Subgroups — #27150481

Order 12416.

492 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	C₂×#15690818	C₂	1	2	yes	maximal, complemented
2	#20261282	C₂	1	1	yes	maximal, characteristic
3	#20260812	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260804	C₂	1	2	yes	maximal, complemented
5	#20260803	C₂	1	2	yes	maximal, complemented
6	C₈×D₃₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₈×D₉₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#20260700	C₂	1	2	yes	maximal, complemented
10	C₄×#13095494	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13095511	C₂²	1	2	yes	—
13	#15690818	—	2	8	no	—
14	C₂×#13095494	C₂²	1	1	yes	characteristic
15	C₂×#13095494	C₂²	1	2	yes	—
16	C₂×#13095494	C₂²	1	1	yes	characteristic
17	C₂²×#11715895	C₂²	1	2	yes	—
18	C₂×C₈×D₉₇	C₂²	1	2	yes	—
19	C₂×C₈×D₉₇	C₂²	1	2	yes	—
20	#15690809	C₄	1	1	yes	characteristic
21	#15690806	C₄	1	1	yes	characteristic
22	#15690796	C₄	1	2	yes	—
23	#15690785	C₄	1	1	yes	characteristic
24	#15690783	C₄	1	2	yes	—
25	#15690781	C₄	1	1	yes	characteristic
26	C₄×D₃₈₈	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₉₇	C₂²	1	2	yes	—
28	C₄²×D₉₇	C₂²	1	1	yes	characteristic
29	C₄⋊#11715895	C₂²	1	1	yes	characteristic
30	#15690739	C₂²	1	2	yes	—
31	C₂²⋊#11715895	C₂²	1	2	yes	—
32	#15690722	C₂²	1	2	yes	—
33	#15690722	C₂²	1	2	yes	—
34	#15690720	C₂²	1	1	yes	characteristic
35	#15690720	C₂²	1	1	yes	characteristic
36	C₄×#11715895	C₂²	1	1	yes	characteristic
37	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
38	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₉₇	C₂²	1	2	yes	—
41	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
42	#13095511	—	2	4	no	—
43	#13095511	—	2	4	no	—
44	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
45	C₂×D₃₈₈	C₂×C₄	1	1	yes	characteristic
46	C₂×D₃₈₈	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×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₉₇	C₂³	1	2	yes	—
53	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
54	#13095494	—	2	2	no	—
55	#13095494	—	2	4	no	—
56	#13095494	D₄	1	4	yes	complemented
57	C₂×#11715895	—	2	4	no	—
58	C₂×#11715895	C₂³	1	2	yes	—
59	C₂×#11715895	C₂³	1	1	yes	characteristic
60	C₂×#11715895	C₂³	1	1	yes	characteristic
61	C₈×D₉₇	—	2	4	no	—
62	C₈×D₉₇	—	2	4	no	—
63	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
64	#13095490	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₉₇	C₂×C₄	1	2	yes	—
66	#13095488	C₂×C₄	1	2	yes	—
67	#13095488	C₂×C₄	1	2	yes	—
68	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
73	M₁₆×C₉₇	—	2	4	no	—
74	C₂×C₈×C₉₇	C₂³	1	2	yes	—
75	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₉₇	C₂×C₄	1	2	yes	—
79	C₄²×C₉₇	C₂³	1	1	yes	characteristic
80	C₂³×C₉₇	C₂²×C₄	1	2	yes	—
81	C₂²×D₉₇	C₂²×C₄	1	2	yes	—
82	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
