Subgroups — #27155226

Order 12416.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₉₇	C₂	1	2	yes	maximal, complemented
2	Q₈×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261392	C₂	1	2	yes	maximal, complemented
4	#20261390	C₂	1	3	yes	maximal, complemented
5	#20261382	C₂	1	6	yes	maximal, complemented
6	#20261370	C₂	1	3	yes	maximal, complemented
7	#20261283	C₂	1	3	yes	maximal, complemented
8	#20261245	C₂	1	6	yes	maximal, complemented
9	C₄×#13095512	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₉₇	C₂	1	3	yes	maximal, complemented
11	#550×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095512	C₂²	1	1	yes	characteristic
13	C₂×#13095511	C₂²	1	6	yes	complemented
14	C₂×#13095510	C₂²	1	6	yes	complemented
15	#41×D₉₇	—	2	16	no	—
16	C₂×Q₈×D₉₇	C₂²	1	2	yes	complemented
17	C₂×D₄×D₉₇	C₂²	1	3	yes	—
18	C₂×D₄×D₉₇	C₂²	1	6	yes	complemented
19	C₂²×C₄×D₉₇	C₂²	1	6	yes	complemented
20	#15690810	C₂²	1	3	yes	complemented
21	#15690797	C₂²	1	6	yes	complemented
22	#15690792	C₂²	1	12	yes	complemented
23	#15690790	C₂²	1	6	yes	complemented
24	#15690789	C₂²	1	6	yes	complemented
25	#15690781	C₂²	1	6	yes	complemented
26	#15690781	C₂²	1	6	yes	complemented
27	#15690781	C₂²	1	6	yes	complemented
28	#15690778	C₂²	1	12	yes	complemented
29	Q₈×Dic₉₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₃₈₈	C₂²	1	3	yes	—
31	C₄×D₃₈₈	C₂²	1	3	yes	complemented
32	D₄×Dic₉₇	C₂²	1	3	yes	complemented
33	D₄×Dic₉₇	C₂²	1	6	yes	complemented
34	#15690772	C₂²	1	1	yes	characteristic
35	#15690772	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₉₇	C₂²	1	6	yes	complemented
37	#15690770	C₂²	1	6	yes	complemented
38	#15690768	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₉₇	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₉₇	C₂²	1	6	yes	complemented
41	C₄²×D₉₇	C₂²	1	3	yes	complemented
42	C₂×#41×C₉₇	C₂²	1	2	yes	complemented
43	#127×C₉₇	C₂²	1	3	yes	complemented
44	#121×C₉₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₉₇	C₂²	1	3	yes	complemented
47	C₂³×D₉₇	C₂³	1	6	yes	—
48	#13095512	D₄	1	8	yes	complemented
49	#13095511	—	2	24	no	—
50	#13095510	—	2	24	no	—
