Subgroups — #27187848

Order 12928.

520 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15692343	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₀₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20266207	C₂	1	2	yes	maximal, complemented
4	#20266206	C₂	1	2	yes	maximal, complemented
5	#20266205	C₂	1	2	yes	maximal, complemented
6	#20266204	C₂	1	2	yes	maximal, complemented
7	#20266184	C₂	1	1	yes	maximal, characteristic, complemented
8	#20266152	C₂	1	1	yes	maximal, characteristic, complemented
9	#20266148	C₂	1	1	yes	maximal, characteristic, complemented
10	#20266142	C₂	1	1	yes	maximal, characteristic, complemented
11	#20265690	C₂	1	1	yes	maximal, characteristic
12	#15692343	—	2	4	no	—
13	C₂×#13108084	C₂²	1	1	yes	characteristic
14	C₂×#13108084	C₂²	1	1	yes	characteristic
15	C₂×#13108084	C₂²	1	2	yes	complemented
16	C₂×D₁₀₁⋊C₈	—	2	4	no	—
17	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₁₀₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×D₁₀₁⋊C₈	C₂²	1	2	yes	complemented
22	C₂²×#11717581	C₂²	1	1	yes	characteristic
23	C₂²×#11717581	C₂²	1	1	yes	characteristic
24	#15692322	C₂²	1	2	yes	complemented
25	C₄⋊#11717581	C₂²	1	2	yes	complemented
26	#15692312	C₂²	1	2	yes	—
27	#15692312	C₂²	1	2	yes	complemented
28	#15692312	C₂²	1	2	yes	complemented
29	C₂²⋊#11717581	C₂²	1	2	yes	—
30	#15692310	C₂²	1	2	yes	complemented
31	C₄×#11717581	C₂²	1	2	yes	complemented
32	C₂×#13108075	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
34	#15692240	C₄	1	2	yes	—
35	#15692231	C₄	1	1	yes	characteristic
36	#15692230	C₄	1	1	yes	characteristic
37	#15692229	C₄	1	1	yes	characteristic
38	#15692225	C₄	1	2	yes	—
39	#15692217	C₂²	1	2	yes	—
40	#15692215	C₂²	1	2	yes	—
41	#15692213	C₂²	1	1	yes	characteristic
42	C₂²⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic
43	#13108084	—	2	2	no	—
44	#13108084	—	2	2	no	—
45	#13108084	—	2	4	no	—
46	D₁₀₁⋊C₈	—	2	2	no	—
47	D₁₀₁⋊C₈	—	2	2	no	—
48	D₁₀₁⋊C₈	—	2	2	no	—
49	D₁₀₁⋊C₈	—	2	4	no	—
50	D₁₀₁⋊C₈	D₄	1	4	yes	complemented
51	C₂×#11717581	—	2	2	no	—
52	C₂×#11717581	—	2	2	no	—
53	C₂×#11717581	C₂³	1	2	yes	—
54	C₂×#11717581	C₂³	1	2	yes	—
55	C₂×#11717581	C₂³	1	1	yes	characteristic
56	C₂×#11717581	C₂³	1	1	yes	characteristic
57	C₂×#11717581	C₂³	1	1	yes	characteristic
58	C₂×#11717581	C₂³	1	1	yes	characteristic
59	C₂³×D₁₀₁	C₂×C₄	1	1	yes	characteristic
60	#13108075	—	2	2	no	—
61	#13108075	—	2	2	no	—
62	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
