Subgroups — #27184429

Order 12928.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265732	C₂	1	2	yes	maximal, complemented
2	#20265728	C₂	1	2	yes	maximal, complemented
3	#20265727	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265714	C₂	1	4	yes	maximal, complemented
5	#20265713	C₂	1	4	yes	maximal, complemented
6	#20265712	C₂	1	4	yes	maximal, complemented
7	#20265690	C₂	1	4	yes	maximal, complemented
8	#20265689	C₂	1	2	yes	maximal, complemented
9	C₄×#13108075	C₂	1	4	yes	maximal, complemented
10	#20265645	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#533×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13108075	C₂²	1	4	yes	—
14	C₂×#13108074	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
17	#15692255	C₂²	1	1	yes	characteristic, complemented
18	#15692252	C₂²	1	1	yes	characteristic, complemented
19	#15692250	C₂²	1	8	yes	complemented
20	#15692246	C₂²	1	4	yes	complemented
21	#15692244	C₂²	1	4	yes	complemented
22	#15692243	C₂²	1	2	yes	complemented
23	#15692240	C₂²	1	8	yes	—
24	#15692239	C₂²	1	4	yes	—
25	#15692235	C₂²	1	4	yes	—
26	#15692231	C₂²	1	4	yes	—
27	#15692230	C₂²	1	4	yes	—
28	#15692230	C₂²	1	4	yes	—
29	#15692229	C₂²	1	4	yes	—
30	#15692225	C₂²	1	4	yes	—
31	Q₈×Dic₁₀₁	C₂²	1	4	yes	complemented
32	#15692220	C₂²	1	4	yes	complemented
33	C₄×D₄₀₄	C₂²	1	8	yes	—
34	C₄×D₄₀₄	C₂²	1	4	yes	complemented
35	#15692217	C₂²	1	8	yes	—
36	#15692217	C₂²	1	4	yes	complemented
37	#15692217	C₂²	1	4	yes	complemented
38	#15692215	C₂²	1	8	yes	—
39	#15692213	C₂²	1	4	yes	—
40	#15692213	C₂²	1	8	yes	complemented
41	#15692212	C₂²	1	4	yes	complemented
42	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
44	C₄²×D₁₀₁	C₂²	1	2	yes	—
45	C₄²×D₁₀₁	C₂²	1	2	yes	—
46	C₄²×D₁₀₁	C₂²	1	2	yes	—
47	C₄²×D₁₀₁	C₂²	1	2	yes	—
48	C₄²×D₁₀₁	C₂²	1	4	yes	complemented
49	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	complemented
50	#125×C₁₀₁	C₂²	1	2	yes	complemented
51	C₂²⋊Q₈×C₁₀₁	C₂²	1	4	yes	—
52	C₄×Q₈×C₁₀₁	C₂²	1	4	yes	complemented
53	#117×C₁₀₁	C₂²	1	2	yes	—
54	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
55	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
56	#13108075	—	2	16	no	—
57	#13108074	—	2	8	no	—
58	C₂×Dic₂₀₂	C₂³	1	2	yes	—
59	C₂×D₄₀₄	C₂³	1	2	yes	—
60	C₂×D₄₀₄	C₂³	1	8	yes	—
61	C₂×D₄₀₄	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₁₀₁	C₂³	1	4	yes	—
63	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
64	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₀₁	—	2	4	no	—
66	C₂×C₄×D₁₀₁	—	2	8	no	—
67	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
68	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
69	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
70	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
71	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
72	C₂×C₄×D₁₀₁	C₂³	1	4	yes	—
