈×C₁₀₁	C₂³	1	4	yes	—
74	C₂³×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	2	no	—
76	C₂³×C₁₀₁	—	4	1	no	—
77	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₁₀₁	—	2	4	no	—
81	C₂²×D₁₀₁	—	2	4	no	—
82	C₂²×D₁₀₁	—	4	2	no	—
83	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
84	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₁	—	2	2	no	—
86	C₂×C₄×C₁₀₁	—	2	2	no	—
87	C₂×C₄×C₁₀₁	—	2	2	no	—
88	C₂×C₄×C₁₀₁	—	2	2	no	—
89	C₂×C₄×C₁₀₁	—	2	2	no	—
90	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₀₁	—	2	4	no	—
96	C₂²⋊D₁₀₁	—	2	4	no	—
97	C₂²⋊D₁₀₁	—	2	4	no	—
98	C₂²⋊D₁₀₁	C₂²⋊C₄	1	8	yes	—
99	C₂×Dic₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
101	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
102	D₄₀₄	—	2	4	no	—
103	D₄₀₄	—	2	2	no	—
104	D₄₀₄	C₂²⋊C₄	1	4	yes	—
105	C₄×D₁₀₁	—	2	4	no	—
106	C₄×D₁₀₁	—	2	4	no	—
107	C₄×D₁₀₁	—	2	4	no	—
108	C₄×D₁₀₁	C₂×D₄	1	8	yes	—
109	Dic₂₀₂	—	2	2	no	—
110	Dic₂₀₂	—	2	4	no	—
111	Dic₂₀₂	C₂²⋊C₄	1	4	yes	—
112	C₈×C₁₀₁	—	2	4	no	—
113	#11717579	—	2	2	no	—
114	#11717579	—	2	2	no	—
115	C₂²×C₁₀₁	—	2	1	no	—
116	C₂²×C₁₀₁	—	2	1	no	—
117	C₂²×C₁₀₁	—	2	1	no	—
118	C₂²×C₁₀₁	—	2	1	no	—
119	C₂²×C₁₀₁	—	2	1	no	—
120	C₂²×C₁₀₁	—	2	1	no	—
121	C₂²×C₁₀₁	—	2	1	no	—
122	C₂²×C₁₀₁	—	4	4	no	—
123	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂²×C₁₀₁	#131	1	2	yes	complemented
125	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	2	no	—
127	C₂×D₁₀₁	—	2	2	no	—
128	C₂×D₁₀₁	—	2	2	no	—
129	C₂×D₁₀₁	—	2	2	no	—
130	C₂×D₁₀₁	—	2	2	no	—
131	C₂×D₁₀₁	—	4	8	no	—
132	C₂×D₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
133	C₄×C₁₀₁	—	2	1	no	—
134	C₄×C₁₀₁	—	2	1	no	—
135	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	2	yes	—
136	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
137	Dic₁₀₁	—	2	2	no	—
138	Dic₁₀₁	C₂×C₂²⋊C₄	1	4	yes	—
139	C₂×C₁₀₁	—	2	2	no	—
140	C₂×C₁₀₁	—	2	2	no	—
141	C₂×C₁₀₁	—	4	1	no	—
142	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
144	C₂×C₁₀₁	#568	1	1	yes	characteristic
145	D₁₀₁	—	2	4	no	—
146	D₁₀₁	—	4	2	no	—
147	#2880	—	101	1	no	maximal, Sylow
148	C₁₀₁	#2880	1	1	yes	characteristic, complemented, Sylow
149	C₂²×#41	—	101	1	no	—
150	C₂×#131	—	101	2	no	—
151	#408	—	101	2	no	—
152	#408	—	101	4	no	—
153	#407	—	101	2	no	—
154	#407	D₁₀₁	1	1	yes	characteristic, complemented
155	C₂×C₂²⋊C₈	—	101	1	no	—
156	C₂×C₂²⋊C₈	—	101	2	no	—
157	C₂×#41	—	101	2	no	—
158	C₂×#41	—	101	1	no	—
159	C₂×#41	—	101	1	no	—
160	C₂×#41	—	101	1	no	—
161	C₂×#41	—	101	2	no	—
162	C₂×#41	—	101	1	no	—
163	C₂×#41	—	202	8	no	—
164	C₂²×Q₈	—	101	1	no	—
165	C₂²×D₄	—	101	2	no	—
166	C₂²×D₄	—	101	1	no	—
167	C₂³×C₄	—	101	2	no	—
168	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
169	#131	—	202	8	no	—
170	C₂×M₁₆	—	101	4	no	—
171	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	complemented
172	C₂²×C₈	—	101	2	no	—
173	C₂²×C₈	—	101	4	no	—
174	C₂²⋊C₈	—	101	1	no	—
175	C₂²⋊C₈	—	101	1	no	—
176	C₂²⋊C₈	—	101	2	no	—
177	C₂²⋊C₈	—	101	4	no	—
178	C₂²⋊C₈	—	101	4	no	—
179	C₂²⋊C₈	C₂×D₁₀₁	1	2	yes	—
180	C₂²⋊C₈	C₂×D₁₀₁	1	2	yes	complemented
181	C₂⁴	—	101	2	no	—
182	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
183	#41	—	101	8	no	—
184	#41	—	202	8	no	—
185	#41	—	202	8	no	—
186	#41	—	202	4	no	—
187	#41	—	202	4	no	—
188	#41	—	202	4	no	—
189	C₂×Q₈	—	101	1	no	—
190	C₂×Q₈	—	101	2	no	—
191	C₂×Q₈	—	101	1	no	—
192	C₂×Q₈	—	202	4	no	—
193	C₂×D₄	—	101	2	no	—
