Subgroups — #27143519

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₉₇	C₂	1	1	yes	maximal, characteristic
2	#20260772	C₂	1	2	yes	maximal
3	#20260563	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260560	C₂	1	2	yes	maximal, complemented
5	#367×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
7	#15690768	—	2	2	no	—
8	C₂×#13095486	C₂²	1	1	yes	characteristic
9	C₁₆×D₉₇	C₄	1	4	yes	complemented
10	#15690744	C₂²	1	2	yes	—
11	#15690741	—	2	2	no	—
12	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
13	#107×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
15	C₂×#11715895	D₄	1	1	yes	characteristic
16	C₈×D₉₇	C₂×C₄	1	2	yes	—
17	C₈×D₉₇	Q₈	1	2	yes	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	#13095488	—	4	2	no	—
20	C₄×Dic₉₇	—	4	2	no	—
21	#13095486	C₂×C₄	1	2	yes	—
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	C₂²×D₉₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	—	4	2	no	—
27	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	2	no	—
29	C₂×Dic₉₇	D₈	1	1	yes	characteristic
30	C₄×D₉₇	C₄⋊C₄	1	2	yes	—
31	C₄×D₉₇	Q₁₆	1	2	yes	—
32	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₉₇	C₂×Q₈	1	1	yes	characteristic
34	#11715895	C₄⋊C₄	1	2	yes	—
35	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
36	C₂×D₉₇	—	2	2	no	—
37	C₂×D₉₇	#107	1	2	yes	—
38	C₄×C₉₇	—	8	2	no	—
39	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₄×C₉₇	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₉₇	—	8	2	no	—
42	Dic₉₇	#107	1	2	yes	—
43	C₂×C₉₇	C₂×#107	1	1	yes	characteristic
44	C₂×C₉₇	#508	1	2	yes	—
45	D₉₇	—	2	2	no	—
46	#2158	—	97	1	no	maximal, Sylow
47	C₉₇	#2158	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₁₆	—	97	1	no	—
49	#427	—	97	2	no	—
50	#367	—	97	1	no	—
51	#367	D₉₇	1	1	yes	characteristic, complemented
52	#366	—	97	2	no	—
53	C₂²×C₈	—	97	1	no	—
54	#117	—	194	2	no	—
55	C₂×C₁₆	—	97	1	no	—
56	C₂×C₁₆	—	97	4	no	—
57	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
58	#107	—	97	2	no	—
59	#107	C₂×D₉₇	1	2	yes	—
60	#106	—	194	2	no	—
61	C₂²×C₄	—	97	1	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	97	2	no	—
64	C₂×C₈	—	97	2	no	—
65	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	194	2	no	—
68	C₂²⋊C₄	—	388	2	no	—
69	C₄²	—	388	2	no	—
70	C₁₆	—	97	2	no	—
71	C₁₆	C₄×D₉₇	1	2	yes	—
72	C₂³	—	97	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	—	97	2	no	—
76	C₂×C₄	—	97	2	no	—
77	C₂×C₄	—	388	2	no	—
78	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
79	C₈	—	97	2	no	—
80	C₈	C₂×C₄×D₉₇	1	1	yes	characteristic
81	C₈	Q₈×D₉₇	1	1	yes	characteristic
82	C₂²	—	97	2	no	—
83	C₂²	—	194	2	no	—
84	C₂²	D₈×D₉₇	1	1	yes	characteristic
85	C₄	—	8	2	no	—
86	C₄	—	776	2	no	—
87	C₄	—	97	2	no	—
88	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
89	C₄	Q₁₆×D₉₇	1	1	yes	characteristic
90	C₂	—	194	2	no	—
91	C₂	#107×D₉₇	1	1	yes	characteristic
92	C₂	#20261166	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₉₇
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#367×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₈
prime radical	C₂²×D₉₇	D₈
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	#367	D₉₇
hypocenter	C₉₇	#2158
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
3. #20260563	C₂	semidirect
4. #20260560	C₂	semidirect
5. #367×C₉₇	C₂	semidirect
9. C₁₆×D₉₇	C₄	semidirect
47. C₉₇	#2158	semidirect
51. #367	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₉₇	2
#20260772	2
#20260563	2
#20260560	2
#367×C₉₇	2
#2158	97