Subgroups — #27143608

Order 12416.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₇₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₈₈	C₂	1	2	yes	maximal
3	#20260563	C₂	1	1	yes	maximal, characteristic
4	#20260540	C₂	1	2	yes	maximal
5	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₈	C₂²	1	2	yes	—
7	C₄×Dic₁₉₄	—	2	2	no	—
8	Dic₇₇₆	C₄	1	4	yes	complemented
9	#15690744	C₂²	1	2	yes	—
10	#15690735	—	2	2	no	—
11	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₉₄	—	2	2	no	—
15	Dic₃₈₈	—	2	2	no	—
16	Dic₃₈₈	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₉₇	—	2	2	no	—
18	#13095490	—	4	2	no	—
19	C₄×Dic₉₇	—	4	2	no	—
20	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
22	C₄²×C₉₇	D₄	1	1	yes	characteristic
23	C₁₆×C₉₇	—	2	1	no	—
24	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₉₇	D₈	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₉₇	—	4	2	no	—
28	C₂×Dic₉₇	—	4	2	no	—
29	Dic₁₉₄	—	2	4	no	—
30	Dic₁₉₄	—	4	2	no	—
31	C₈×C₉₇	—	2	1	no	—
32	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₉₇	#41	1	1	yes	characteristic
34	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₉₇	—	2	2	no	—
36	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₉₇	#135	1	1	yes	characteristic
38	Dic₉₇	—	4	4	no	—
39	Dic₉₇	—	8	2	no	—
40	C₂×C₉₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₉₇	#510	1	2	yes	—
42	#2176	—	97	1	no	maximal, Sylow
43	C₉₇	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	97	1	no	—
45	C₄×Q₁₆	—	97	2	no	—
46	#367	—	97	1	no	—
47	#358	—	97	2	no	—
48	#346	D₉₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	97	2	no	—
50	C₄×Q₈	—	194	2	no	—
51	Q₃₂	—	97	4	no	—
52	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
54	#107	—	97	2	no	—
55	#103	—	194	2	no	—
56	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
57	C₂×Q₈	—	194	2	no	—
58	Q₁₆	—	97	4	no	—
59	Q₁₆	—	194	2	no	—
60	C₂×C₈	D₃₈₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
62	C₄⋊C₄	—	388	2	no	—
63	C₄⋊C₄	—	194	2	no	—
64	C₄²	—	388	2	no	—
65	C₄²	D₃₈₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₉₇	1	2	yes	—
68	Q₈	—	388	2	no	—
69	Q₈	—	194	4	no	—
70	C₂×C₄	—	388	2	no	—
71	C₂×C₄	—	388	2	no	—
72	C₂×C₄	D₇₇₆	1	2	yes	—
73	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
76	C₈	#13095511	1	1	yes	characteristic
77	C₂²	C₂×D₇₇₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	776	2	no	—
80	C₄	—	388	4	no	—
81	C₄	C₄×D₃₈₈	1	1	yes	characteristic
82	C₄	#15690832	1	1	yes	characteristic
83	C₂	C₄×D₇₇₆	1	1	yes	characteristic
84	C₂	#20261178	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₇₆
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#346	D₉₇
hypocenter	C₉₇	#2176
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₇₇₆	C₄	semidirect
43. C₉₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₇₆	2
C₄×Dic₃₈₈	2
#20260563	2
#20260540	2
#346×C₉₇	2
#2176	97