73	C₂×C₄×D₁₀₁	C₂³	1	8	yes	—
74	C₂×C₄×D₁₀₁	C₂³	1	2	yes	—
75	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
76	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
77	#13108053	C₂³	1	8	yes	—
78	#13108051	C₂³	1	8	yes	—
79	#13108051	C₂³	1	8	yes	—
80	#13108051	C₂³	1	8	yes	—
81	#13108051	C₂³	1	8	yes	—
82	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₀₁	C₂³	1	2	yes	—
85	C₄×Dic₁₀₁	C₂³	1	2	yes	—
86	C₄×Dic₁₀₁	C₂³	1	2	yes	—
87	C₄×Dic₁₀₁	C₂³	1	2	yes	—
88	C₄×Dic₁₀₁	C₂³	1	2	yes	—
89	C₄×Dic₁₀₁	C₂³	1	8	yes	—
90	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
91	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
92	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
94	C₄⋊C₄×C₁₀₁	C₂³	1	8	yes	—
95	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
96	C₂²⋊C₄×C₁₀₁	C₂³	1	4	yes	—
97	C₄²×C₁₀₁	C₂³	1	2	yes	—
98	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
99	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
100	C₄²×C₁₀₁	C₂³	1	4	yes	—
101	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
102	C₂²×D₁₀₁	—	2	2	no	—
103	C₂²×D₁₀₁	—	2	4	no	—
104	C₂²×D₁₀₁	C₂⁴	1	4	yes	—
105	C₂²×D₁₀₁	C₂⁴	1	2	yes	—
106	Q₈×C₁₀₁	—	2	8	no	—
107	C₂×C₄×C₁₀₁	—	2	4	no	—
108	C₂×C₄×C₁₀₁	#41	1	4	yes	complemented
109	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₀₁	C₂⁴	1	8	yes	—
111	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₁₀₁	C₂⁴	1	2	yes	—
113	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₁₀₁	—	2	8	no	—
115	C₂×Dic₁₀₁	—	2	2	no	—
116	C₂×Dic₁₀₁	—	2	4	no	—
117	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
118	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
119	C₂×Dic₁₀₁	C₂⁴	1	4	yes	—
120	C₂×Dic₁₀₁	C₂⁴	1	2	yes	—
121	D₄₀₄	—	2	16	no	—
122	D₄₀₄	—	2	4	no	—
123	D₄₀₄	—	2	8	no	—
124	C₄×D₁₀₁	—	2	4	no	—
125	C₄×D₁₀₁	—	2	4	no	—
126	C₄×D₁₀₁	—	2	4	no	—
127	C₄×D₁₀₁	—	2	4	no	—
128	C₄×D₁₀₁	—	2	8	no	—
129	C₄×D₁₀₁	—	2	4	no	—
130	C₄×D₁₀₁	—	2	16	no	—
131	C₄×D₁₀₁	#41	1	8	yes	complemented
132	Dic₂₀₂	—	2	4	no	—
133	C₂²×C₁₀₁	—	2	2	no	—
134	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
135	C₂²×C₁₀₁	C₂⁵	1	1	yes	characteristic
136	C₂×D₁₀₁	—	2	8	no	—
137	C₂×D₁₀₁	—	2	2	no	—
138	C₂×D₁₀₁	—	2	2	no	—
139	C₂×D₁₀₁	—	2	8	no	—
140	C₂×D₁₀₁	—	2	4	no	—
141	C₂×D₁₀₁	C₂×#41	1	4	yes	—
142	C₄×C₁₀₁	—	2	2	no	—
143	C₄×C₁₀₁	—	2	8	no	—
144	C₄×C₁₀₁	C₂×#41	1	4	yes	—
145	C₄×C₁₀₁	C₂×#41	1	2	yes	—
146	C₄×C₁₀₁	C₂×#41	1	2	yes	—
147	Dic₁₀₁	—	2	4	no	—
148	Dic₁₀₁	—	2	2	no	—
149	Dic₁₀₁	—	2	2	no	—
150	Dic₁₀₁	C₂×#41	1	4	yes	—
151	C₂×C₁₀₁	—	2	2	no	—
152	C₂×C₁₀₁	C₂²×#41	1	2	yes	—
153	C₂×C₁₀₁	C₂²×#41	1	1	yes	characteristic
154	D₁₀₁	—	2	4	no	—
155	D₁₀₁	—	4	4	no	—
156	#3444	—	101	1	no	maximal, Sylow
157	C₁₀₁	#3444	1	1	yes	characteristic, complemented, Sylow
158	#533	—	101	2	no	—
159	#533	D₁₀₁	1	1	yes	characteristic, complemented
