Subgroups — #27150784

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260813	C₂	1	2	yes	maximal, complemented
2	#20260807	C₂	1	2	yes	maximal, complemented
3	#20260807	C₂	1	2	yes	maximal, complemented
4	D₄×#11715895	C₂	1	2	yes	maximal, complemented
5	#20260796	C₂	1	2	yes	maximal, complemented
6	#20260702	C₂	1	2	yes	maximal, complemented
7	#20260699	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊#11715895	C₂	1	1	yes	maximal, characteristic, complemented
9	#525×C₉₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#13095495	C₂²	1	1	yes	characteristic
11	C₂×#13095495	C₂²	1	1	yes	characteristic
12	C₂×#13095495	C₂²	1	2	yes	complemented
13	C₂²×#11715895	C₂²	1	1	yes	characteristic
14	C₂²×#11715895	C₂²	1	1	yes	characteristic
15	C₂²×#11715895	C₂²	1	2	yes	complemented
16	C₄⋊#11715895	C₂²	1	2	yes	complemented
17	C₄⋊#11715895	C₂²	1	2	yes	complemented
18	C₂²⋊#11715895	C₂²	1	2	yes	—
19	C₂²⋊#11715895	C₂²	1	2	yes	—
20	C₂²⋊#11715895	C₂²	1	2	yes	—
21	C₂²⋊#11715895	C₂²	1	2	yes	—
22	C₂²⋊#11715895	C₂²	1	2	yes	complemented
23	C₂²⋊#11715895	C₂²	1	2	yes	complemented
24	#15690721	C₂²	1	2	yes	complemented
25	C₄×#11715895	C₂²	1	2	yes	complemented
26	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
27	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
28	#123×C₉₇	C₄	1	2	yes	—
29	C₂²⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
30	C₂²⋊Q₈×C₉₇	C₄	1	1	yes	characteristic
31	#121×C₉₇	C₄	1	1	yes	characteristic
32	#121×C₉₇	C₄	1	1	yes	characteristic
33	#120×C₉₇	C₄	1	2	yes	—
34	C₄×D₄×C₉₇	C₂²	1	2	yes	—
35	C₄×D₄×C₉₇	C₂²	1	2	yes	—
36	#117×C₉₇	C₂²	1	1	yes	characteristic
37	#13095495	—	4	1	no	—
38	#13095495	—	4	1	no	—
39	#13095495	—	4	2	no	—
40	C₂×#11715895	—	4	1	no	—
41	C₂×#11715895	—	4	1	no	—
42	C₂×#11715895	—	4	2	no	—
43	C₂×#11715895	C₂³	1	2	yes	—
44	C₂×#11715895	C₂³	1	2	yes	—
45	C₂×#11715895	C₂³	1	2	yes	—
46	C₂×#11715895	C₂³	1	2	yes	—
47	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
48	#41×C₉₇	—	4	1	no	—
49	#41×C₉₇	—	4	1	no	—
50	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
52	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
53	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
54	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₉₇	—	4	1	no	—
56	C₂²×C₄×C₉₇	—	4	1	no	—
57	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
69	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
70	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
71	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
72	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
73	C₄²×C₉₇	C₂³	1	2	yes	—
74	C₂³×C₉₇	—	2	2	no	—
75	C₂³×C₉₇	—	4	1	no	—
76	C₂³×C₉₇	—	4	1	no	—
77	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
80	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₉₇	—	4	1	no	—
82	D₄×C₉₇	—	2	4	no	—
83	D₄×C₉₇	—	4	2	no	—
84	D₄×C₉₇	—	4	1	no	—
85	D₄×C₉₇	—	4	2	no	—
86	C₂×C₄×C₉₇	—	2	1	no	—
87	C₂×C₄×C₉₇	—	2	1	no	—
88	C₂×C₄×C₉₇	—	2	1	no	—
89	C₂×C₄×C₉₇	—	2	1	no	—
90	C₂×C₄×C₉₇	—	2	1	no	—
91	C₂×C₄×C₉₇	—	2	1	no	—
92	C₂×C₄×C₉₇	—	2	2	no	—
93	C₂×C₄×C₉₇	—	2	2	no	—
94	C₂×C₄×C₉₇	—	4	1	no	—
95	C₂×C₄×C₉₇	—	4	1	no	—
96	C₂×C₄×C₉₇	—	4	1	no	—
97	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
104	#11715895	—	2	2	no	—
105	#11715895	—	2	2	no	—
106	#11715895	—	2	2	no	—
107	#11715895	—	2	2	no	—
108	C₂²×C₉₇	—	2	1	no	—
109	C₂²×C₉₇	—	2	1	no	—
110	C₂²×C₉₇	—	2	1	no	—
111	C₂²×C₉₇	—	2	1	no	—
112	C₂²×C₉₇	—	2	2	no	—
113	C₂²×C₉₇	—	2	2	no	—
114	C₂²×C₉₇	—	2	1	no	—
115	C₂²×C₉₇	—	2	1	no	—
116	C₂²×C₉₇	—	2	1	no	—
117	C₂²×C₉₇	—	2	2	no	—
118	C₂²×C₉₇	—	2	1	no	—
119	C₂²×C₉₇	—	4	4	no	—
120	C₂²×C₉₇	#131	1	2	yes	—
121	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₉₇	—	2	1	no	—
123	C₄×C₉₇	—	2	1	no	—
124	C₄×C₉₇	—	2	1	no	—
125	C₄×C₉₇	—	2	2	no	—
126	C₄×C₉₇	—	2	2	no	—
127	C₄×C₉₇	—	2	2	no	—
128	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
129	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂×C₉₇	—	2	2	no	—
131	C₂×C₉₇	—	4	2	no	—
132	C₂×C₉₇	—	4	1	no	—
133	C₂×C₉₇	—	4	1	no	—
134	C₂×C₉₇	#518	1	1	yes	characteristic
135	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
136	C₂×C₉₇	#568	1	1	yes	characteristic
137	#2976	—	97	1	no	maximal, Sylow
