Subgroups — #27143606

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	#20260829	C₂	1	2	yes	maximal
3	#20260538	C₂	1	2	yes	maximal
4	#20260510	C₂	1	1	yes	maximal, characteristic
5	#367×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₈	C₂²	1	2	yes	—
7	#15690775	—	2	2	no	—
8	Dic₇₇₆	C₄	1	4	yes	complemented
9	C₂×#13095486	C₂²	1	1	yes	characteristic
10	#15690733	—	2	2	no	—
11	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	#107×C₉₇	C₂²	1	2	yes	—
14	C₂×Dic₁₉₄	—	2	2	no	—
15	Dic₃₈₈	—	2	2	no	—
16	Dic₃₈₈	C₂×C₄	1	4	yes	—
17	C₂×#11715895	D₄	1	1	yes	characteristic
18	#13095490	—	4	2	no	—
19	C₄×Dic₉₇	—	4	2	no	—
20	C₄×Dic₉₇	D₄	1	1	yes	characteristic
21	#13095486	—	2	1	no	—
22	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₉₇	—	4	2	no	—
26	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₉₇	—	4	2	no	—
28	C₂×Dic₉₇	D₈	1	2	yes	—
29	Dic₁₉₄	—	2	4	no	—
30	Dic₁₉₄	—	4	2	no	—
31	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₉₇	#41	1	1	yes	characteristic
33	#11715895	—	2	1	no	—
34	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₉₇	—	8	2	no	—
36	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₉₇	#135	1	1	yes	characteristic
38	Dic₉₇	—	2	2	no	—
39	Dic₉₇	—	4	4	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	D₉₇	1	1	yes	characteristic
47	#358	—	97	2	no	—
48	#346	—	97	1	no	—
49	C₂×Q₁₆	—	97	2	no	—
50	C₄×Q₈	—	194	2	no	—
51	Q₃₂	—	97	4	no	—
52	C₂×C₁₆	—	97	1	no	—
53	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
54	#107	C₂×D₉₇	1	2	yes	—
55	#103	—	194	2	no	—
56	C₄×C₈	—	97	1	no	—
57	C₂×Q₈	—	194	2	no	—
58	Q₁₆	—	97	4	no	—
59	Q₁₆	—	194	2	no	—
60	C₂×C₈	—	97	1	no	—
61	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	388	2	no	—
64	C₄²	—	97	1	no	—
65	C₄²	—	388	2	no	—
66	C₁₆	—	194	1	no	—
67	C₁₆	C₄×D₉₇	1	2	yes	—
68	Q₈	—	388	2	no	—
69	Q₈	—	194	4	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	97	2	no	—
72	C₂×C₄	—	388	2	no	—
73	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
74	C₈	—	194	1	no	—
75	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
76	C₈	#13095512	1	1	yes	characteristic
77	C₂²	D₈×D₉₇	1	1	yes	characteristic
78	C₄	—	8	2	no	—
79	C₄	—	388	4	no	—
80	C₄	—	194	2	no	—
81	C₄	#15690772	1	1	yes	characteristic
82	C₄	#15690833	1	1	yes	characteristic
83	C₂	#20260817	1	1	yes	characteristic
84	C₂	#20261174	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#367×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#367	D₉₇
hypocenter	C₉₇	#2176
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

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
#20260829	2
#20260538	2
#20260510	2
#367×C₉₇	2
#2176	97