Subgroups — #27158183

Order 12416.

568 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261680	C₂	1	2	yes	maximal, complemented
2	#20261679	C₂	1	2	yes	maximal, complemented
3	#20261678	C₂	1	2	yes	maximal
4	#20261677	C₂	1	2	yes	maximal
5	C₄×C₄⋊#87604	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊#87604	C₂	1	2	yes	maximal
7	#20261671	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₉₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊#87604	C₂²	1	4	yes	—
10	C₂×C₂²⋊#87604	C₂²	1	4	yes	—
11	C₂×C₄×#87604	C₂²	1	4	yes	—
12	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
13	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
14	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
15	(C₂×#87604)⋊C₄	C₂²	1	4	yes	—
16	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
21	C₂×C₄×Dic₉₇	C₄	1	1	yes	characteristic, complemented
22	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
23	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
24	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
25	C₄²×D₉₇	C₄	1	1	yes	characteristic, complemented
26	C₄²×D₉₇	C₂²	1	1	yes	characteristic
27	C₄²×D₉₇	C₂²	1	1	yes	characteristic
28	C₄²×D₉₇	C₂²	1	1	yes	characteristic
29	C₄²×D₉₇	C₂²	1	1	yes	characteristic
30	C₂×C₄²×C₉₇	C₄	1	1	yes	characteristic, complemented
31	C₂²×#87604	C₂³	1	8	yes	—
32	C₄⋊#87604	—	2	8	no	—
33	C₂²⋊#87604	—	2	8	no	—
34	C₄×#87604	—	2	8	no	—
35	C₂³×D₉₇	C₂³	1	1	yes	characteristic
36	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×D₉₇	—	2	4	no	—
40	C₂×C₄×D₉₇	—	2	4	no	—
41	C₂×C₄×D₉₇	—	2	4	no	—
42	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
51	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
54	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
55	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
73	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₉₇	—	2	2	no	—
75	C₂²×D₉₇	—	2	2	no	—
76	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×D₉₇	#41	1	4	yes	—
78	C₂²×D₉₇	C₂⁴	1	1	yes	characteristic
79	C₂×#87604	—	2	8	no	—
80	C₂×#87604	—	2	8	no	—
81	C₂×#87604	—	2	8	no	—
82	C₂×C₄×C₉₇	—	2	2	no	—
83	C₂×C₄×C₉₇	—	2	2	no	—
84	C₂×C₄×C₉₇	—	2	2	no	—
85	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₉₇	—	2	2	no	—
92	C₂×Dic₉₇	—	2	2	no	—
93	C₂×Dic₉₇	—	2	2	no	—
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
100	C₄×D₉₇	—	2	2	no	—
101	C₄×D₉₇	—	2	2	no	—
102	C₄×D₉₇	—	2	2	no	—
103	C₄×D₉₇	—	2	2	no	—
104	C₄×D₉₇	—	2	2	no	—
105	C₄×D₉₇	—	2	2	no	—
106	C₄×D₉₇	—	2	2	no	—
107	C₄×D₉₇	—	2	2	no	—
108	C₄×D₉₇	—	2	2	no	—
109	C₄×D₉₇	—	2	2	no	—
110	C₄×D₉₇	#41	1	4	yes	—
111	C₄×D₉₇	#41	1	4	yes	—
112	C₂²×C₉₇	—	2	1	no	—