160	#532	—	101	2	no	—
161	#532	—	101	1	no	—
162	#529	—	101	4	no	—
163	#529	—	101	4	no	—
164	#529	—	101	4	no	—
165	#525	—	101	4	no	—
166	#525	—	101	2	no	—
167	C₄×#41	—	101	4	no	—
168	C₄×#41	—	101	2	no	—
169	C₂²×C₄²	—	101	1	no	—
170	C₂×#41	—	101	4	no	—
171	C₂×#41	—	101	2	no	—
172	C₂³×C₄	—	101	2	no	—
173	C₂³×C₄	—	101	1	no	—
174	C₄⋊Q₈	—	101	1	no	—
175	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	complemented
176	#127	—	101	1	no	—
177	#126	—	101	8	no	—
178	#125	—	101	4	no	—
179	#125	C₂×D₁₀₁	1	2	yes	complemented
180	#124	—	101	4	no	—
181	#124	—	101	2	no	—
182	#123	—	101	8	no	—
183	#123	—	101	4	no	—
184	C₂²⋊Q₈	—	101	4	no	—
185	C₂²⋊Q₈	—	101	4	no	—
186	C₂²⋊Q₈	C₂×D₁₀₁	1	4	yes	—
187	#121	—	101	4	no	—
188	#121	—	101	4	no	—
189	#121	—	101	4	no	—
190	#120	—	101	4	no	—
191	C₄×Q₈	—	101	4	no	—
192	C₄×Q₈	—	101	4	no	—
193	C₄×Q₈	C₂×D₁₀₁	1	4	yes	complemented
194	C₄×D₄	—	101	8	no	—
195	C₄×D₄	—	101	4	no	—
196	C₄×D₄	—	101	4	no	—
197	C₄×D₄	—	101	8	no	—
198	C₄×D₄	—	101	8	no	—
199	C₄×D₄	—	101	4	no	—
200	#117	—	101	4	no	—
201	#117	—	101	8	no	—
202	#117	—	101	4	no	—
203	#117	C₂×D₁₀₁	1	2	yes	—
204	C₂×C₄²	—	101	1	no	—
205	C₂×C₄²	—	101	2	no	—
206	C₂×C₄²	—	101	4	no	—
207	C₂×C₄²	—	101	2	no	—
208	C₂×C₄²	—	101	2	no	—
209	C₂×C₄²	—	101	2	no	—
210	C₂×C₄²	—	101	2	no	—
211	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
212	C₂⁴	—	101	1	no	—
213	#41	—	202	16	no	—
214	#41	—	202	8	no	—
215	C₂×Q₈	—	101	2	no	—
216	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
217	C₂×D₄	—	101	4	no	—
218	C₂×D₄	—	101	8	no	—
219	C₂×D₄	—	101	4	no	—
220	C₂×D₄	—	101	2	no	—
221	C₂²×C₄	—	101	2	no	—
222	C₂²×C₄	—	101	2	no	—
223	C₂²×C₄	—	101	1	no	—
224	C₂²×C₄	—	101	2	no	—
225	C₂²×C₄	—	101	2	no	—
226	C₂²×C₄	—	101	2	no	—
227	C₂²×C₄	—	101	2	no	—
228	C₂²×C₄	—	101	8	no	—
229	C₂²×C₄	—	101	2	no	—
230	C₂²×C₄	—	101	4	no	—
231	C₂²×C₄	—	202	8	no	—
232	C₂²×C₄	—	202	4	no	—
233	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
235	C₄⋊C₄	—	101	8	no	—
236	C₄⋊C₄	—	101	4	no	—
237	C₄⋊C₄	—	101	8	no	—
238	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
239	C₄⋊C₄	C₂²×D₁₀₁	1	8	yes	—
240	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
241	C₂²⋊C₄	—	101	8	no	—
242	C₂²⋊C₄	—	101	8	no	—
243	C₂²⋊C₄	—	101	8	no	—
244	C₂²⋊C₄	—	101	8	no	—
245	C₂²⋊C₄	C₂²×D₁₀₁	1	4	yes	—
246	C₄²	—	101	2	no	—
247	C₄²	—	101	2	no	—
248	C₄²	—	101	2	no	—
249	C₄²	—	101	2	no	—
250	C₄²	—	101	2	no	—
251	C₄²	—	101	8	no	—
252	C₄²	—	101	1	no	—
253	C₄²	—	101	1	no	—
254	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
255	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
256	C₄²	C₂²×D₁₀₁	1	2	yes	—
257	C₄²	C₂²×D₁₀₁	1	4	yes	—
258	C₂³	—	101	2	no	—
259	C₂³	—	101	4	no	—
260	C₂³	—	202	4	no	—
261	C₂³	—	202	2	no	—
262	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
263	Q₈	—	2	8	no	—
264	Q₈	—	202	4	no	—
