Subgroups — #27158962

Order 12416.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261838	C₂	1	2	yes	maximal
2	C₄⋊#13095523	C₂	1	1	yes	maximal, characteristic
3	#20261829	C₂	1	2	yes	maximal
4	C₄×#13095523	C₂	1	1	yes	maximal, characteristic
5	Q₈×#11715897	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095523	C₂²	1	1	yes	characteristic
7	C₂×#13095523	C₂²	1	2	yes	—
8	C₂×#13095523	C₂²	1	1	yes	characteristic
9	C₄⋊#11715897	C₄	1	2	yes	—
10	C₄⋊#11715897	C₄	1	1	yes	characteristic
11	C₄×#11715897	C₂²	1	1	yes	characteristic
12	C₄×#11715897	C₂²	1	2	yes	—
13	Q₈×Dic₉₇	C₄	1	1	yes	characteristic
14	#13095523	—	2	1	no	—
15	#13095523	—	2	2	no	—
16	#13095523	Q₈	1	2	yes	complemented
17	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
18	C₂×#11715897	C₂×C₄	1	2	yes	—
19	C₂×#11715897	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₉₇	C₈	1	2	yes	—
21	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
22	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
23	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
25	Q₈×C₉₇	—	2	2	no	—
26	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₉₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
30	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
31	#11715897	—	2	1	no	—
32	#11715897	—	2	2	no	—
33	#11715897	C₂×Q₈	1	1	yes	characteristic
34	#11715897	#41	1	1	yes	characteristic
35	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₉₇	—	2	2	no	—
37	C₄×C₉₇	M₃₂	1	2	yes	—
38	Dic₉₇	—	2	2	no	—
39	Dic₉₇	—	2	1	no	—
40	Dic₉₇	C₄×Q₈	1	1	yes	characteristic
41	Dic₉₇	#131	1	1	yes	characteristic
42	C₂×C₉₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₉₇	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₉₇	#504	1	1	yes	characteristic
45	#2183	—	97	1	no	maximal, Sylow
46	C₉₇	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	97	1	no	—
48	C₄⋊C₁₆	—	97	1	no	—
49	C₄⋊C₁₆	—	97	2	no	—
50	#346	—	97	2	no	—
51	C₄×C₁₆	—	97	1	no	—
52	C₄×Q₈	—	97	1	no	—
53	C₂×C₁₆	—	97	1	no	—
54	C₂×C₁₆	—	97	1	no	—
55	C₂×C₁₆	—	97	2	no	—
56	C₄⋊C₈	—	97	2	no	—
57	C₄⋊C₈	—	97	1	no	—
58	C₄×C₈	—	97	1	no	—
59	C₄×C₈	—	97	2	no	—
60	C₂×Q₈	#11715906	1	1	yes	characteristic
61	C₂×C₈	—	97	1	no	—
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	—	97	2	no	—
64	C₄⋊C₄	—	97	2	no	—
65	C₄⋊C₄	—	97	1	no	—
66	C₄²	—	97	2	no	—
67	C₄²	—	97	1	no	—
68	C₁₆	—	97	2	no	—
69	C₁₆	—	194	1	no	—
70	C₁₆	—	194	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	—	97	1	no	—
73	C₂×C₄	—	97	2	no	—
74	C₂×C₄	—	97	1	no	—
75	C₂×C₄	C₂×#11715906	1	2	yes	—
76	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
77	C₈	—	97	1	no	—
78	C₈	—	97	1	no	—
79	C₈	—	194	1	no	—
80	C₈	—	194	2	no	—
81	C₂²	C₂²×#11715906	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	97	1	no	—
84	C₄	—	97	1	no	—
85	C₄	—	194	2	no	—
86	C₄	—	194	1	no	—
87	C₄	#15690906	1	2	yes	—
88	C₂	Q₈×#11715906	1	1	yes	characteristic
89	C₂	C₂×#15690906	1	1	yes	characteristic
90	C₂	#20261857	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715906
commutator, socle, prime radical	C₂²×C₉₇	C₂²×C₈
Fitting	C₂×Q₈×C₉₇	C₈
cosocle	C₂×#11715897	C₂³
hypercenter	C₂×Q₈	#11715906
hypocenter	C₉₇	#2183
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
16. #13095523	Q₈	semidirect
46. C₉₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
#20261838	2
C₄⋊#13095523	2
#20261829	2
C₄×#13095523	2
Q₈×#11715897	2
#2183	97