Subgroups — #27150799

Order 12416.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261282	C₂	1	1	yes	maximal, characteristic
2	#20260814	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260813	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260810	C₂	1	2	yes	maximal, complemented
5	#20260800	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₉₇	C₂	1	2	yes	maximal, complemented
7	#20260787	C₂	1	1	yes	maximal, characteristic, complemented
8	#20260786	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260701	C₂	1	2	yes	maximal, complemented
10	#20260700	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	2	yes	—
13	C₂×#13095495	C₂²	1	2	yes	—
14	C₂×#13095494	C₂²	1	2	yes	—
15	C₂×C₈×D₉₇	C₂²	1	2	yes	—
16	#15690809	C₄	1	1	yes	characteristic
17	#15690806	C₄	1	1	yes	characteristic
18	#15690796	C₄	1	2	yes	—
19	#15690785	C₄	1	1	yes	characteristic
20	#15690783	C₄	1	2	yes	—
21	#15690781	C₄	1	1	yes	characteristic
22	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
24	C₄²×D₉₇	C₂²	1	1	yes	characteristic
25	C₄⋊#11715895	C₂²	1	1	yes	characteristic
26	#15690739	C₂²	1	1	yes	characteristic
27	#15690739	C₂²	1	1	yes	characteristic
28	C₂²⋊#11715895	C₂²	1	2	yes	—
29	#15690722	C₂²	1	1	yes	characteristic
30	#15690722	C₂²	1	2	yes	—
31	#15690722	C₂²	1	1	yes	characteristic
32	#15690722	C₂²	1	2	yes	—
33	#15690722	C₂²	1	2	yes	—
34	C₄×#11715895	C₂²	1	1	yes	characteristic
35	C₈×Dic₉₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
37	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
40	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
41	#13095511	—	4	2	no	—
42	#13095511	—	4	2	no	—
43	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
44	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₃₈₈	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
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	2	yes	—
53	#13095495	—	4	2	no	—
54	#13095494	—	4	2	no	—
55	C₂×#11715895	C₂³	1	1	yes	characteristic
56	C₂×#11715895	C₂³	1	2	yes	—
57	C₂×#11715895	C₂³	1	1	yes	characteristic
58	C₈×D₉₇	—	4	2	no	—
59	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
60	#13095490	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
62	#13095488	C₂×C₄	1	2	yes	—
63	#13095488	C₂×C₄	1	2	yes	—
64	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
69	C₂×C₈×C₉₇	—	4	2	no	—
70	C₂×C₈×C₉₇	C₂³	1	2	yes	—
71	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
75	C₄²×C₉₇	C₂³	1	1	yes	characteristic
76	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
78	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
79	D₄×C₉₇	—	2	2	no	—
80	C₂×C₄×C₉₇	—	2	2	no	—
81	C₂×C₄×C₉₇	—	2	2	no	—
82	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₉₇	—	2	4	no	—
87	C₂²⋊D₉₇	—	4	2	no	—
88	C₂²⋊D₉₇	—	4	2	no	—
89	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
92	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
93	D₃₈₈	—	2	2	no	—
94	D₃₈₈	—	4	2	no	—
95	C₄×D₉₇	—	2	2	no	—
96	C₄×D₉₇	—	2	2	no	—
97	C₄×D₉₇	—	2	2	no	—
98	C₄×D₉₇	—	2	1	no	—
99	C₄×D₉₇	—	2	1	no	—
100	C₄×D₉₇	—	2	2	no	—
101	Dic₁₉₄	—	4	2	no	—
102	C₈×C₉₇	—	2	1	no	—
103	C₈×C₉₇	—	2	2	no	—
104	C₈×C₉₇	—	2	1	no	—
105	#11715895	—	2	1	no	—
106	#11715895	—	2	2	no	—
107	#11715895	—	2	1	no	—
108	C₂²×C₉₇	—	2	2	no	—
109	C₂²×C₉₇	—	2	2	no	—
110	C₂²×C₉₇	—	2	2	no	—
111	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₉₇	—	2	1	no	—
113	C₂×D₉₇	—	2	2	no	—
114	C₂×D₉₇	—	2	1	no	—
115	C₂×D₉₇	—	2	1	no	—
116	C₂×D₉₇	—	2	2	no	—
117	C₂×D₉₇	—	2	2	no	—
118	C₄×C₉₇	—	2	2	no	—
119	C₄×C₉₇	—	2	1	no	—
120	C₄×C₉₇	#131	1	2	yes	—
121	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₉₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₉₇	—	2	1	no	—
124	Dic₉₇	—	2	2	no	—
125	Dic₉₇	—	2	1	no	—
126	Dic₉₇	#131	1	2	yes	—
127	C₂×C₉₇	—	4	2	no	—
128	C₂×C₉₇	#518	1	1	yes	characteristic
129	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
131	D₉₇	—	4	1	no	—
132	D₉₇	—	4	2	no	—
133	#2979	—	97	1	no	maximal, Sylow
134	C₉₇	#2979	1	1	yes	characteristic, complemented, Sylow
