Subgroups — #27150466

Order 12416.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690817	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261242	C₂	1	1	yes	maximal, characteristic
4	#20260810	C₂	1	2	yes	maximal, complemented
5	#20260809	C₂	1	2	yes	maximal, complemented
6	#20260803	C₂	1	2	yes	maximal, complemented
7	#20260798	C₂	1	2	yes	maximal, complemented
8	#20260763	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15690722	C₂	1	1	yes	maximal, characteristic, complemented
10	#20260690	C₂	1	1	yes	maximal, characteristic, complemented
11	#408×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₉₇	C₂²	1	1	yes	characteristic
13	#15690817	—	2	4	no	—
14	C₂×#13095495	C₂²	1	1	yes	characteristic
15	C₂×#13095495	C₂²	1	2	yes	complemented
16	C₂×#13095494	C₂²	1	2	yes	—
17	M₁₆×D₉₇	—	2	4	no	—
18	C₂²×#11715895	C₂²	1	1	yes	characteristic
19	C₂²×#11715895	C₂²	1	2	yes	complemented
20	C₂×C₈×D₉₇	C₂²	1	2	yes	—
21	#15690791	C₄	1	2	yes	—
22	#15690786	C₄	1	1	yes	characteristic
23	#15690786	C₄	1	1	yes	characteristic
24	#15690781	C₄	1	1	yes	characteristic
25	#15690781	C₄	1	1	yes	characteristic
26	#15690778	C₄	1	2	yes	—
27	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
29	#15690769	C₂²	1	1	yes	characteristic
30	#15690739	C₂²	1	2	yes	complemented
31	#15690739	C₂²	1	2	yes	complemented
32	#15690722	C₂²	1	1	yes	characteristic
33	#15690722	C₂²	1	1	yes	characteristic
34	#15690722	C₂²	1	1	yes	characteristic
35	#15690722	C₂²	1	1	yes	characteristic
36	#15690720	C₂²	1	2	yes	complemented
37	C₈×Dic₉₇	C₂²	1	2	yes	complemented
38	C₂×#41×C₉₇	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
40	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	complemented
42	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	complemented
43	C₂³×D₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
45	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₉₇	—	2	2	no	—
48	C₂×C₄×D₉₇	—	2	2	no	—
49	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
53	#13095495	—	2	4	no	—
54	#13095495	D₄	1	4	yes	complemented
55	#13095494	—	2	4	no	—
56	C₂×#11715895	—	2	2	no	—
57	C₂×#11715895	—	2	4	no	—
58	C₂×#11715895	C₂³	1	1	yes	characteristic
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₂×#11715895	C₂³	1	2	yes	—
61	C₈×D₉₇	—	2	4	no	—
62	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
63	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
64	#13095490	C₂×C₄	1	2	yes	—
65	#13095490	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
67	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
69	#13095488	C₂×C₄	1	2	yes	—
70	#13095488	C₂×C₄	1	2	yes	—
71	C₄×Dic₉₇	C₂³	1	2	yes	—
72	#41×C₉₇	—	2	2	no	—
73	#41×C₉₇	—	2	2	no	—
74	C₂×Q₈×C₉₇	C₂×C₄	1	1	yes	characteristic
75	C₂×D₄×C₉₇	C₂×C₄	1	2	yes	—
76	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
78	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
79	M₁₆×C₉₇	—	2	2	no	—
80	C₂×C₈×C₉₇	—	2	2	no	—
81	C₂×C₈×C₉₇	C₂³	1	2	yes	—
82	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
84	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
85	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₉₇	—	2	2	no	—
87	C₂²×D₉₇	—	2	2	no	—