63	C₂×D₄₀₄	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄₀₄	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊D₁₀₁	C₂×C₄	1	2	yes	—
66	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₀₁	—	2	2	no	—
68	C₂×C₄×D₁₀₁	—	2	2	no	—
69	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
73	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
75	#13108053	C₂×C₄	1	2	yes	—
76	#13108051	C₂×C₄	1	2	yes	—
77	#13108051	C₂×C₄	1	2	yes	—
78	#13108051	C₂×C₄	1	2	yes	—
79	#13108051	C₂×C₄	1	2	yes	—
80	C₄×Dic₁₀₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
85	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	2	yes	—
86	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₀₁	—	2	2	no	—
88	C₂²×D₁₀₁	—	2	2	no	—
89	C₂²×D₁₀₁	—	2	2	no	—
90	C₂²×D₁₀₁	C₂²×C₄	1	2	yes	—
91	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×D₁₀₁	C₂²×C₄	1	1	yes	characteristic
93	Q₈×C₁₀₁	—	2	2	no	—
94	C₂×C₄×C₁₀₁	—	2	2	no	—
95	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²⋊D₁₀₁	—	2	4	no	—
100	C₂×Dic₁₀₁	—	2	1	no	—
101	C₂×Dic₁₀₁	—	2	1	no	—
102	C₂×Dic₁₀₁	C₂²×C₄	1	2	yes	—
103	C₂×Dic₁₀₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₁	C₂⁴	1	1	yes	characteristic
105	D₄₀₄	—	2	4	no	—
106	D₄₀₄	—	2	4	no	—
107	D₄₀₄	—	2	2	no	—
108	C₄×D₁₀₁	—	2	2	no	—
109	C₄×D₁₀₁	—	2	2	no	—
110	C₄×D₁₀₁	—	2	1	no	—
111	C₄×D₁₀₁	—	2	2	no	—
112	C₄×D₁₀₁	—	2	1	no	—
113	C₄×D₁₀₁	—	2	2	no	—
114	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
115	C₄×D₁₀₁	#41	1	2	yes	—
116	#11717581	—	2	1	no	—
117	#11717581	—	2	1	no	—
118	#11717581	—	2	2	no	—
119	#11717581	—	2	2	no	—
120	#11717581	C₂×D₄	1	2	yes	—
121	#11717581	C₂×D₄	1	2	yes	—
122	C₂²×C₁₀₁	—	2	1	no	—
123	C₂²×C₁₀₁	—	2	1	no	—
124	C₂²×C₁₀₁	#131	1	2	yes	complemented
125	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×D₁₀₁	—	2	8	no	—
127	C₂×D₁₀₁	—	2	1	no	—
128	C₂×D₁₀₁	—	2	1	no	—
129	C₂×D₁₀₁	—	2	1	no	—
130	C₂×D₁₀₁	—	2	1	no	—
131	C₂×D₁₀₁	—	2	2	no	—
132	C₂×D₁₀₁	—	2	2	no	—
133	C₂×D₁₀₁	—	2	2	no	—
134	C₂×D₁₀₁	C₄×D₄	1	2	yes	—
135	C₂×D₁₀₁	#131	1	2	yes	complemented
136	C₄×C₁₀₁	—	2	1	no	—
137	C₄×C₁₀₁	—	2	2	no	—
138	C₄×C₁₀₁	—	2	2	no	—
139	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
140	Dic₁₀₁	—	2	1	no	—
141	Dic₁₀₁	—	2	2	no	—
142	Dic₁₀₁	C₂²×D₄	1	1	yes	characteristic
143	Dic₁₀₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₁₀₁	—	2	2	no	—
145	C₂×C₁₀₁	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
147	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
148	D₁₀₁	—	2	2	no	—
149	D₁₀₁	—	2	2	no	—
