Subgroups — #27143529

Order 12416.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690749	C₂	1	1	yes	maximal, characteristic
2	#108×D₉₇	C₂	1	2	yes	maximal
3	#20260565	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260564	C₂	1	2	yes	maximal, complemented
5	#368×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×D₉₇	—	2	2	no	—
7	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
8	#15690749	C₄	1	4	yes	—
9	C₂×#13095486	C₂²	1	1	yes	characteristic
10	#15690746	C₂²	1	2	yes	—
11	#15690745	—	2	2	no	—
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	#108×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
15	#13095495	—	2	2	no	—
16	#13095494	—	4	2	no	—
17	C₂×#11715895	D₄	1	1	yes	characteristic
18	C₈×D₉₇	—	4	2	no	—
19	C₈×D₉₇	C₂×C₄	1	2	yes	—
20	C₈×D₉₇	Q₈	1	2	yes	—
21	#13095486	C₂×C₄	1	2	yes	—
22	M₁₆×C₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×D₉₇	Q₁₆	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₉₇	Q₁₆	1	1	yes	characteristic
28	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
29	C₄×D₉₇	D₈	1	2	yes	—
30	C₈×C₉₇	—	4	2	no	—
31	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
33	#11715895	—	4	2	no	—
34	#11715895	C₄⋊C₄	1	2	yes	—
35	C₂²×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×D₉₇	—	4	1	no	—
37	C₂×D₉₇	#107	1	2	yes	—
38	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
40	Dic₉₇	#107	1	2	yes	—
41	C₂×C₉₇	—	2	1	no	—
42	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
43	D₉₇	—	2	2	no	—
44	#2161	—	97	1	no	maximal, Sylow
45	C₉₇	#2161	1	1	yes	characteristic, complemented, Sylow
46	C₂×M₃₂	—	97	1	no	—
47	C₂×#108	—	97	2	no	—
48	#368	—	97	1	no	—
49	#368	—	97	2	no	—
50	#368	D₉₇	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	194	2	no	—
52	C₂²×C₈	—	97	1	no	—
53	M₃₂	—	97	4	no	—
54	C₂×C₁₆	—	97	1	no	—
55	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
56	#108	—	97	2	no	—
57	#108	—	194	2	no	—
58	#108	C₂×D₉₇	1	2	yes	—
59	C₂²×C₄	—	97	1	no	—
60	M₁₆	—	2	2	no	—
61	M₁₆	—	388	2	no	—
62	M₁₆	—	194	2	no	—
63	C₂×C₈	—	97	1	no	—
64	C₂×C₈	—	97	2	no	—
65	C₂×C₈	—	97	2	no	—
66	C₂×C₈	—	388	2	no	—
67	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
68	C₁₆	—	97	2	no	—
69	C₁₆	C₄×D₉₇	1	2	yes	—
70	C₂³	—	97	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	97	2	no	—
73	C₂×C₄	—	97	2	no	—
74	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
75	C₈	—	4	2	no	—
76	C₈	—	97	2	no	—
77	C₈	—	388	2	no	—
78	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
79	C₈	Q₈×D₉₇	1	1	yes	characteristic
80	C₂²	—	97	2	no	—
81	C₂²	—	388	1	no	—
82	C₂²	Q₁₆×D₉₇	1	1	yes	characteristic
83	C₄	—	97	2	no	—
84	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
85	C₄	D₈×D₉₇	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	194	2	no	—
88	C₂	#107×D₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#368×C₉₇	C₂
socle	C₂×C₉₇	C₂×#107
prime radical	C₂²×D₉₇	Q₁₆
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#368	D₉₇
hypocenter	C₉₇	#2161
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
3. #20260565	C₂	semidirect
4. #20260564	C₂	semidirect
5. #368×C₉₇	C₂	semidirect
45. C₉₇	#2161	semidirect
50. #368	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690749	2
#108×D₉₇	2
#20260565	2
#20260564	2
#368×C₉₇	2
#2161	97