51	C₂×Dic₁₉₄	C₂³	1	6	yes	—
52	Q₈×D₉₇	—	2	8	no	—
53	C₂×D₃₈₈	C₂³	1	3	yes	—
54	C₂×D₃₈₈	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₉₇	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₉₇	C₂³	1	12	yes	—
57	D₄×D₉₇	—	2	24	no	—
58	D₄×D₉₇	—	2	12	no	—
59	C₂²×Dic₉₇	C₂³	1	6	yes	—
60	C₂×C₄×D₉₇	—	2	24	no	—
61	C₂×C₄×D₉₇	C₂³	1	3	yes	—
62	C₂×C₄×D₉₇	C₂³	1	6	yes	—
63	C₂×C₄×D₉₇	C₂³	1	2	yes	—
64	C₂×C₄×D₉₇	C₂³	1	6	yes	—
65	C₂×C₄×D₉₇	C₂³	1	3	yes	—
66	C₂×C₄×D₉₇	C₂³	1	3	yes	—
67	C₄⋊Dic₉₇	C₂³	1	3	yes	—
68	C₄⋊Dic₉₇	C₂³	1	3	yes	—
69	C₄⋊Dic₉₇	C₂³	1	6	yes	—
70	#13095490	C₂³	1	6	yes	—
71	C₂²⋊Dic₉₇	C₂³	1	6	yes	—
72	C₂²⋊Dic₉₇	C₂³	1	12	yes	—
73	#13095488	C₂³	1	2	yes	—
74	#13095488	C₂³	1	6	yes	—
75	#13095488	C₂³	1	6	yes	—
76	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
77	C₄×Dic₉₇	C₂³	1	3	yes	—
78	C₄×Dic₉₇	C₂³	1	3	yes	—
79	#41×C₉₇	—	2	8	no	—
80	C₂×Q₈×C₉₇	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₉₇	C₂³	1	3	yes	—
82	C₂×D₄×C₉₇	C₂³	1	6	yes	—
83	C₂×D₄×C₉₇	C₂³	1	6	yes	—
84	C₂²×C₄×C₉₇	C₂³	1	6	yes	—
85	C₄⋊C₄×C₉₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₉₇	C₂³	1	3	yes	—
87	C₂²⋊C₄×C₉₇	C₂³	1	6	yes	—
88	C₄²×C₉₇	C₂³	1	3	yes	—
89	C₂³×C₉₇	C₂⁴	1	6	yes	—
90	C₂²×D₉₇	—	2	12	no	—
91	C₂²×D₉₇	—	2	12	no	—
92	C₂²×D₉₇	—	4	6	no	—
93	C₂²×D₉₇	C₂⁴	1	3	yes	—
94	C₂²×D₉₇	C₂⁴	1	2	yes	—
95	Q₈×C₉₇	C₂×D₄	1	4	yes	complemented
96	D₄×C₉₇	—	2	12	no	—
97	D₄×C₉₇	—	2	6	no	—
98	D₄×C₉₇	—	4	6	no	—
99	C₂×C₄×C₉₇	—	2	12	no	—
100	C₂×C₄×C₉₇	C₂⁴	1	3	yes	—
101	C₂×C₄×C₉₇	C₂⁴	1	2	yes	—
102	C₂×C₄×C₉₇	C₂⁴	1	3	yes	—
103	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₉₇	—	2	24	no	—
105	C₂²⋊D₉₇	—	2	12	no	—
106	C₂×Dic₉₇	—	2	12	no	—
107	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₉₇	C₂⁴	1	3	yes	—
109	C₂×Dic₉₇	C₂⁴	1	6	yes	—
110	C₂×Dic₉₇	C₂⁴	1	1	yes	characteristic
111	D₃₈₈	C₂×D₄	1	12	yes	complemented
112	D₃₈₈	#41	1	4	yes	complemented
113	C₄×D₉₇	—	2	12	no	—
114	C₄×D₉₇	—	2	12	no	—
115	C₄×D₉₇	—	2	6	no	—
116	C₄×D₉₇	—	2	4	no	—
117	C₄×D₉₇	—	4	3	no	—
118	C₄×D₉₇	C₂×D₄	1	12	yes	complemented
119	Dic₁₉₄	—	2	12	no	—
120	C₂²×C₉₇	—	2	6	no	—
121	C₂²×C₉₇	—	2	6	no	—
122	C₂²×C₉₇	—	2	6	no	—
