Subgroups — #27158077

Order 12416.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690884	C₂	1	2	yes	maximal
2	#20261657	C₂	1	4	yes	maximal, complemented
3	#20260693	C₂	1	1	yes	maximal, characteristic
4	#15690884	—	2	2	no	—
5	#15690884	C₂²	1	4	yes	—
6	C₂×#13095514	—	2	2	no	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	#15690769	C₄	1	1	yes	characteristic
9	#15690720	C₄	1	2	yes	—
10	C₈×Dic₉₇	C₂²	1	2	yes	—
11	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic
12	#13095514	—	2	4	no	—
13	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
14	C₂×#11715895	D₄	1	2	yes	—
15	C₂×#11715895	Q₈	1	2	yes	—
16	C₂×#11715895	C₂³	1	1	yes	characteristic
17	C₄⋊Dic₉₇	—	2	1	no	—
18	C₂²⋊Dic₉₇	—	2	1	no	—
19	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
21	M₁₆×C₉₇	C₈	1	4	yes	—
22	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
23	C₂³×C₉₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
26	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₉₇	—	2	2	no	—
29	C₈×C₉₇	C₂×C₈	1	4	yes	—
30	#11715895	C₄⋊C₄	1	2	yes	—
31	#11715895	C₂×D₄	1	1	yes	characteristic
32	#11715895	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₉₇	—	4	1	no	—
34	C₂²×C₉₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₉₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₉₇	C₄⋊C₈	1	2	yes	—
37	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
39	Dic₉₇	—	4	2	no	—
40	C₂×C₉₇	—	2	1	no	—
41	C₂×C₉₇	—	2	2	no	—
42	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
43	#2153	—	97	1	no	maximal, Sylow
44	C₉₇	#2153	1	1	yes	characteristic, complemented, Sylow
45	C₂×M₃₂	—	97	2	no	—
46	#405	—	97	1	no	—
47	#364	—	97	4	no	—
48	C₂×M₁₆	#87604	1	1	yes	characteristic
49	C₂²×C₈	—	97	1	no	—
50	#117	—	97	1	no	—
51	M₃₂	—	97	4	no	—
52	M₃₂	—	194	2	no	—
53	C₂×C₁₆	—	194	2	no	—
54	#97	—	97	2	no	—
55	C₄×C₈	—	97	2	no	—
56	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
57	M₁₆	#11715897	1	4	yes	—
58	C₂×C₈	—	97	2	no	—
59	C₂×C₈	—	97	2	no	—
60	C₂×C₈	—	97	1	no	—
61	C₂×C₈	—	97	1	no	—
62	C₂×C₈	C₂×#87604	1	2	yes	—
63	C₄⋊C₄	—	194	1	no	—
64	C₂²⋊C₄	—	194	1	no	—
65	C₄²	—	97	2	no	—
66	C₁₆	—	194	4	no	—
67	C₂³	#13095521	1	1	yes	characteristic
68	C₂×C₄	—	194	2	no	—
69	C₂×C₄	C₄⋊#87604	1	2	yes	—
70	C₂×C₄	C₂×#11715897	1	2	yes	—
71	C₂×C₄	#13095521	1	1	yes	characteristic
72	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
73	C₈	—	97	1	no	—
74	C₈	—	97	2	no	—
75	C₈	—	97	1	no	—
76	C₈	C₂×#11715897	1	4	yes	—
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊#11715897	1	2	yes	—
79	C₂²	C₂×#13095521	1	1	yes	characteristic
80	C₄	—	388	2	no	—
81	C₄	C₄⋊#11715897	1	2	yes	—
82	C₄	C₂×C₄⋊#87604	1	1	yes	characteristic
83	C₄	C₂²×#11715897	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊#11715897	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂×M₁₆×C₉₇	C₄
socle	C₂×C₉₇	C₂×C₄⋊C₈
prime radical	C₂³×C₉₇	M₁₆
cosocle	C₂×#11715895	C₂³
hypercenter	C₂×M₁₆	#87604
hypocenter	C₉₇	#2153
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20261657	C₂	semidirect
44. C₉₇	#2153	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690884	2
#20261657	2
#20260693	2
#2153	97