Subgroups — #27156751

Order 12416.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261656	C₂	1	1	yes	maximal, characteristic
2	#20261599	C₂	1	1	yes	maximal, characteristic
3	#20261113	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₄⋊#11715897	—	2	1	no	—
6	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
7	C₂×#13095501	—	2	1	no	—
8	#15690808	—	2	1	no	—
9	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₉₇	C₄	1	1	yes	characteristic
11	C₂×#11715897	—	4	1	no	—
12	#13095514	—	4	1	no	—
13	C₂×Dic₁₉₄	—	2	1	no	—
14	Dic₃₈₈	—	4	1	no	—
15	#13095501	—	4	1	no	—
16	#13095501	—	4	1	no	—
17	C₂×#11715895	D₄	1	1	yes	characteristic
18	#13095490	—	4	1	no	—
19	#13095490	—	4	1	no	—
20	C₄×Dic₉₇	D₄	1	1	yes	characteristic
21	C₂×Q₈×C₉₇	—	2	1	no	—
22	Q₁₆×C₉₇	—	4	1	no	—
23	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₉₇	—	4	1	no	—
25	Q₈×C₉₇	—	4	1	no	—
26	C₂×C₄×C₉₇	—	4	1	no	—
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	4	1	no	—
29	C₂×Dic₉₇	D₈	1	1	yes	characteristic
30	C₂×Dic₉₇	SD₁₆	1	1	yes	characteristic
31	Dic₁₉₄	—	4	1	no	—
32	Dic₁₉₄	—	4	1	no	—
33	#11715897	—	8	1	no	—
34	C₈×C₉₇	—	2	1	no	—
35	#11715895	D₈	1	1	yes	characteristic
36	#11715895	SD₁₆	1	1	yes	characteristic
37	C₂²×C₉₇	#102	1	1	yes	characteristic
38	C₄×C₉₇	—	8	1	no	—
39	C₄×C₉₇	#100	1	1	yes	characteristic
40	C₄×C₉₇	#102	1	1	yes	characteristic
41	Dic₉₇	—	2	1	no	—
42	Dic₉₇	—	8	1	no	—
43	Dic₉₇	SD₃₂	1	1	yes	characteristic
44	Dic₉₇	Q₃₂	1	1	yes	characteristic
45	C₂×C₉₇	#331	1	1	yes	characteristic
46	C₂×C₉₇	#358	1	1	yes	characteristic
47	C₂×C₉₇	#362	1	1	yes	characteristic
48	#1362	—	97	1	no	maximal, Sylow
49	C₉₇	#1362	1	1	yes	characteristic, complemented, Sylow
50	#494	—	97	1	no	—
51	C₄⋊C₁₆	—	97	1	no	—
52	#335	—	97	1	no	—
53	C₂×Q₁₆	—	97	1	no	—
54	C₂×Q₁₆	—	194	1	no	—
55	C₂×Q₁₆	#87604	1	1	yes	characteristic
56	C₄⋊Q₈	—	194	1	no	—
57	C₂×C₁₆	—	194	1	no	—
58	C₄⋊C₈	—	194	1	no	—
59	C₄×C₈	—	97	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	194	1	no	—
62	Q₁₆	—	4	1	no	—
63	Q₁₆	—	388	1	no	—
64	Q₁₆	—	388	1	no	—
65	Q₁₆	—	388	1	no	—
66	C₂×C₈	—	97	1	no	—
67	C₂×C₈	—	388	1	no	—
68	C₂×C₈	C₂×#87604	1	1	yes	characteristic
69	C₄⋊C₄	—	388	1	no	—
70	C₄⋊C₄	—	388	1	no	—
71	C₄²	—	97	1	no	—
72	C₁₆	—	388	1	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	Q₈	—	388	1	no	—
76	Q₈	—	388	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	97	1	no	—
79	C₂×C₄	—	97	1	no	—
80	C₂×C₄	—	388	1	no	—
81	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	776	1	no	—
84	C₈	—	97	1	no	—
85	C₈	—	97	1	no	—
86	C₂²	#15690872	1	1	yes	characteristic
87	C₄	—	8	1	no	—
88	C₄	—	776	1	no	—
89	C₄	—	97	1	no	—
90	C₄	—	97	1	no	—
91	C₄	—	194	1	no	—
92	C₄	#15690870	1	1	yes	characteristic
93	C₄	#15690872	1	1	yes	characteristic
94	C₂	#20261593	1	1	yes	characteristic
95	C₂	#20261648	1	1	yes	characteristic
96	C₂	#20261654	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690872
commutator	C₂×C₈×C₉₇	C₂×C₄
Frattini	C₂×C₈	C₂×#87604
Fitting	C₂×Q₁₆×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	#102
cosocle	C₈×Dic₉₇	C₂²
hypercenter	C₂×Q₁₆	#87604
hypocenter	C₉₇	#1362
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
49. C₉₇	#1362	semidirect

Maximal subgroups

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

subgroup	index
#20261656	2
#20261599	2
#20261113	2
#1362	97