123	C₂²×C₉₇	C₂⁵	1	1	yes	characteristic
124	C₂×D₉₇	—	2	2	no	—
125	C₂×D₉₇	—	2	3	no	—
126	C₂×D₉₇	—	2	3	no	—
127	C₂×D₉₇	—	2	2	no	—
128	C₂×D₉₇	—	4	24	no	—
129	C₂×D₉₇	C₂²×D₄	1	6	yes	—
130	C₂×D₉₇	C₂×#41	1	4	yes	complemented
131	C₄×C₉₇	—	2	2	no	—
132	C₄×C₉₇	—	2	3	no	—
133	C₄×C₉₇	C₂²×D₄	1	6	yes	—
134	C₄×C₉₇	C₂×#41	1	2	yes	complemented
135	Dic₉₇	—	2	6	no	—
136	Dic₉₇	—	2	3	no	—
137	Dic₉₇	—	2	1	no	—
138	Dic₉₇	C₂²×D₄	1	2	yes	—
139	C₂×C₉₇	—	4	6	no	—
140	C₂×C₉₇	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₉₇	C₂²×#41	1	1	yes	characteristic
142	C₂×C₉₇	#585	1	1	yes	characteristic
143	D₉₇	—	2	4	no	—
144	D₉₇	—	2	6	no	—
145	D₄×#41	—	97	1	no	maximal, Sylow
146	C₉₇	D₄×#41	1	1	yes	characteristic, complemented, Sylow
147	C₂²×#41	—	97	2	no	—
148	#550	D₉₇	1	1	yes	characteristic, complemented
149	D₄×Q₈	—	97	1	no	—
150	#548	—	97	2	no	—
151	#547	—	97	3	no	—
152	#546	—	97	6	no	—
153	D₄²	—	97	3	no	—
154	#532	—	97	3	no	—
155	#525	—	97	6	no	—
156	C₄×#41	—	97	1	no	—
157	C₂×C₄×D₄	—	97	3	no	—
158	C₂×#41	—	97	1	no	—
159	C₂×#41	—	97	6	no	—
160	C₂×#41	—	97	6	no	—
161	C₂×#41	—	194	16	no	—
162	C₂×#41	C₂×D₉₇	1	2	yes	complemented
163	C₂²×Q₈	—	97	2	no	—
164	C₂²×D₄	—	97	3	no	—
165	C₂²×D₄	—	97	6	no	—
166	C₂³×C₄	—	97	6	no	—
167	C₄⋊Q₈	—	97	3	no	—
168	#127	C₂×D₉₇	1	3	yes	complemented
169	#124	—	97	6	no	—
170	#123	—	97	12	no	—
171	C₂²⋊Q₈	—	97	6	no	—
172	C₂²⋊Q₈	—	97	6	no	—
173	#121	—	97	6	no	—
174	#121	—	97	6	no	—
175	#121	—	97	6	no	—
176	#121	C₂×D₉₇	1	6	yes	complemented
177	#120	—	97	12	no	—
178	C₄×Q₈	—	97	1	no	—
179	C₄×Q₈	C₂×D₉₇	1	1	yes	characteristic, complemented
180	C₄×D₄	—	97	3	no	—
181	C₄×D₄	—	97	6	no	—
182	C₄×D₄	—	97	6	no	—
183	C₄×D₄	—	97	1	no	—
184	C₄×D₄	—	97	3	no	—
185	C₄×D₄	—	97	6	no	—
186	C₄×D₄	—	97	3	no	—
187	C₄×D₄	—	97	3	no	—
188	C₄×D₄	C₂×D₉₇	1	3	yes	complemented
189	#117	—	97	3	no	—
190	C₂×C₄⋊C₄	—	97	3	no	—
191	C₂×C₂²⋊C₄	—	97	6	no	—
192	C₂×C₄²	—	97	3	no	—
193	C₂⁴	—	97	6	no	—
194	#41	—	2	8	no	—
195	#41	—	97	8	no	—
196	#41	—	194	24	no	—
197	#41	—	194	24	no	—
198	C₂×Q₈	—	97	6	no	—
199	C₂×Q₈	—	194	8	no	—
200	C₂×Q₈	C₂²×D₉₇	1	1	yes	characteristic
201	C₂×D₄	—	97	6	no	—