83	D₄×C₉₇	—	2	2	no	—
84	C₂×C₄×C₉₇	—	2	2	no	—
85	C₂×C₄×C₉₇	—	2	2	no	—
86	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₉₇	—	2	4	no	—
91	C₂²⋊D₉₇	—	2	4	no	—
92	C₂²⋊D₉₇	—	4	2	no	—
93	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
96	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
97	D₃₈₈	—	2	4	no	—
98	D₃₈₈	—	2	2	no	—
99	C₄×D₉₇	—	2	2	no	—
100	C₄×D₉₇	—	2	2	no	—
101	C₄×D₉₇	—	2	2	no	—
102	C₄×D₉₇	—	4	1	no	—
103	C₄×D₉₇	C₂×D₄	1	2	yes	—
104	C₄×D₉₇	#41	1	2	yes	—
105	Dic₁₉₄	—	2	4	no	—
106	C₈×C₉₇	—	2	2	no	—
107	C₈×C₉₇	—	2	1	no	—
108	C₈×C₉₇	C₂×D₄	1	2	yes	—
109	#11715895	—	2	2	no	—
110	#11715895	—	2	1	no	—
111	#11715895	C₂×D₄	1	2	yes	—
112	C₂²×C₉₇	—	2	2	no	—
113	C₂²×C₉₇	—	2	2	no	—
114	C₂²×C₉₇	—	2	2	no	—
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	2	no	—
119	C₂×D₉₇	—	2	2	no	—
120	C₂×D₉₇	—	2	2	no	—
121	C₂×D₉₇	C₄×D₄	1	2	yes	—
122	C₄×C₉₇	—	2	2	no	—
123	C₄×C₉₇	—	2	1	no	—
124	C₄×C₉₇	#131	1	2	yes	complemented
125	C₄×C₉₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₉₇	C₂×#41	1	1	yes	characteristic
127	Dic₉₇	—	2	2	no	—
128	Dic₉₇	—	2	1	no	—
129	Dic₉₇	—	2	1	no	—
130	Dic₉₇	C₄×D₄	1	2	yes	—
131	C₂×C₉₇	—	4	2	no	—
132	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₉₇	C₂×#131	1	1	yes	characteristic
134	C₂×C₉₇	#568	1	1	yes	characteristic
135	D₉₇	—	2	2	no	—
136	D₉₇	—	4	2	no	—
137	#2935	—	97	1	no	maximal, Sylow
138	C₉₇	#2935	1	1	yes	characteristic, complemented, Sylow
139	C₂×#131	—	97	2	no	—
140	#532	—	97	1	no	—
141	#436	—	97	1	no	—
142	#436	D₉₇	1	1	yes	characteristic, complemented
143	#435	—	97	2	no	—
144	#435	—	97	2	no	—
145	C₈×D₄	—	97	1	no	—
146	C₈×D₄	—	97	1	no	—
147	C₂×C₄⋊C₈	—	97	1	no	—
148	#408	—	97	2	no	—
149	C₄×M₁₆	—	97	1	no	—
150	C₂×#41	—	97	2	no	—
151	#131	—	194	8	no	—
152	C₂×M₁₆	—	97	1	no	—
153	C₂×M₁₆	—	97	2	no	—
154	C₂×M₁₆	—	97	1	no	—
155	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
156	C₂²×C₈	—	97	2	no	—
157	C₂²×C₈	—	97	2	no	—
158	C₂²×C₈	—	97	2	no	—
159	C₄⋊Q₈	—	97	1	no	—
160	#127	—	97	1	no	—
161	#124	—	97	2	no	—
162	#121	—	97	1	no	—
163	#121	—	97	1	no	—
164	#121	—	97	2	no	—
165	C₄×D₄	—	97	2	no	—
166	C₄×D₄	—	97	1	no	—
167	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic
168	C₂×C₄²	—	97	1	no	—
169	C₄⋊C₈	—	97	2	no	—
170	C₄⋊C₈	—	97	1	no	—
171	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
172	C₂²⋊C₈	—	97	2	no	—
173	C₂²⋊C₈	—	97	2	no	—
174	C₂²⋊C₈	—	97	2	no	—
175	C₂²⋊C₈	C₂×D₉₇	1	2	yes	—
176	#97	—	97	1	no	—
177	#97	—	97	1	no	—
178	C₄×C₈	—	97	1	no	—
179	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
180	#41	—	194	4	no	—
181	#41	—	194	4	no	—
182	C₂×Q₈	—	97	2	no	—
183	C₂×D₄	—	97	2	no	—
184	C₂×D₄	—	97	2	no	—
185	C₂×D₄	—	97	2	no	—
186	C₂×D₄	—	97	1	no	—
187	C₂×D₄	—	97	2	no	—
188	C₂×D₄	C₄×D₉₇	1	1	yes	characteristic
189	C₂²×C₄	—	97	1	no	—
190	C₂²×C₄	—	97	1	no	—