265	D₄	—	202	16	no	—
266	D₄	—	202	8	no	—
267	D₄	—	202	8	no	—
268	D₄	—	202	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	101	2	no	—
271	C₂×C₄	—	101	8	no	—
272	C₂×C₄	—	101	2	no	—
273	C₂×C₄	—	101	2	no	—
274	C₂×C₄	—	101	4	no	—
275	C₂×C₄	—	202	4	no	—
276	C₂×C₄	—	202	16	no	—
277	C₂×C₄	—	202	4	no	—
278	C₂×C₄	—	202	4	no	—
279	C₂×C₄	—	202	2	no	—
280	C₂×C₄	—	202	4	no	—
281	C₂×C₄	—	202	4	no	—
282	C₂×C₄	—	202	4	no	—
283	C₂×C₄	—	202	8	no	—
284	C₂×C₄	#13108075	1	4	yes	complemented
285	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
286	C₂×C₄	C₂³×D₁₀₁	1	8	yes	—
287	C₂×C₄	C₂³×D₁₀₁	1	2	yes	—
288	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	101	4	no	—
292	C₂²	—	202	8	no	—
293	C₂²	—	202	8	no	—
294	C₂²	—	202	4	no	—
295	C₂²	—	202	2	no	—
296	C₂²	—	202	2	no	—
297	C₂²	C₂×#13108075	1	2	yes	—
298	C₂²	C₂⁴×D₁₀₁	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	8	no	—
301	C₄	—	101	4	no	—
302	C₄	—	202	2	no	—
303	C₄	—	202	2	no	—
304	C₄	—	202	4	no	—
305	C₄	#41×D₁₀₁	1	4	yes	—
306	C₄	C₂×#13108075	1	2	yes	—
307	C₄	C₂×#13108075	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	404	4	no	—
310	C₂	—	202	4	no	—
311	C₂	C₂×#41×D₁₀₁	1	2	yes	—
312	C₂	C₂²×#13108075	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₀₁
commutator, socle, cosocle	C₂²×C₁₀₁	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₁
Fitting	#533×C₁₀₁	C₂
hypercenter	#533	D₁₀₁
hypocenter	C₁₀₁	#3444
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #20265732	C₂	semidirect
2. #20265728	C₂	semidirect
3. #20265727	C₂	semidirect
4. #20265714	C₂	semidirect
5. #20265713	C₂	semidirect
6. #20265712	C₂	semidirect
7. #20265690	C₂	semidirect
8. #20265689	C₂	semidirect
9. C₄×#13108075	C₂	semidirect
10. #20265645	C₂	semidirect
11. C₂×C₄²×D₁₀₁	C₂	semidirect
12. #533×C₁₀₁	C₂	semidirect
17. #15692255	C₂²	semidirect
18. #15692252	C₂²	semidirect
19. #15692250	C₂²	semidirect
20. #15692246	C₂²	semidirect
21. #15692244	C₂²	semidirect
22. #15692243	C₂²	semidirect
31. Q₈×Dic₁₀₁	C₂²	semidirect
32. #15692220	C₂²	semidirect
34. C₄×D₄₀₄	C₂²	semidirect
36. #15692217	C₂²	semidirect
37. #15692217	C₂²	semidirect
40. #15692213	C₂²	semidirect
41. #15692212	C₂²	semidirect
48. C₄²×D₁₀₁	C₂²	semidirect
49. C₄⋊Q₈×C₁₀₁	C₂²	semidirect
50. #125×C₁₀₁	C₂²	semidirect
52. C₄×Q₈×C₁₀₁	C₂²	semidirect
108. C₂×C₄×C₁₀₁	#41	semidirect
131. C₄×D₁₀₁	#41	semidirect
157. C₁₀₁	#3444	semidirect
159. #533	D₁₀₁	semidirect
175. C₄⋊Q₈	C₂×D₁₀₁	semidirect
179. #125	C₂×D₁₀₁	semidirect
193. C₄×Q₈	C₂×D₁₀₁	semidirect
284. C₂×C₄	#13108075	semidirect

Maximal subgroups

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

subgroup	index
#20265732	2
#20265728	2
#20265727	2
#20265714	2
#20265713	2
#20265712	2
#20265690	2
#20265689	2
C₄×#13108075	2
#20265645	2
C₂×C₄²×D₁₀₁	2
#533×C₁₀₁	2
#3444	101