Subgroups — #27139708

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260853	C₂	1	1	yes	maximal, characteristic
2	C₈×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690832	C₄	1	1	yes	characteristic
5	#15690816	—	2	1	no	—
6	#15690746	C₄	1	1	yes	characteristic
7	#15690738	—	2	1	no	—
8	C₄×#11715895	—	2	1	no	—
9	M₃₂×C₉₇	—	2	1	no	—
10	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
11	#13095511	—	2	1	no	—
12	Dic₃₈₈	C₈	1	1	yes	characteristic, complemented
13	#13095500	—	2	1	no	—
14	D₇₇₆	C₈	1	1	yes	characteristic, complemented
15	#13095495	—	2	1	no	—
16	#13095494	—	4	1	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₈×D₉₇	—	4	1	no	—
19	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
21	C₄²×C₉₇	D₄	1	1	yes	characteristic
22	C₁₆×C₉₇	—	4	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₉₇	—	4	1	no	—
26	D₃₈₈	—	2	1	no	—
27	C₄×D₉₇	—	4	1	no	—
28	Dic₁₉₄	—	2	1	no	—
29	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
30	C₈×C₉₇	M₁₆	1	1	yes	characteristic
31	C₈×C₉₇	D₈	1	1	yes	characteristic
32	C₈×C₉₇	SD₁₆	1	1	yes	characteristic
33	#11715895	—	2	4	no	—
34	#11715895	—	4	1	no	—
35	C₂²×C₉₇	#104	1	1	yes	characteristic
36	C₂×D₉₇	—	4	1	no	—
37	C₄×C₉₇	—	2	2	no	—
38	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₉₇	#102	1	1	yes	characteristic
40	Dic₉₇	—	4	1	no	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
43	D₉₇	—	8	1	no	—
44	#1335	—	97	1	no	maximal, Sylow
45	C₉₇	#1335	1	1	yes	characteristic, complemented, Sylow
46	#443	—	97	1	no	—
47	#364	D₉₇	1	1	yes	characteristic, complemented
48	C₈²	—	97	1	no	—
49	#135	—	97	1	no	—
50	#131	—	194	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	97	1	no	—
53	#104	—	194	1	no	—
54	C₄×C₈	—	194	1	no	—
55	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
56	#41	—	194	1	no	—
57	Q₁₆	—	97	1	no	—
58	SD₁₆	—	194	1	no	—
59	D₈	—	97	1	no	—
60	M₁₆	—	388	1	no	—
61	M₁₆	—	194	1	no	—
62	C₂×C₈	—	388	1	no	—
63	C₂×C₈	—	194	2	no	—
64	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
66	C₄²	D₃₈₈	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	194	1	no	—
69	D₄	—	388	1	no	—
70	D₄	—	194	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	388	1	no	—
73	C₂×C₄	#13095488	1	1	yes	characteristic
74	C₈	—	388	1	no	—
75	C₈	—	194	4	no	—
76	C₈	C₈×D₉₇	1	1	yes	characteristic
77	C₈	#13095494	1	1	yes	characteristic
78	C₈	#13095496	1	1	yes	characteristic
79	C₈	#13095498	1	1	yes	characteristic
80	C₂²	—	388	1	no	—
81	C₂²	#15690736	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	388	1	no	—
84	C₄	#15690722	1	1	yes	characteristic
85	C₄	#15690730	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	776	1	no	—
88	C₂	#20260462	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13095496
commutator	C₈×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#364×C₉₇	C₂
socle	C₂×C₉₇	D₄⋊C₈
prime radical	#15690832	C₄
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#364	D₉₇
hypocenter	C₉₇	#1335
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×#11715895	C₂	semidirect
3. #364×C₉₇	C₂	semidirect
12. Dic₃₈₈	C₈	semidirect
14. D₇₇₆	C₈	semidirect
45. C₉₇	#1335	semidirect
47. #364	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260853	2
C₈×#11715895	2
#364×C₉₇	2
#1335	97