113	C₂²×C₉₇	—	2	1	no	—
114	C₂²×C₉₇	#117	1	2	yes	—
115	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₉₇	—	2	1	no	—
117	C₂×D₉₇	—	2	1	no	—
118	C₂×D₉₇	—	2	8	no	—
119	C₂×D₉₇	#117	1	2	yes	—
120	C₂×D₉₇	C₂×#41	1	2	yes	—
121	C₂×D₉₇	C₂×#41	1	2	yes	—
122	C₂×D₉₇	C₂×#41	1	2	yes	—
123	#87604	—	4	8	no	—
124	C₄×C₉₇	—	2	1	no	—
125	C₄×C₉₇	—	2	1	no	—
126	C₄×C₉₇	—	2	1	no	—
127	C₄×C₉₇	—	2	1	no	—
128	C₄×C₉₇	#117	1	2	yes	—
129	C₄×C₉₇	#117	1	2	yes	—
130	Dic₉₇	—	2	1	no	—
131	Dic₉₇	—	2	1	no	—
132	Dic₉₇	—	2	1	no	—
133	Dic₉₇	—	2	1	no	—
134	Dic₉₇	#117	1	2	yes	—
135	Dic₉₇	#117	1	2	yes	—
136	C₂×C₉₇	—	2	2	no	—
137	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
138	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
139	C₂×C₉₇	C₂×#117	1	1	yes	characteristic
140	D₉₇	—	2	2	no	—
141	D₉₇	#529	1	4	yes	—
142	#2283	—	97	1	no	maximal, Sylow
143	C₉₇	#2283	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₄²	—	97	1	no	—
145	#383	—	97	2	no	—
146	#382	—	97	2	no	—
147	#381	—	97	2	no	—
148	#380	—	97	2	no	—
149	C₄×C₄⋊C₄	—	97	2	no	—
150	C₄×C₂²⋊C₄	—	97	2	no	—
151	#376	—	97	2	no	—
152	C₂³×C₄	—	97	1	no	—
153	C₂³×C₄	—	97	1	no	—
154	C₂³×C₄	—	97	1	no	—
155	C₂×C₄⋊C₄	—	97	4	no	—
156	C₂×C₂²⋊C₄	—	97	4	no	—
157	C₂×C₄²	—	97	1	no	—
158	C₂×C₄²	—	97	1	no	—
159	C₂×C₄²	—	97	1	no	—
160	C₂×C₄²	—	97	1	no	—
161	C₂×C₄²	—	97	1	no	—
162	C₂×C₄²	—	97	1	no	—
163	C₂×C₄²	—	97	1	no	—
164	C₂×C₄²	—	97	1	no	—
165	C₂×C₄²	—	97	1	no	—
166	C₂×C₄²	—	97	1	no	—
167	C₂×C₄²	—	97	1	no	—
168	C₂×C₄²	—	97	4	no	—
169	C₂×C₄²	#87604	1	1	yes	characteristic, complemented
170	#95	—	97	4	no	—
171	#95	—	97	4	no	—
172	#95	—	97	4	no	—
173	#95	—	97	4	no	—
174	C₂⁴	—	97	1	no	—
175	C₂²×C₄	—	97	1	no	—
176	C₂²×C₄	—	97	1	no	—
177	C₂²×C₄	—	97	1	no	—
178	C₂²×C₄	—	97	1	no	—
179	C₂²×C₄	—	97	1	no	—
180	C₂²×C₄	—	97	1	no	—
181	C₂²×C₄	—	97	1	no	—
182	C₂²×C₄	—	97	1	no	—
183	C₂²×C₄	—	97	1	no	—
184	C₂²×C₄	—	97	1	no	—
185	C₂²×C₄	—	97	1	no	—
186	C₂²×C₄	—	97	1	no	—
187	C₂²×C₄	—	97	1	no	—
188	C₂²×C₄	—	97	1	no	—
189	C₂²×C₄	—	97	1	no	—
190	C₂²×C₄	—	97	8	no	—
191	C₂²×C₄	—	194	4	no	—
192	C₂²×C₄	—	194	4	no	—
193	C₂²×C₄	—	194	4	no	—
194	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
195	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
196	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
197	C₄⋊C₄	—	194	8	no	—
198	C₂²⋊C₄	—	194	8	no	—
199	C₄²	—	97	1	no	—
200	C₄²	—	97	1	no	—
201	C₄²	—	97	1	no	—
202	C₄²	—	97	1	no	—
203	C₄²	—	97	1	no	—
204	C₄²	—	97	1	no	—
205	C₄²	—	97	1	no	—
206	C₄²	—	97	1	no	—
207	C₄²	—	97	1	no	—
208	C₄²	—	97	1	no	—
209	C₄²	—	97	1	no	—
210	C₄²	—	97	1	no	—
211	C₄²	—	194	8	no	—
212	C₄²	C₂×#87604	1	1	yes	characteristic