88	C₂²×D₉₇	—	4	1	no	—
89	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₉₇	—	2	2	no	—
92	D₄×C₉₇	—	2	4	no	—
93	D₄×C₉₇	—	2	2	no	—
94	C₂×C₄×C₉₇	—	2	2	no	—
95	C₂×C₄×C₉₇	—	2	2	no	—
96	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₉₇	#41	1	2	yes	—
100	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₉₇	—	2	4	no	—
102	C₂²⋊D₉₇	—	4	2	no	—
103	C₂×Dic₉₇	—	2	2	no	—
104	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
105	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
106	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
108	C₄×D₉₇	—	2	2	no	—
109	C₄×D₉₇	—	2	2	no	—
110	C₄×D₉₇	—	2	1	no	—
111	C₄×D₉₇	—	2	1	no	—
112	C₄×D₉₇	—	2	1	no	—
113	C₄×D₉₇	—	2	1	no	—
114	C₈×C₉₇	—	2	2	no	—
115	C₈×C₉₇	—	2	1	no	—
116	C₈×C₉₇	—	2	1	no	—
117	#11715895	—	2	1	no	—
118	#11715895	—	2	1	no	—
119	#11715895	C₂×D₄	1	4	yes	—
120	C₂²×C₉₇	—	2	1	no	—
121	C₂²×C₉₇	—	2	1	no	—
122	C₂²×C₉₇	—	2	2	no	—
123	C₂²×C₉₇	—	2	2	no	—
124	C₂²×C₉₇	—	2	2	no	—
125	C₂²×C₉₇	C₄×D₄	1	2	yes	—
126	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×D₉₇	—	2	1	no	—
128	C₂×D₉₇	—	2	1	no	—
129	C₂×D₉₇	—	2	1	no	—
130	C₂×D₉₇	—	2	1	no	—
131	C₂×D₉₇	—	2	1	no	—
132	C₂×D₉₇	—	4	4	no	—
133	C₂×D₉₇	#131	1	2	yes	complemented
134	C₄×C₉₇	—	2	2	no	—
135	C₄×C₉₇	C₄×D₄	1	2	yes	—
136	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
138	Dic₉₇	—	2	1	no	—
139	Dic₉₇	—	2	1	no	—
140	Dic₉₇	—	2	2	no	—
141	Dic₉₇	—	2	2	no	—
142	C₂×C₉₇	—	2	2	no	—
143	C₂×C₉₇	—	4	2	no	—
144	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
146	C₂×C₉₇	#568	1	1	yes	characteristic
147	D₉₇	—	2	2	no	—
148	D₉₇	—	4	1	no	—
149	#2933	—	97	1	no	maximal, Sylow
150	C₉₇	#2933	1	1	yes	characteristic, complemented, Sylow
151	C₂×#131	—	97	1	no	—
152	C₂²×M₁₆	—	97	1	no	—
153	#525	—	97	1	no	—
154	#436	—	97	2	no	—
155	#435	—	97	2	no	—
156	#435	—	97	2	no	—
157	C₈×D₄	—	97	2	no	—
158	#424	—	97	1	no	—
159	#408	D₉₇	1	1	yes	characteristic, complemented
160	C₂×C₂²⋊C₈	—	97	1	no	—
161	#405	—	97	1	no	—
162	C₂×#41	C₂×D₉₇	1	1	yes	characteristic
163	C₂³×C₄	—	97	1	no	—
164	#131	—	194	4	no	—
165	C₂×M₁₆	—	97	1	no	—
166	C₂×M₁₆	—	97	2	no	—
167	C₂×M₁₆	—	97	2	no	—
168	C₂×M₁₆	—	194	4	no	—
169	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
170	C₂²×C₈	—	97	1	no	—
171	C₂²×C₈	—	97	2	no	—
172	C₂²×C₈	—	97	2	no	—
173	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
174	#123	—	97	2	no	—
175	C₂²⋊Q₈	—	97	1	no	—
176	C₂²⋊Q₈	—	97	1	no	—
177	#121	—	97	1	no	—
178	#121	—	97	1	no	—
179	#120	—	97	2	no	—
180	C₄×D₄	—	97	2	no	—
181	C₄×D₄	—	97	2	no	—
182	#117	—	97	1	no	—
183	C₄⋊C₈	—	97	2	no	—
184	C₄⋊C₈	—	97	2	no	—
185	C₂²⋊C₈	—	97	1	no	—
186	C₂²⋊C₈	—	97	1	no	—
187	C₂²⋊C₈	—	97	1	no	—
188	C₂²⋊C₈	—	97	1	no	—
189	C₂²⋊C₈	C₂×D₉₇	1	2	yes	complemented
190	C₂²⋊C₈	C₂×D₉₇	1	2	yes	complemented
191	#97	—	97	2	no	—
192	C₄×C₈	—	97	2	no	—
193	C₂⁴	—	97	1	no	—
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₄×D₉₇	1	1	yes	characteristic
197	C₂×D₄	—	97	2	no	—
198	C₂×D₄	—	97	2	no	—
199	C₂×D₄	C₄×D₉₇	1	2	yes	—
200	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
201	C₂²×C₄	—	97	1	no	—
202	C₂²×C₄	—	97	1	no	—
203	C₂²×C₄	—	97	1	no	—
204	C₂²×C₄	—	97	1	no	—
205	C₂²×C₄	—	97	1	no	—
206	C₂²×C₄	—	194	2	no	—