150	D₁₀₁	—	4	2	no	—
151	#2929	—	101	1	no	maximal, Sylow
152	C₁₀₁	#2929	1	1	yes	characteristic, complemented, Sylow
153	C₂×#131	—	101	1	no	—
154	C₂³×C₈	—	101	1	no	—
155	#525	—	101	1	no	—
156	#435	—	101	2	no	—
157	#435	—	101	2	no	—
158	C₈×D₄	—	101	2	no	—
159	C₈×D₄	—	101	2	no	—
160	#424	—	101	1	no	—
161	#408	—	101	1	no	—
162	#407	—	101	1	no	—
163	#405	—	101	1	no	—
164	C₂×#41	—	101	1	no	—
165	C₂³×C₄	—	101	1	no	—
166	#131	—	202	4	no	—
167	C₂×M₁₆	—	101	1	no	—
168	C₂×M₁₆	—	101	1	no	—
169	C₂×M₁₆	—	101	2	no	—
170	C₂²×C₈	—	101	1	no	—
171	C₂²×C₈	—	101	1	no	—
172	C₂²×C₈	—	101	1	no	—
173	C₂²×C₈	—	101	1	no	—
174	C₂²×C₈	—	101	1	no	—
175	C₂²×C₈	—	101	1	no	—
176	C₂²×C₈	—	101	2	no	—
177	C₂²×C₈	—	202	4	no	—
178	#123	—	101	2	no	—
179	C₂²⋊Q₈	—	101	1	no	—
180	C₂²⋊Q₈	#87938	1	1	yes	characteristic
181	#121	—	101	1	no	—
182	#121	—	101	1	no	—
183	#120	—	101	2	no	—
184	C₄×D₄	—	101	2	no	—
185	C₄×D₄	—	101	2	no	—
186	#117	—	101	1	no	—
187	C₄⋊C₈	—	101	2	no	—
188	C₄⋊C₈	—	101	2	no	—
189	C₂²⋊C₈	—	101	2	no	—
190	C₂²⋊C₈	—	101	2	no	—
191	C₂²⋊C₈	—	101	2	no	—
192	C₂²⋊C₈	—	101	2	no	—
193	#97	—	101	2	no	—
194	C₄×C₈	—	101	2	no	—
195	C₂⁴	—	101	1	no	—
196	#41	—	202	2	no	—
197	#41	—	202	2	no	—
198	C₂×Q₈	C₂×#87938	1	1	yes	characteristic
199	C₂×D₄	—	101	2	no	—
200	C₂×D₄	—	101	2	no	—
201	C₂×D₄	—	101	2	no	—
202	C₂×D₄	—	101	1	no	—
203	C₂²×C₄	—	101	1	no	—
204	C₂²×C₄	—	101	1	no	—
205	C₂²×C₄	—	101	1	no	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	1	no	—
209	C₂²×C₄	—	202	2	no	—
210	C₂²×C₄	—	202	2	no	—
211	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
212	M₁₆	—	202	4	no	—
213	M₁₆	—	202	2	no	—
214	M₁₆	—	202	2	no	—
215	C₂×C₈	—	101	1	no	—
216	C₂×C₈	—	101	1	no	—
217	C₂×C₈	—	101	4	no	—
218	C₂×C₈	—	101	2	no	—
219	C₂×C₈	—	101	1	no	—
220	C₂×C₈	—	101	1	no	—
221	C₂×C₈	—	101	2	no	—
222	C₂×C₈	—	202	2	no	—
223	C₂×C₈	—	202	2	no	—
224	C₂×C₈	—	202	2	no	—
225	C₂×C₈	—	202	2	no	—
226	C₂×C₈	—	202	4	no	—
227	C₂×C₈	—	202	2	no	—
228	C₄⋊C₄	—	101	2	no	—
229	C₄⋊C₄	—	101	1	no	—
230	C₄⋊C₄	C₂×#87938	1	1	yes	characteristic
231	C₄⋊C₄	C₂×#87938	1	2	yes	—
232	C₂²⋊C₄	—	101	2	no	—
233	C₂²⋊C₄	—	101	2	no	—
234	C₂²⋊C₄	—	101	2	no	—
235	C₂²⋊C₄	—	101	2	no	—
236	C₂²⋊C₄	C₂×#87938	1	2	yes	—
237	C₄²	—	101	2	no	—
238	C₂³	—	101	1	no	—
239	C₂³	—	101	1	no	—
240	C₂³	—	101	2	no	—
241	C₂³	—	202	2	no	—
242	C₂³	—	202	2	no	—
243	C₂³	—	202	2	no	—
244	C₂³	C₂²×#87938	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	D₄	—	202	4	no	—
247	D₄	—	202	4	no	—
248	D₄	—	202	4	no	—