213	C₄²	C₂×#87604	1	1	yes	characteristic
214	C₄²	C₂×#87604	1	1	yes	characteristic
215	C₄²	C₂×#87604	1	1	yes	characteristic
216	C₂³	—	97	1	no	—
217	C₂³	—	97	4	no	—
218	C₂³	—	97	1	no	—
219	C₂³	—	194	2	no	—
220	C₂³	—	194	2	no	—
221	C₂³	C₂²×#87604	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	97	1	no	—
226	C₂×C₄	—	97	1	no	—
227	C₂×C₄	—	97	4	no	—
228	C₂×C₄	—	97	4	no	—
229	C₂×C₄	—	97	1	no	—
230	C₂×C₄	—	97	1	no	—
231	C₂×C₄	—	97	1	no	—
232	C₂×C₄	—	97	1	no	—
233	C₂×C₄	—	194	2	no	—
234	C₂×C₄	—	194	2	no	—
235	C₂×C₄	—	194	2	no	—
236	C₂×C₄	—	194	2	no	—
237	C₂×C₄	—	194	2	no	—
238	C₂×C₄	—	194	2	no	—
239	C₂×C₄	—	194	2	no	—
240	C₂×C₄	—	194	2	no	—
241	C₂×C₄	—	194	2	no	—
242	C₂×C₄	—	194	2	no	—
243	C₂×C₄	—	194	8	no	—
244	C₂×C₄	—	194	8	no	—
245	C₂×C₄	—	194	8	no	—
246	C₂×C₄	—	194	2	no	—
247	C₂×C₄	—	194	2	no	—
248	C₂×C₄	—	194	2	no	—
249	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
250	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
251	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
252	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
253	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
254	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	97	2	no	—
258	C₂²	—	97	2	no	—
259	C₂²	—	97	2	no	—
260	C₂²	—	97	2	no	—
261	C₂²	—	194	8	no	—
262	C₂²	—	194	1	no	—
263	C₂²	—	194	1	no	—
264	C₂²	#15690889	1	2	yes	—
265	C₂²	C₂³×#87604	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	97	2	no	—
271	C₄	—	97	2	no	—
272	C₄	—	388	8	no	—
273	C₄	—	194	1	no	—
274	C₄	—	194	1	no	—
275	C₄	—	194	1	no	—
276	C₄	—	194	1	no	—
277	C₄	#15690889	1	2	yes	—
278	C₄	#15690889	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	97	4	no	—
281	C₂	—	194	2	no	—
282	C₂	C₂×#15690889	1	1	yes	characteristic
283	C₂	C₂×#15690889	1	1	yes	characteristic
284	C₂	C₂×#15690889	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂³×C₄
Frattini	C₂²	C₂³×#87604
Fitting	C₂×C₄²×C₉₇	C₄
prime radical	C₂³×D₉₇	C₂³
cosocle	C₂²×D₉₇	C₂⁴
hypercenter	C₂×C₄²	#87604
hypocenter	C₉₇	#2283
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #20261680	C₂	semidirect
2. #20261679	C₂	semidirect
5. C₄×C₄⋊#87604	C₂	semidirect
7. #20261671	C₂	semidirect
19. C₂×C₄×Dic₉₇	C₄	semidirect
20. C₂×C₄×Dic₉₇	C₄	semidirect
21. C₂×C₄×Dic₉₇	C₄	semidirect
22. C₄²×D₉₇	C₄	semidirect
23. C₄²×D₉₇	C₄	semidirect
24. C₄²×D₉₇	C₄	semidirect
25. C₄²×D₉₇	C₄	semidirect
30. C₂×C₄²×C₉₇	C₄	semidirect
143. C₉₇	#2283	semidirect
169. C₂×C₄²	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261680	2
#20261679	2
#20261678	2
#20261677	2
C₄×C₄⋊#87604	2
C₄×C₂²⋊#87604	2
#20261671	2
C₂×C₄²×D₉₇	2
#2283	97