202	C₂×D₄	—	97	12	no	—
203	C₂×D₄	—	97	1	no	—
204	C₂×D₄	—	97	3	no	—
205	C₂×D₄	—	194	12	no	—
206	C₂×D₄	—	194	24	no	—
207	C₂×D₄	C₂²×D₉₇	1	3	yes	—
208	C₂×D₄	C₂²×D₉₇	1	6	yes	—
209	C₂×D₄	C₂²×D₉₇	1	6	yes	—
210	C₂²×C₄	—	97	3	no	—
211	C₂²×C₄	—	97	6	no	—
212	C₂²×C₄	—	97	3	no	—
213	C₂²×C₄	—	97	3	no	—
214	C₂²×C₄	—	97	6	no	—
215	C₂²×C₄	—	97	2	no	—
216	C₂²×C₄	—	97	6	no	—
217	C₂²×C₄	—	194	24	no	—
218	C₂²×C₄	C₂²×D₉₇	1	6	yes	—
219	C₄⋊C₄	—	97	6	no	—
220	C₄⋊C₄	—	97	3	no	—
221	C₄⋊C₄	—	97	3	no	—
222	C₄⋊C₄	—	97	6	no	—
223	C₄⋊C₄	C₂²×D₉₇	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₉₇	1	3	yes	—
225	C₂²⋊C₄	—	97	6	no	—
226	C₂²⋊C₄	—	97	12	no	—
227	C₂²⋊C₄	—	97	2	no	—
228	C₂²⋊C₄	—	97	6	no	—
229	C₂²⋊C₄	—	97	6	no	—
230	C₂²⋊C₄	C₂²×D₉₇	1	6	yes	—
231	C₄²	—	97	1	no	—
232	C₄²	—	97	3	no	—
233	C₄²	—	97	3	no	—
234	C₄²	C₂²×D₉₇	1	3	yes	—
235	C₂³	—	97	2	no	—
236	C₂³	—	97	3	no	—
237	C₂³	—	388	6	no	—
238	C₂³	—	194	12	no	—
239	C₂³	—	194	12	no	—
240	C₂³	C₂³×D₉₇	1	6	yes	—
241	Q₈	—	194	12	no	—
242	Q₈	D₄×D₉₇	1	4	yes	complemented
243	D₄	—	2	12	no	—
244	D₄	—	2	6	no	—
245	D₄	—	4	6	no	—
246	D₄	—	97	12	no	—
247	D₄	—	97	4	no	—
248	D₄	—	194	24	no	—
249	D₄	—	194	12	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	97	1	no	—
252	C₂×C₄	—	97	12	no	—
253	C₂×C₄	—	97	1	no	—
254	C₂×C₄	—	97	3	no	—
255	C₂×C₄	—	97	6	no	—
256	C₂×C₄	—	388	3	no	—
257	C₂×C₄	—	194	4	no	—
258	C₂×C₄	—	194	12	no	—
259	C₂×C₄	—	194	12	no	—
260	C₂×C₄	—	194	12	no	—
261	C₂×C₄	—	194	6	no	—
262	C₂×C₄	C₂³×D₉₇	1	3	yes	—
263	C₂×C₄	C₂³×D₉₇	1	2	yes	—
264	C₂×C₄	C₂³×D₉₇	1	3	yes	—
265	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
266	C₂²	—	2	6	no	—
267	C₂²	—	2	6	no	—
268	C₂²	—	2	6	no	—
269	C₂²	—	97	6	no	—
270	C₂²	—	97	4	no	—
271	C₂²	—	388	24	no	—
272	C₂²	—	194	2	no	—
273	C₂²	—	194	3	no	—
274	C₂²	—	194	3	no	—
275	C₂²	—	194	2	no	—
276	C₂²	C₂⁴×D₉₇	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	3	no	—
279	C₄	—	97	2	no	—
280	C₄	—	194	6	no	—
281	C₄	—	194	3	no	—
282	C₄	—	194	1	no	—
283	C₄	C₂×D₄×D₉₇	1	6	yes	—
284	C₄	#41×D₉₇	1	2	yes	complemented
285	C₂	—	4	6	no	—
286	C₂	—	194	4	no	—
287	C₂	—	194	6	no	—
