Subgroups — #27158133

Order 12416.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×#87604	C₂	1	2	yes	maximal
2	#20261662	C₂	1	1	yes	maximal, characteristic
3	#20261648	C₂	1	2	yes	maximal
4	#20261622	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×D₉₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×#87604	—	2	2	no	—
7	#15690883	C₂²	1	1	yes	characteristic
8	#15690879	C₂²	1	2	yes	—
9	#15690873	—	2	2	no	—
10	C₈×#87604	C₂²	1	1	yes	characteristic
11	Q₁₆×D₉₇	C₂²	1	2	yes	—
12	Dic₇₇₆	C₄	1	1	yes	characteristic, complemented
13	#15690756	C₄	1	2	yes	complemented
14	C₁₆×D₉₇	C₂²	1	1	yes	characteristic
15	Q₃₂×C₉₇	C₄	1	1	yes	characteristic, complemented
16	#13095519	D₄	1	1	yes	characteristic
17	C₄⋊#87604	—	2	2	no	—
18	C₄⋊#87604	—	4	2	no	—
19	C₄×#87604	—	4	2	no	—
20	C₄×#87604	D₄	1	1	yes	characteristic
21	#13095514	—	2	1	no	—
22	Q₈×D₉₇	—	2	2	no	—
23	Dic₃₈₈	C₂×C₄	1	2	yes	—
24	#13095501	—	2	2	no	—
25	C₈×D₉₇	C₂³	1	1	yes	characteristic
26	#13095486	C₂×C₄	1	1	yes	characteristic
27	Q₁₆×C₉₇	C₂×C₄	1	2	yes	—
28	C₁₆×C₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#87604	—	4	2	no	—
30	C₂×#87604	D₈	1	2	yes	—
31	Q₈×C₉₇	—	2	2	no	—
32	C₄×D₉₇	—	4	2	no	—
33	C₄×D₉₇	C₂×D₄	1	1	yes	characteristic
34	Dic₁₉₄	—	2	2	no	—
35	Dic₁₉₄	—	4	2	no	—
36	#11715897	—	2	1	no	—
37	C₈×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	#11715895	#41	1	1	yes	characteristic
39	C₂×D₉₇	C₂×D₈	1	1	yes	characteristic
40	#87604	—	2	2	no	—
41	#87604	—	8	2	no	—
42	C₄×C₉₇	—	4	2	no	—
43	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
44	Dic₉₇	—	4	2	no	—
45	Dic₉₇	#135	1	1	yes	characteristic
46	C₂×C₉₇	C₄×D₈	1	1	yes	characteristic
47	D₉₇	#510	1	2	yes	—
48	#2176	—	97	1	no	maximal, Sylow
49	C₉₇	#2176	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	97	1	no	—
51	C₄×Q₁₆	—	97	2	no	—
52	#367	—	97	1	no	—
53	#358	—	97	2	no	—
54	#346	—	97	1	no	—
55	C₂×Q₁₆	—	97	2	no	—
56	C₄×Q₈	—	194	2	no	—
57	Q₃₂	—	97	2	no	—
58	Q₃₂	—	97	1	no	—
59	Q₃₂	#87604	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	97	1	no	—
61	C₂×C₁₆	—	97	1	no	—
62	#107	—	97	2	no	—
63	#103	—	194	2	no	—
64	C₄×C₈	—	97	1	no	—
65	C₂×Q₈	—	194	2	no	—
66	Q₁₆	—	97	2	no	—
67	Q₁₆	—	194	2	no	—
68	Q₁₆	C₂×#87604	1	2	yes	—
69	C₂×C₈	—	97	1	no	—
70	C₂×C₈	—	97	1	no	—
71	C₄⋊C₄	—	388	2	no	—
72	C₄⋊C₄	—	194	2	no	—
73	C₄²	—	97	1	no	—
74	C₄²	—	388	2	no	—
75	C₁₆	—	97	1	no	—
76	C₁₆	—	194	1	no	—
77	C₁₆	C₂×#87604	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	Q₈	—	388	2	no	—
80	Q₈	—	194	2	no	—
81	C₂×C₄	—	97	1	no	—
82	C₂×C₄	—	97	2	no	—
83	C₂×C₄	—	388	2	no	—
84	C₂×C₄	—	388	2	no	—
85	C₈	—	97	1	no	—
86	C₈	—	194	1	no	—
87	C₈	C₂²×#87604	1	1	yes	characteristic
88	C₂²	—	97	1	no	—
89	C₄	—	4	2	no	—
90	C₄	—	776	2	no	—
91	C₄	—	97	1	no	—
92	C₄	—	388	2	no	—
93	C₄	—	194	2	no	—
94	C₄	D₄×#87604	1	1	yes	characteristic
95	C₂	—	97	2	no	—
96	C₂	D₈×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×#87604
commutator	C₈×C₉₇	C₂²×C₄
Frattini	C₈	C₂²×#87604
Fitting	Q₃₂×C₉₇	C₄
socle	C₂×C₉₇	C₄×D₈
prime radical	C₂×D₉₇	C₂×D₈
cosocle	C₈×D₉₇	C₂³
hypercenter	Q₃₂	#87604
hypocenter	C₉₇	#2176
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
12. Dic₇₇₆	C₄	semidirect
13. #15690756	C₄	semidirect
15. Q₃₂×C₉₇	C₄	semidirect
49. C₉₇	#2176	semidirect
59. Q₃₂	#87604	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×#87604	2
#20261662	2
#20261648	2
#20261622	2
Q₃₂×D₉₇	2
#2176	97