191	C₂²×C₄	—	97	1	no	—
192	C₂²×C₄	—	97	2	no	—
193	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	97	4	no	—
196	M₁₆	—	194	4	no	—
197	M₁₆	—	194	2	no	—
198	C₂×C₈	—	97	1	no	—
199	C₂×C₈	—	97	1	no	—
200	C₂×C₈	—	97	2	no	—
201	C₂×C₈	—	194	4	no	—
202	C₂×C₈	—	194	4	no	—
203	C₂×C₈	—	194	4	no	—
204	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₉₇	1	2	yes	—
206	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
207	C₄⋊C₄	—	97	2	no	—
208	C₄⋊C₄	—	97	1	no	—
209	C₄⋊C₄	C₄×D₉₇	1	1	yes	characteristic
210	C₂²⋊C₄	—	97	2	no	—
211	C₂²⋊C₄	—	97	2	no	—
212	C₂²⋊C₄	—	97	2	no	—
213	C₂²⋊C₄	C₄×D₉₇	1	2	yes	—
214	C₄²	—	97	1	no	—
215	C₄²	—	97	1	no	—
216	C₄²	—	97	1	no	—
217	C₄²	C₂²×D₉₇	1	1	yes	characteristic
218	C₂³	—	97	1	no	—
219	C₂³	—	97	2	no	—
220	C₂³	C₂×C₄×D₉₇	1	2	yes	—
221	Q₈	—	194	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	388	2	no	—
224	D₄	—	194	4	no	—
225	D₄	—	194	4	no	—
226	D₄	—	194	4	no	—
227	D₄	—	194	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	97	1	no	—
231	C₂×C₄	—	97	2	no	—
232	C₂×C₄	—	97	2	no	—
233	C₂×C₄	—	97	1	no	—
234	C₂×C₄	—	97	1	no	—
235	C₂×C₄	—	97	2	no	—
236	C₂×C₄	—	388	1	no	—
237	C₂×C₄	—	194	2	no	—
238	C₂×C₄	—	194	2	no	—
239	C₂×C₄	—	194	2	no	—
240	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	97	2	no	—
247	C₈	—	194	2	no	—
248	C₈	—	194	1	no	—
249	C₈	D₄×D₉₇	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	97	2	no	—
254	C₂²	—	194	2	no	—
255	C₂²	—	194	2	no	—
256	C₂²	—	194	2	no	—
257	C₂²	—	194	1	no	—
258	C₂²	—	194	1	no	—
259	C₂²	C₂²×C₄×D₉₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	97	2	no	—
263	C₄	—	194	2	no	—
264	C₄	—	194	1	no	—
265	C₄	—	194	1	no	—
266	C₄	#15690818	1	2	yes	complemented
267	C₄	C₂×D₄×D₉₇	1	1	yes	characteristic
268	C₄	#41×D₉₇	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	388	2	no	—
271	C₂	—	194	2	no	—
272	C₂	C₄×D₄×D₉₇	1	1	yes	characteristic
273	C₂	C₂×#15690818	1	1	yes	characteristic
274	C₂	#20261500	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₉₇
commutator, socle	C₂²×C₉₇	C₂³×C₄
Fitting	#436×C₉₇	C₂
prime radical	#20261282	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#436	D₉₇
hypocenter	C₉₇	#2935
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×#15690818	C₂	semidirect
3. #20260812	C₂	semidirect
4. #20260804	C₂	semidirect
5. #20260803	C₂	semidirect
6. C₈×D₃₈₈	C₂	semidirect
7. D₄×#11715895	C₂	semidirect
8. C₄⋊C₈×D₉₇	C₂	semidirect
9. #20260700	C₂	semidirect
10. C₄×#13095494	C₂	semidirect
11. #436×C₉₇	C₂	semidirect
56. #13095494	D₄	semidirect
124. C₄×C₉₇	#131	semidirect
138. C₉₇	#2935	semidirect
142. #436	D₉₇	semidirect
266. C₄	#15690818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690818	2
#20261282	2
#20260812	2
#20260804	2
#20260803	2
C₈×D₃₈₈	2
D₄×#11715895	2
C₄⋊C₈×D₉₇	2
#20260700	2
C₄×#13095494	2
#436×C₉₇	2
#2935	97