138	C₉₇	#2976	1	1	yes	characteristic, complemented, Sylow
139	#525	D₉₇	1	1	yes	characteristic
140	#436	—	97	2	no	—
141	#435	—	97	2	no	—
142	#435	—	97	2	no	—
143	C₈×D₄	—	97	2	no	—
144	#433	—	97	2	no	—
145	#408	—	97	2	no	—
146	#407	—	97	1	no	—
147	C₂×C₂²⋊C₈	—	97	1	no	—
148	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
149	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
150	C₂×M₁₆	—	97	1	no	—
151	C₂×M₁₆	—	97	1	no	—
152	C₂×M₁₆	—	97	2	no	—
153	C₂²×C₈	—	97	1	no	—
154	C₂²×C₈	—	97	1	no	—
155	C₂²×C₈	—	97	2	no	—
156	#123	Dic₉₇	1	2	yes	—
157	C₂²⋊Q₈	Dic₉₇	1	1	yes	characteristic
158	C₂²⋊Q₈	Dic₉₇	1	1	yes	characteristic
159	#121	Dic₉₇	1	1	yes	characteristic
160	#121	Dic₉₇	1	1	yes	characteristic
161	#120	Dic₉₇	1	2	yes	—
162	C₄×D₄	C₂×D₉₇	1	2	yes	—
163	C₄×D₄	C₂×D₉₇	1	2	yes	—
164	#117	C₂×D₉₇	1	1	yes	characteristic
165	C₄⋊C₈	—	97	2	no	—
166	C₄⋊C₈	—	97	2	no	—
167	C₂²⋊C₈	—	97	2	no	—
168	C₂²⋊C₈	—	97	2	no	—
169	C₂²⋊C₈	—	97	2	no	—
170	C₂²⋊C₈	—	97	2	no	—
171	C₂²⋊C₈	—	97	2	no	—
172	C₂²⋊C₈	—	97	2	no	—
173	#97	—	97	2	no	—
174	C₄×C₈	—	97	2	no	—
175	C₂⁴	C₂×Dic₉₇	1	1	yes	characteristic
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	C₂×Dic₉₇	1	1	yes	characteristic
179	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
180	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
181	C₂×D₄	C₂×Dic₉₇	1	2	yes	—
182	C₂×D₄	C₂×Dic₉₇	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
186	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
191	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
192	M₁₆	—	388	1	no	—
193	M₁₆	—	388	2	no	—
194	M₁₆	—	388	1	no	—
195	C₂×C₈	—	97	2	no	—
196	C₂×C₈	—	97	2	no	—
197	C₂×C₈	—	97	2	no	—
198	C₂×C₈	—	97	2	no	—
199	C₂×C₈	—	388	1	no	—
200	C₂×C₈	—	388	2	no	—
201	C₂×C₈	—	388	1	no	—
202	C₄⋊C₄	C₂×Dic₉₇	1	1	yes	characteristic
203	C₄⋊C₄	C₂×Dic₉₇	1	1	yes	characteristic
204	C₄⋊C₄	C₂×Dic₉₇	1	2	yes	—
205	C₄⋊C₄	C₂×Dic₉₇	1	2	yes	—
206	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
207	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
208	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
209	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
210	C₂²⋊C₄	C₂×Dic₉₇	1	2	yes	—
211	C₄²	C₂²×D₉₇	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
216	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
217	C₂³	C₂²×Dic₉₇	1	2	yes	—
218	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
219	Q₈	—	4	1	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	1	no	—
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
236	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
237	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
238	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×Dic₉₇	1	2	yes	—
240	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
242	C₈	—	194	2	no	—
243	C₈	—	194	2	no	—
244	C₈	—	194	2	no	—
245	C₈	—	194	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	#15690817	1	2	yes	—
259	C₂²	C₂³×Dic₉₇	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	#15690855	1	1	yes	characteristic
267	C₄	#15690855	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	2	no	—
272	C₂	#20261210	1	1	yes	characteristic
273	C₂	C₂×#15690817	1	1	yes	characteristic
274	C₂	#20261502	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting, prime radical	#525×C₉₇	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#525	D₉₇
hypocenter	C₉₇	#2976
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. #20260813	C₂	semidirect
2. #20260807	C₂	semidirect
3. #20260807	C₂	semidirect
4. D₄×#11715895	C₂	semidirect
5. #20260796	C₂	semidirect
6. #20260702	C₂	semidirect
7. #20260699	C₂	semidirect
8. C₂×C₂²⋊#11715895	C₂	semidirect
12. C₂×#13095495	C₂²	semidirect
15. C₂²×#11715895	C₂²	semidirect
16. C₄⋊#11715895	C₂²	semidirect
17. C₄⋊#11715895	C₂²	semidirect
22. C₂²⋊#11715895	C₂²	semidirect
23. C₂²⋊#11715895	C₂²	semidirect
24. #15690721	C₂²	semidirect
25. C₄×#11715895	C₂²	semidirect
138. C₉₇	#2976	semidirect

Maximal subgroups

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

subgroup	index
#20260813	2
#20260807	2
D₄×#11715895	2
#20260796	2
#20260702	2
#20260699	2
C₂×C₂²⋊#11715895	2
#525×C₉₇	2
#2976	97