207	C₂²×C₄	—	194	2	no	—
208	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
209	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	97	4	no	—
212	M₁₆	—	194	4	no	—
213	M₁₆	—	194	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	97	2	no	—
216	C₂×C₈	—	97	1	no	—
217	C₂×C₈	—	97	1	no	—
218	C₂×C₈	—	194	4	no	—
219	C₂×C₈	—	194	4	no	—
220	C₂×C₈	—	194	2	no	—
221	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₉₇	1	2	yes	—
224	C₄⋊C₄	—	97	2	no	—
225	C₄⋊C₄	—	97	2	no	—
226	C₄⋊C₄	—	97	1	no	—
227	C₄⋊C₄	—	97	1	no	—
228	C₂²⋊C₄	—	97	2	no	—
229	C₂²⋊C₄	—	97	2	no	—
230	C₂²⋊C₄	—	97	2	no	—
231	C₂²⋊C₄	—	97	2	no	—
232	C₂²⋊C₄	—	97	2	no	—
233	C₄²	—	97	2	no	—
234	C₂³	—	97	1	no	—
235	C₂³	—	97	1	no	—
236	C₂³	—	388	1	no	—
237	C₂³	—	194	2	no	—
238	C₂³	—	194	2	no	—
239	C₂³	C₂×C₄×D₉₇	1	2	yes	—
240	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	388	2	no	—
245	D₄	—	194	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	97	2	no	—
249	C₂×C₄	—	97	2	no	—
250	C₂×C₄	—	97	1	no	—
251	C₂×C₄	—	97	1	no	—
252	C₂×C₄	—	194	2	no	—
253	C₂×C₄	—	194	2	no	—
254	C₂×C₄	—	194	2	no	—
255	C₂×C₄	—	194	1	no	—
256	C₂×C₄	—	194	1	no	—
257	C₂×C₄	—	194	1	no	—
258	C₂×C₄	—	194	1	no	—
259	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
261	C₂×C₄	D₄×D₉₇	1	2	yes	—
262	C₂×C₄	#13095510	1	2	yes	—
263	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	97	4	no	—
268	C₈	—	194	1	no	—
269	C₈	—	194	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	97	2	no	—
276	C₂²	—	388	4	no	—
277	C₂²	—	194	1	no	—
278	C₂²	—	194	1	no	—
279	C₂²	—	194	1	no	—
280	C₂²	—	194	1	no	—
281	C₂²	—	194	1	no	—
282	C₂²	#15690770	1	2	yes	—
283	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	194	1	no	—
286	C₄	—	194	1	no	—
287	C₄	—	194	2	no	—
288	C₄	—	194	2	no	—
289	C₄	#15690770	1	2	yes	—
290	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
291	C₄	C₂×#13095510	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	4	2	no	—
294	C₂	—	388	1	no	—
295	C₂	—	194	2	no	—
296	C₂	C₂×#15690770	1	1	yes	characteristic
297	C₂	#131×D₉₇	1	1	yes	characteristic
298	C₂	#20261500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#408×C₉₇	C₂
prime radical	#20261242	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#408	D₉₇
hypocenter	C₉₇	#2933
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
1. C₂×#15690817	C₂	semidirect
2. C₂×M₁₆×D₉₇	C₂	semidirect
4. #20260810	C₂	semidirect
5. #20260809	C₂	semidirect
6. #20260803	C₂	semidirect
7. #20260798	C₂	semidirect
8. #20260763	C₂	semidirect
9. C₂×#15690722	C₂	semidirect
10. #20260690	C₂	semidirect
11. #408×C₉₇	C₂	semidirect
15. C₂×#13095495	C₂²	semidirect
19. C₂²×#11715895	C₂²	semidirect
30. #15690739	C₂²	semidirect
31. #15690739	C₂²	semidirect
36. #15690720	C₂²	semidirect
37. C₈×Dic₉₇	C₂²	semidirect
41. C₂²⋊C₈×C₉₇	C₂²	semidirect
42. C₂²⋊C₈×C₉₇	C₂²	semidirect
54. #13095495	D₄	semidirect
133. C₂×D₉₇	#131	semidirect
150. C₉₇	#2933	semidirect
159. #408	D₉₇	semidirect
189. C₂²⋊C₈	C₂×D₉₇	semidirect
190. C₂²⋊C₈	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690817	2
C₂×M₁₆×D₉₇	2
#20261242	2
#20260810	2
#20260809	2
#20260803	2
#20260798	2
#20260763	2
C₂×#15690722	2
#20260690	2
#408×C₉₇	2
#2933	97