288	C₂	C₂²×D₄×D₉₇	1	1	yes	characteristic
289	C₂	C₂×#41×D₉₇	1	1	yes	characteristic
290	C₂	#20261570	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₉₇
commutator, socle, cosocle	C₂²×C₉₇	C₂⁵
Fitting	#550×C₉₇	C₂
hypercenter	#550	D₉₇
hypocenter	C₉₇	D₄×#41
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₉₇	C₂	semidirect
2. Q₈×D₃₈₈	C₂	semidirect
3. #20261392	C₂	semidirect
4. #20261390	C₂	semidirect
5. #20261382	C₂	semidirect
6. #20261370	C₂	semidirect
7. #20261283	C₂	semidirect
8. #20261245	C₂	semidirect
9. C₄×#13095512	C₂	semidirect
10. C₄×D₄×D₉₇	C₂	semidirect
11. #550×C₉₇	C₂	semidirect
13. C₂×#13095511	C₂²	semidirect
14. C₂×#13095510	C₂²	semidirect
16. C₂×Q₈×D₉₇	C₂²	semidirect
18. C₂×D₄×D₉₇	C₂²	semidirect
19. C₂²×C₄×D₉₇	C₂²	semidirect
20. #15690810	C₂²	semidirect
21. #15690797	C₂²	semidirect
22. #15690792	C₂²	semidirect
23. #15690790	C₂²	semidirect
24. #15690789	C₂²	semidirect
25. #15690781	C₂²	semidirect
26. #15690781	C₂²	semidirect
27. #15690781	C₂²	semidirect
28. #15690778	C₂²	semidirect
29. Q₈×Dic₉₇	C₂²	semidirect
31. C₄×D₃₈₈	C₂²	semidirect
32. D₄×Dic₉₇	C₂²	semidirect
33. D₄×Dic₉₇	C₂²	semidirect
35. #15690772	C₂²	semidirect
36. C₄×C₂²⋊D₉₇	C₂²	semidirect
37. #15690770	C₂²	semidirect
38. #15690768	C₂²	semidirect
39. C₄⋊C₄×D₉₇	C₂²	semidirect
40. C₂²⋊C₄×D₉₇	C₂²	semidirect
41. C₄²×D₉₇	C₂²	semidirect
42. C₂×#41×C₉₇	C₂²	semidirect
43. #127×C₉₇	C₂²	semidirect
44. #121×C₉₇	C₂²	semidirect
45. C₄×Q₈×C₉₇	C₂²	semidirect
46. C₄×D₄×C₉₇	C₂²	semidirect
48. #13095512	D₄	semidirect
95. Q₈×C₉₇	C₂×D₄	semidirect
111. D₃₈₈	C₂×D₄	semidirect
112. D₃₈₈	#41	semidirect
118. C₄×D₉₇	C₂×D₄	semidirect
130. C₂×D₉₇	C₂×#41	semidirect
134. C₄×C₉₇	C₂×#41	semidirect
146. C₉₇	D₄×#41	semidirect
148. #550	D₉₇	semidirect
162. C₂×#41	C₂×D₉₇	semidirect
168. #127	C₂×D₉₇	semidirect
176. #121	C₂×D₉₇	semidirect
179. C₄×Q₈	C₂×D₉₇	semidirect
188. C₄×D₄	C₂×D₉₇	semidirect
242. Q₈	D₄×D₉₇	semidirect
284. C₄	#41×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₉₇	2
Q₈×D₃₈₈	2
#20261392	2
#20261390	2
#20261382	2
#20261370	2
#20261283	2
#20261245	2
C₄×#13095512	2
C₄×D₄×D₉₇	2
#550×C₉₇	2
D₄×#41	97