Subgroups — #27139707

Order 12416.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
2	#20260486	C₂	1	1	yes	maximal, characteristic
3	#346×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
5	C₄×Dic₁₉₄	—	2	1	no	—
6	#15690744	C₄	1	1	yes	characteristic
7	C₄⋊#11715895	—	2	1	no	—
8	#15690735	—	2	1	no	—
9	C₂×C₁₆×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₄	—	2	1	no	—
12	Dic₃₈₈	—	2	1	no	—
13	Dic₃₈₈	C₈	1	2	yes	complemented
14	C₂×#11715895	—	4	1	no	—
15	C₄⋊Dic₉₇	—	2	1	no	—
16	#13095490	—	4	1	no	—
17	C₄×Dic₉₇	—	4	1	no	—
18	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
20	C₄²×C₉₇	D₄	1	1	yes	characteristic
21	C₁₆×C₉₇	—	2	2	no	—
22	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₉₇	—	4	1	no	—
26	C₂×Dic₉₇	—	4	1	no	—
27	Dic₁₉₄	—	2	2	no	—
28	Dic₁₉₄	—	4	1	no	—
29	C₈×C₉₇	—	2	1	no	—
30	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₉₇	M₁₆	1	1	yes	characteristic
32	#11715895	—	8	1	no	—
33	C₂²×C₉₇	#102	1	1	yes	characteristic
34	C₄×C₉₇	—	2	1	no	—
35	C₄×C₉₇	—	2	1	no	—
36	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₉₇	#104	1	1	yes	characteristic
38	Dic₉₇	—	4	2	no	—
39	Dic₉₇	—	8	1	no	—
40	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₉₇	#360	1	1	yes	characteristic
42	C₂×C₉₇	#362	1	1	yes	characteristic
43	#1334	—	97	1	no	maximal, Sylow
44	C₉₇	#1334	1	1	yes	characteristic, complemented, Sylow
45	C₄×Q₁₆	—	97	1	no	—
46	#346	D₉₇	1	1	yes	characteristic
47	#334	—	97	1	no	—
48	C₂×Q₁₆	—	97	1	no	—
49	C₄×Q₈	—	194	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	97	1	no	—
52	C₄⋊C₈	—	194	1	no	—
53	#103	—	194	1	no	—
54	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
55	C₂×Q₈	—	194	1	no	—
56	Q₁₆	—	97	2	no	—
57	Q₁₆	—	194	1	no	—
58	C₂×C₈	—	388	1	no	—
59	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
61	C₄⋊C₄	—	388	1	no	—
62	C₄⋊C₄	—	194	1	no	—
63	C₄²	—	388	1	no	—
64	C₄²	D₃₈₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	388	1	no	—
67	Q₈	—	194	2	no	—
68	C₂×C₄	—	388	1	no	—
69	C₂×C₄	—	388	1	no	—
70	C₂×C₄	#13095488	1	1	yes	characteristic
71	C₂×C₄	D₇₇₆	1	1	yes	characteristic
72	C₂×C₄	#13095500	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	776	1	no	—
75	C₈	C₈×D₉₇	1	1	yes	characteristic
76	C₈	#13095494	1	1	yes	characteristic
77	C₂²	#15690732	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	776	1	no	—
81	C₄	—	388	2	no	—
82	C₄	#15690722	1	1	yes	characteristic
83	C₄	#15690738	1	1	yes	characteristic
84	C₂	#20260464	1	1	yes	characteristic
85	C₂	#20260546	1	1	yes	characteristic
86	C₂	#20260552	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690732
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1334
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
13. Dic₃₈₈	C₈	semidirect
44. C₉₇	#1334	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₈₈	2
#20260486	2
#346×C₉₇	2
#1334	97