Subgroups — #27156670

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261631	C₂	1	2	yes	maximal, complemented
2	#20260702	C₂	1	1	yes	maximal, characteristic
3	#15690884	—	2	2	no	—
4	C₂×#13095495	C₂²	1	1	yes	characteristic
5	C₂²×#11715895	C₄	1	1	yes	characteristic
6	C₂²⋊#11715895	—	2	2	no	—
7	C₂×#41×C₉₇	C₄	1	1	yes	characteristic
8	#13095514	—	4	2	no	—
9	#13095495	—	4	1	no	—
10	C₂×#11715895	—	2	1	no	—
11	C₂×#11715895	—	4	1	no	—
12	C₂×#11715895	D₄	1	2	yes	—
13	#41×C₉₇	—	4	1	no	—
14	#41×C₉₇	—	4	1	no	—
15	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
16	C₂×D₄×C₉₇	—	2	1	no	—
17	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
18	C₂²×C₄×C₉₇	—	2	1	no	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₉₇	—	2	1	no	—
21	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
22	Q₈×C₉₇	—	4	1	no	—
23	D₄×C₉₇	—	4	1	no	—
24	D₄×C₉₇	—	4	2	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	2	1	no	—
27	C₂×C₄×C₉₇	—	2	1	no	—
28	C₂×C₄×C₉₇	—	4	1	no	—
29	C₂×C₄×C₉₇	—	4	1	no	—
30	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
32	#11715895	—	2	2	no	—
33	#11715895	—	4	1	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	—	4	2	no	—
37	C₂²×C₉₇	—	4	1	no	—
38	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	—	4	1	no	—
41	C₄×C₉₇	#99	1	1	yes	characteristic
42	C₄×C₉₇	#100	1	1	yes	characteristic
43	C₂×C₉₇	—	2	1	no	—
44	C₂×C₉₇	—	4	1	no	—
45	C₂×C₉₇	—	8	1	no	—
46	C₂×C₉₇	#323	1	1	yes	characteristic
47	#1321	—	97	1	no	maximal, Sylow
48	C₉₇	#1321	1	1	yes	characteristic, complemented, Sylow
49	#408	—	97	1	no	—
50	#349	—	97	2	no	—
51	C₂×#41	#87604	1	1	yes	characteristic
52	C₂×M₁₆	—	97	1	no	—
53	C₂²×C₈	—	97	1	no	—
54	M₃₂	—	194	2	no	—
55	C₂²⋊C₈	—	194	2	no	—
56	#41	—	4	1	no	—
57	#41	—	4	1	no	—
58	C₂×Q₈	#11715897	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	#11715897	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
63	M₁₆	—	388	1	no	—
64	C₂×C₈	—	97	2	no	—
65	C₂×C₈	—	388	1	no	—
66	C₂×C₈	—	194	1	no	—
67	C₁₆	—	388	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	#13095520	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	D₄	—	4	1	no	—
72	D₄	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
79	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
80	C₈	—	388	1	no	—
81	C₈	—	194	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#15690868	1	1	yes	characteristic
90	C₄	#15690870	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	4	1	no	—
93	C₂	—	8	1	no	—
94	C₂	#20261578	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690868
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#87604
Fitting, prime radical	C₂×#41×C₉₇	C₄
socle	C₂×C₉₇	#323
cosocle	C₂×#13095495	C₂²
hypercenter	C₂×#41	#87604
hypocenter	C₉₇	#1321
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20261631	C₂	semidirect
48. C₉₇	#1321	semidirect

Maximal subgroups

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

subgroup	index
#20261631	2
#20260702	2
#1321	97