Subgroups — #27156704

Order 12416.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095514	C₂	1	1	yes	maximal, characteristic
2	#20261623	C₂	1	1	yes	maximal, characteristic
3	#20260866	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₂×#13095514	—	2	1	no	—
6	C₄⋊#11715895	—	2	1	no	—
7	C₄⋊#11715895	C₄	1	1	yes	characteristic
8	#15690721	—	2	1	no	—
9	C₄×#11715895	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₉₇	C₄	1	1	yes	characteristic
11	#13095514	—	2	2	no	—
12	#13095514	—	4	1	no	—
13	C₂×#11715895	—	2	1	no	—
14	C₂×#11715895	D₄	1	1	yes	characteristic
15	C₂×#11715895	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
17	C₄⋊C₄×C₉₇	—	2	1	no	—
18	C₄⋊C₄×C₉₇	C₈	1	1	yes	characteristic
19	C₄²×C₉₇	—	2	1	no	—
20	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
21	Q₈×C₉₇	—	4	1	no	—
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
27	#11715895	—	2	1	no	—
28	#11715895	—	4	1	no	—
29	#11715895	SD₁₆	1	1	yes	characteristic
30	#11715895	Q₁₆	1	1	yes	characteristic
31	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₉₇	—	2	1	no	—
33	C₄×C₉₇	—	2	1	no	—
34	C₄×C₉₇	—	4	1	no	—
35	C₄×C₉₇	—	4	1	no	—
36	C₄×C₉₇	#103	1	1	yes	characteristic
37	C₄×C₉₇	#104	1	1	yes	characteristic
38	C₂×C₉₇	Q₈⋊C₈	1	1	yes	characteristic
39	C₂×C₉₇	#349	1	1	yes	characteristic
40	C₂×C₉₇	#350	1	1	yes	characteristic
41	#1338	—	97	1	no	maximal, Sylow
42	C₉₇	#1338	1	1	yes	characteristic, complemented, Sylow
43	#446	—	97	1	no	—
44	C₄⋊C₁₆	—	97	1	no	—
45	#346	—	97	1	no	—
46	C₄×Q₈	#87604	1	1	yes	characteristic
47	C₂×C₁₆	—	194	1	no	—
48	C₂×C₁₆	—	194	1	no	—
49	C₄⋊C₈	—	97	1	no	—
50	C₄⋊C₈	—	194	1	no	—
51	#97	—	194	1	no	—
52	C₄×C₈	—	97	1	no	—
53	C₂×Q₈	#11715897	1	1	yes	characteristic
54	C₂×C₈	—	97	1	no	—
55	C₂×C₈	—	97	1	no	—
56	C₂×C₈	—	194	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	#11715897	1	1	yes	characteristic
59	C₄²	—	2	1	no	—
60	C₄²	C₂×#87604	1	1	yes	characteristic
61	C₁₆	—	388	1	no	—
62	C₁₆	—	194	2	no	—
63	Q₈	—	4	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
67	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
68	C₂×C₄	#13095520	1	1	yes	characteristic
69	C₈	—	97	1	no	—
70	C₈	—	97	1	no	—
71	C₈	—	388	1	no	—
72	C₈	—	194	1	no	—
73	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	#15690873	1	1	yes	characteristic
79	C₄	#15690875	1	1	yes	characteristic
80	C₂	Q₈⋊#11715897	1	1	yes	characteristic
81	C₂	#20261631	1	1	yes	characteristic
82	C₂	#20261633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87604
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₄²	C₂×#87604
Fitting	C₄×Q₈×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₂²⋊C₈
cosocle	C₄×#11715895	C₂²
hypercenter	C₄×Q₈	#87604
hypocenter	C₉₇	#1338
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
42. C₉₇	#1338	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095514	2
#20261623	2
#20260866	2
#1338	97