194	C₂×D₄	—	101	2	no	—
195	C₂×D₄	—	101	2	no	—
196	C₂×D₄	—	101	2	no	—
197	C₂×D₄	—	101	1	no	—
198	C₂×D₄	—	101	2	no	—
199	C₂×D₄	—	101	1	no	—
200	C₂×D₄	—	202	4	no	—
201	C₂×D₄	—	202	8	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	101	2	no	—
204	C₂²×C₄	—	101	2	no	—
205	C₂²×C₄	—	101	2	no	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	101	2	no	—
208	C₂²×C₄	—	202	8	no	—
209	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
210	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
211	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
214	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	202	8	no	—
217	C₂×C₈	—	101	2	no	—
218	C₂×C₈	—	101	2	no	—
219	C₂×C₈	—	202	8	no	—
220	C₂×C₈	—	202	4	no	—
221	C₂×C₈	C₂²×D₁₀₁	1	4	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	101	2	no	—
226	C₂³	—	101	2	no	—
227	C₂³	—	404	2	no	—
228	C₂³	—	202	4	no	—
229	C₂³	—	202	4	no	—
230	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
231	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
232	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
233	Q₈	—	101	4	no	—
234	Q₈	—	202	4	no	—
235	Q₈	—	202	2	no	—
236	D₄	—	101	8	no	—
237	D₄	—	101	4	no	—
238	D₄	—	202	4	no	—
239	D₄	—	202	4	no	—
240	D₄	—	202	4	no	—
241	D₄	—	202	4	no	—
242	D₄	—	202	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	101	2	no	—
249	C₂×C₄	—	101	8	no	—
250	C₂×C₄	—	101	2	no	—
251	C₂×C₄	—	202	4	no	—
252	C₂×C₄	—	202	4	no	—
253	C₂×C₄	—	202	4	no	—
254	C₂×C₄	—	202	4	no	—
255	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
258	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
259	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	202	2	no	—
262	C₈	—	202	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	101	4	no	—
272	C₂²	—	404	8	no	—
273	C₂²	—	202	2	no	—
274	C₂²	—	202	2	no	—
275	C₂²	—	202	2	no	—
276	C₂²	—	202	2	no	—
277	C₂²	—	202	2	no	—
278	C₂²	C₂²⋊C₄×D₁₀₁	1	2	yes	—
279	C₂²	#15692263	1	2	yes	complemented
280	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	101	4	no	—
284	C₄	—	202	2	no	—
285	C₄	C₂²⋊C₄×D₁₀₁	1	2	yes	—
286	C₄	C₂×D₄×D₁₀₁	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	—	4	1	no	—
290	C₂	—	404	2	no	—
291	C₂	—	202	4	no	—
292	C₂	C₂×C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
293	C₂	C₂×#15692263	1	1	yes	characteristic
294	C₂	#20265943	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	#407×C₁₀₁	C₂
prime radical	C₂²×#13108074	C₂
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	#407	D₁₀₁
hypocenter	C₁₀₁	#2880
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. C₂×#15692263	C₂	semidirect
3. #20265145	C₂	semidirect
4. #20265142	C₂	semidirect
5. #20265139	C₂	semidirect
6. C₂×C₂²⋊#11717579	C₂	semidirect
7. C₂²⋊C₈×D₁₀₁	C₂	semidirect
8. #407×C₁₀₁	C₂	semidirect
21. C₂×#13108057	C₂²	semidirect
22. C₂²×#11717579	C₂²	semidirect
23. C₂×C₈×D₁₀₁	C₂²	semidirect
26. C₂²⋊#11717579	C₂²	semidirect
27. #15692167	C₂²	semidirect
28. #15692167	C₂²	semidirect
30. C₂×M₁₆×C₁₀₁	C₂²	semidirect
32. C₂²⋊C₈×C₁₀₁	C₂²	semidirect
124. C₂²×C₁₀₁	#131	semidirect
148. C₁₀₁	#2880	semidirect
154. #407	D₁₀₁	semidirect
171. C₂×M₁₆	C₂×D₁₀₁	semidirect
180. C₂²⋊C₈	C₂×D₁₀₁	semidirect
279. C₂²	#15692263	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13108074	2
C₂×#15692263	2
#20265145	2
#20265142	2
#20265139	2
C₂×C₂²⋊#11717579	2
C₂²⋊C₈×D₁₀₁	2
#407×C₁₀₁	2
#2880	101