249	D₄	—	202	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	101	1	no	—
252	C₂×C₄	—	101	2	no	—
253	C₂×C₄	—	101	2	no	—
254	C₂×C₄	—	101	2	no	—
255	C₂×C₄	—	101	1	no	—
256	C₂×C₄	—	202	1	no	—
257	C₂×C₄	—	202	2	no	—
258	C₂×C₄	—	202	1	no	—
259	C₂×C₄	—	202	2	no	—
260	C₂×C₄	—	202	2	no	—
261	C₂×C₄	—	202	1	no	—
262	C₂×C₄	—	202	1	no	—
263	C₂×C₄	—	202	2	no	—
264	C₂×C₄	C₂²×#87938	1	2	yes	—
265	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
266	C₂×C₄	C₂²×#87938	1	2	yes	—
267	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
268	C₈	—	101	2	no	—
269	C₈	—	101	2	no	—
270	C₈	—	202	1	no	—
271	C₈	—	202	1	no	—
272	C₈	—	202	2	no	—
273	C₈	—	202	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	101	2	no	—
277	C₂²	—	101	2	no	—
278	C₂²	—	202	8	no	—
279	C₂²	—	202	2	no	—
280	C₂²	—	202	2	no	—
281	C₂²	—	202	2	no	—
282	C₂²	—	202	1	no	—
283	C₂²	—	202	1	no	—
284	C₂²	—	202	1	no	—
285	C₂²	—	202	1	no	—
286	C₂²	#15692343	1	2	yes	complemented
287	C₂²	C₂³×#87938	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	101	1	no	—
292	C₄	—	101	1	no	—
293	C₄	—	202	1	no	—
294	C₄	—	202	2	no	—
295	C₄	D₄×#87938	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	404	2	no	—
298	C₂	—	202	2	no	—
299	C₂	—	202	2	no	—
300	C₂	C₂×D₄×#87938	1	1	yes	characteristic
301	C₂	D₁₀₁⋊#131	1	1	yes	characteristic
302	C₂	C₂×#15692343	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#87938
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂²⋊Q₈×C₁₀₁	C₄
prime radical	#20265690	C₂
cosocle	C₂×Dic₁₀₁	C₂⁴
hypercenter	C₂²⋊Q₈	#87938
hypocenter	C₁₀₁	#2929
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15692343	C₂	semidirect
2. C₂²×D₁₀₁⋊C₈	C₂	semidirect
3. #20266207	C₂	semidirect
4. #20266206	C₂	semidirect
5. #20266205	C₂	semidirect
6. #20266204	C₂	semidirect
7. #20266184	C₂	semidirect
8. #20266152	C₂	semidirect
9. #20266148	C₂	semidirect
10. #20266142	C₂	semidirect
15. C₂×#13108084	C₂²	semidirect
21. C₂×D₁₀₁⋊C₈	C₂²	semidirect
24. #15692322	C₂²	semidirect
25. C₄⋊#11717581	C₂²	semidirect
27. #15692312	C₂²	semidirect
28. #15692312	C₂²	semidirect
30. #15692310	C₂²	semidirect
31. C₄×#11717581	C₂²	semidirect
50. D₁₀₁⋊C₈	D₄	semidirect
124. C₂²×C₁₀₁	#131	semidirect
135. C₂×D₁₀₁	#131	semidirect
152. C₁₀₁	#2929	semidirect
286. C₂²	#15692343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15692343	2
C₂²×D₁₀₁⋊C₈	2
#20266207	2
#20266206	2
#20266205	2
#20266204	2
#20266184	2
#20266152	2
#20266148	2
#20266142	2
#20265690	2
#2929	101