Subgroups — #27144031

Order 12416.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261151	C₂	1	1	yes	maximal, characteristic
2	#20260568	C₂	1	2	yes	maximal, complemented
3	C₂×#15690716	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₉₇	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#15690816	C₄	1	1	yes	characteristic
7	#15690750	C₄	1	1	yes	characteristic
8	#15690749	C₄	1	2	yes	—
9	C₁₆×D₉₇	C₂²	1	2	yes	—
10	#15690716	C₂²	1	1	yes	characteristic
11	#15690716	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	C₃₂×C₉₇	C₂²	1	2	yes	—
14	#13095511	C₈	1	1	yes	characteristic
15	#13095495	C₈	1	1	yes	characteristic
16	#13095494	C₈	1	2	yes	—
17	C₈×D₉₇	C₂×C₄	1	2	yes	—
18	#13095486	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
23	C₂²⋊D₉₇	C₁₆	1	2	yes	—
24	D₃₈₈	C₁₆	1	1	yes	characteristic
25	C₄×D₉₇	C₂×C₈	1	2	yes	—
26	Dic₁₉₄	C₁₆	1	1	yes	characteristic
27	C₈×C₉₇	C₂×C₈	1	1	yes	characteristic
28	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
29	#11715895	C₂×C₈	1	2	yes	—
30	C₂²×C₉₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₉₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₉₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₉₇	—	2	1	no	—
36	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₉₇	—	2	2	no	—
38	#2258	—	97	1	no	maximal, Sylow
39	C₉₇	#2258	1	1	yes	characteristic, complemented, Sylow
40	#504	—	97	1	no	—
41	M₆₄	—	97	2	no	—
42	M₆₄	D₉₇	1	1	yes	characteristic, complemented
43	C₂×C₃₂	—	97	1	no	—
44	C₂×C₃₂	—	97	2	no	—
45	#131	—	97	1	no	—
46	M₃₂	—	97	1	no	—
47	M₃₂	—	97	2	no	—
48	C₂×C₁₆	—	97	2	no	—
49	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
50	C₃₂	—	97	1	no	—
51	C₃₂	—	97	1	no	—
52	C₃₂	C₂×D₉₇	1	2	yes	—
53	#41	—	97	1	no	—
54	M₁₆	—	97	1	no	—
55	M₁₆	—	97	2	no	—
56	C₂×C₈	—	97	2	no	—
57	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
58	C₁₆	—	97	2	no	—
59	C₁₆	C₄×D₉₇	1	1	yes	characteristic
60	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
61	Q₈	—	97	1	no	—
62	D₄	—	97	2	no	—
63	D₄	—	97	1	no	—
64	C₂×C₄	—	97	2	no	—
65	C₂×C₄	C₈×D₉₇	1	1	yes	characteristic
66	C₈	—	97	2	no	—
67	C₈	C₈×D₉₇	1	1	yes	characteristic
68	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
69	C₂²	—	97	2	no	—
70	C₂²	C₁₆×D₉₇	1	1	yes	characteristic
71	C₄	—	97	2	no	—
72	C₄	C₁₆×D₉₇	1	1	yes	characteristic
73	C₄	C₂×C₈×D₉₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	194	2	no	—
76	C₂	C₂×C₁₆×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₉₇
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	M₆₄×C₉₇	C₂
prime radical	#13095511	C₈
cosocle	C₁₆×C₉₇	C₂³
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #20260568	C₂	semidirect
3. C₂×#15690716	C₂	semidirect
4. C₃₂×D₉₇	C₂	semidirect
5. M₆₄×C₉₇	C₂	semidirect
39. C₉₇	#2258	semidirect
42. M₆₄	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261151	2
#20260568	2
C₂×#15690716	2
C₃₂×D₉₇	2
M₆₄×C₉₇	2
#2258	97