Subgroups — #27143594

Order 12416.

150 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	C₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
5	#366×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	1	yes	characteristic
29	C₂×Dic₉₇	D₈	1	1	yes	characteristic
30	Dic₁₉₄	—	2	4	no	—
31	Dic₁₉₄	—	4	2	no	—
32	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₉₇	#41	1	1	yes	characteristic
34	#11715895	—	2	1	no	—
35	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₉₇	—	8	2	no	—
37	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₉₇	#135	1	1	yes	characteristic
39	Dic₉₇	—	2	1	no	—
40	Dic₉₇	—	4	4	no	—
41	Dic₉₇	Q₃₂	1	2	yes	complemented
42	C₂×C₉₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₉₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₉₇	#508	1	1	yes	characteristic
45	C₄×Q₃₂	—	97	1	no	maximal, Sylow
46	C₉₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	—	97	1	no	—
48	C₄×Q₁₆	—	97	2	no	—
49	#366	D₉₇	1	1	yes	characteristic
50	#358	—	97	2	no	—
51	C₄×C₁₆	—	97	1	no	—
52	C₂×Q₁₆	—	97	2	no	—
53	C₄×Q₈	—	194	2	no	—
54	Q₃₂	—	97	4	no	—
55	C₂×C₁₆	—	97	1	no	—
56	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
57	#107	C₂×D₉₇	1	2	yes	—
58	#103	—	194	2	no	—
59	C₄×C₈	—	97	1	no	—
60	C₂×Q₈	—	194	2	no	—
61	Q₁₆	—	97	4	no	—
62	Q₁₆	—	194	2	no	—
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	388	2	no	—
67	C₄²	—	97	1	no	—
68	C₄²	—	388	2	no	—
69	C₁₆	—	194	1	no	—
70	C₁₆	C₄×D₉₇	1	2	yes	—
71	Q₈	—	388	2	no	—
72	Q₈	—	194	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	—	97	1	no	—
76	C₂×C₄	—	388	2	no	—
77	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
78	C₈	—	194	1	no	—
79	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
80	C₈	#13095512	1	1	yes	characteristic
81	C₂²	D₈×D₉₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	97	2	no	—
84	C₄	—	388	4	no	—
85	C₄	—	194	1	no	—
86	C₄	#15690772	1	1	yes	characteristic
87	C₄	#15690833	1	1	yes	characteristic
88	C₂	#20260817	1	1	yes	characteristic
89	C₂	Q₃₂×D₉₇	1	1	yes	characteristic
90	C₂	#20261164	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#366×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂×D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#366	D₉₇
hypocenter	C₉₇	C₄×Q₃₂
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₇₇₆	C₄	semidirect
41. Dic₉₇	Q₃₂	semidirect
46. C₉₇	C₄×Q₃₂	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
C₁₆×Dic₉₇	2
#366×C₉₇	2
C₄×Q₃₂	97