135	#532	—	97	1	no	—
136	#436	—	97	1	no	—
137	#436	—	97	2	no	—
138	#436	—	97	1	no	—
139	C₈×D₄	—	97	2	no	—
140	C₈×D₄	—	97	1	no	—
141	C₈×D₄	D₉₇	1	1	yes	characteristic, complemented
142	#431	—	97	1	no	—
143	#431	—	97	1	no	—
144	#408	—	97	2	no	—
145	#408	—	97	2	no	—
146	C₂×#41	—	97	2	no	—
147	C₂×M₁₆	—	97	2	no	—
148	C₂×M₁₆	—	97	2	no	—
149	C₂²×C₈	—	97	2	no	—
150	C₂²×C₈	C₂×D₉₇	1	2	yes	—
151	C₄⋊Q₈	—	97	1	no	—
152	#127	—	97	1	no	—
153	#124	—	97	2	no	—
154	#121	—	97	1	no	—
155	#121	—	97	1	no	—
156	#121	—	97	2	no	—
157	C₄×D₄	—	97	2	no	—
158	C₄×D₄	—	97	1	no	—
159	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic
160	C₂×C₄²	—	97	1	no	—
161	C₄⋊C₈	—	97	1	no	—
162	C₄⋊C₈	—	97	1	no	—
163	C₄⋊C₈	—	97	1	no	—
164	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
165	C₂²⋊C₈	—	97	2	no	—
166	C₂²⋊C₈	—	97	1	no	—
167	C₂²⋊C₈	—	97	2	no	—
168	C₂²⋊C₈	—	97	1	no	—
169	C₂²⋊C₈	—	97	2	no	—
170	C₂²⋊C₈	—	97	2	no	—
171	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
172	C₄×C₈	—	97	2	no	—
173	C₄×C₈	—	97	1	no	—
174	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
175	#41	—	388	2	no	—
176	#41	—	388	2	no	—
177	C₂×Q₈	—	97	2	no	—
178	C₂×D₄	—	97	2	no	—
179	C₂×D₄	—	97	2	no	—
180	C₂×D₄	—	97	2	no	—
181	C₂×D₄	—	97	2	no	—
182	C₂×D₄	—	97	1	no	—
183	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
184	C₂²×C₄	—	97	1	no	—
185	C₂²×C₄	—	97	1	no	—
186	C₂²×C₄	—	97	1	no	—
187	C₂²×C₄	—	97	2	no	—
188	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
189	M₁₆	—	388	2	no	—
190	M₁₆	—	388	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	97	1	no	—
193	C₂×C₈	—	97	2	no	—
194	C₂×C₈	—	97	1	no	—
195	C₂×C₈	—	388	2	no	—
196	C₂×C₈	C₂²×D₉₇	1	2	yes	—
197	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
199	C₄⋊C₄	—	97	2	no	—
200	C₄⋊C₄	—	97	1	no	—
201	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	97	2	no	—
203	C₂²⋊C₄	—	97	2	no	—
204	C₂²⋊C₄	—	97	2	no	—
205	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
206	C₄²	—	97	1	no	—
207	C₄²	—	97	1	no	—
208	C₄²	—	97	1	no	—
209	C₄²	C₂²×D₉₇	1	1	yes	characteristic
210	C₂³	—	97	2	no	—
211	C₂³	—	97	1	no	—
212	C₂³	C₂×C₄×D₉₇	1	2	yes	—
213	Q₈	—	388	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	388	2	no	—
216	D₄	—	388	2	no	—
217	D₄	—	388	2	no	—
218	D₄	—	194	4	no	—
219	D₄	—	194	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	97	1	no	—
223	C₂×C₄	—	97	1	no	—
224	C₂×C₄	—	97	1	no	—
225	C₂×C₄	—	97	2	no	—
226	C₂×C₄	—	194	2	no	—
227	C₂×C₄	—	194	2	no	—
228	C₂×C₄	—	194	2	no	—
229	C₂×C₄	—	194	1	no	—
230	C₂×C₄	—	194	2	no	—
231	C₂×C₄	—	194	1	no	—
232	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	194	1	no	—
240	C₈	—	194	2	no	—
241	C₈	—	194	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	194	1	no	—
246	C₂²	—	194	2	no	—
247	C₂²	—	194	1	no	—
248	C₂²	—	194	2	no	—
249	C₂²	—	194	1	no	—
250	C₂²	—	194	2	no	—
251	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	97	2	no	—
255	C₄	—	194	1	no	—
256	C₄	—	194	2	no	—
257	C₄	—	194	1	no	—
258	C₄	#15690816	1	2	yes	—
259	C₄	#15690855	1	1	yes	characteristic
260	C₄	#15690856	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	388	1	no	—
263	C₂	—	388	2	no	—
264	C₂	#20261207	1	1	yes	characteristic
265	C₂	#131×D₉₇	1	1	yes	characteristic
266	C₂	C₂×#15690816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	C₈×D₄×C₉₇	C₂
prime radical	#20261282	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₈×D₄	D₉₇
hypocenter	C₉₇	#2979
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. #20260814	C₂	semidirect
3. #20260813	C₂	semidirect
4. #20260810	C₂	semidirect
5. #20260800	C₂	semidirect
6. C₈×C₂²⋊D₉₇	C₂	semidirect
7. #20260787	C₂	semidirect
8. #20260786	C₂	semidirect
9. #20260701	C₂	semidirect
10. #20260700	C₂	semidirect
11. C₈×D₄×C₉₇	C₂	semidirect
134. C₉₇	#2979	semidirect
141. C₈×D₄	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261282	2
#20260814	2
#20260813	2
#20260810	2
#20260800	2
C₈×C₂²⋊D₉₇	2
#20260787	2
#20260786	2
#20260701	2
#20260700	2
C₈×D₄×C₉₇	2
#2979	97