Subgroups — #27156688

Order 12416.

100 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, complemented
2	#20261623	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260813	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	—	2	1	no	—
5	C₂×#13095514	—	2	1	no	—
6	C₂×#13095495	—	2	1	no	—
7	C₄⋊#11715895	C₄	1	1	yes	characteristic
8	C₂²⋊#11715895	—	2	1	no	—
9	C₄×#11715895	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₉₇	C₄	1	1	yes	characteristic
11	#13095514	—	2	2	no	—
12	#13095514	—	4	1	no	—
13	#13095495	—	4	1	no	—
14	#13095495	—	4	1	no	—
15	C₂×#11715895	—	2	1	no	—
16	C₂×#11715895	D₄	1	1	yes	characteristic
17	C₂×#11715895	D₄	1	1	yes	characteristic
18	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	—	2	1	no	—
20	C₄⋊C₄×C₉₇	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₉₇	—	2	1	no	—
22	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₉₇	—	2	1	no	—
24	D₄×C₉₇	—	4	1	no	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	—	4	1	no	—
27	C₂×C₄×C₉₇	—	4	1	no	—
28	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₉₇	M₁₆	1	1	yes	characteristic
31	#11715895	—	2	1	no	—
32	#11715895	—	4	1	no	—
33	#11715895	D₈	1	1	yes	characteristic
34	#11715895	SD₁₆	1	1	yes	characteristic
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	—	4	1	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	—	2	1	no	—
41	C₄×C₉₇	—	4	1	no	—
42	C₄×C₉₇	#102	1	1	yes	characteristic
43	C₄×C₉₇	#104	1	1	yes	characteristic
44	C₂×C₉₇	—	8	1	no	—
45	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₉₇	#349	1	1	yes	characteristic
47	C₂×C₉₇	#350	1	1	yes	characteristic
48	#1330	—	97	1	no	maximal, Sylow
49	C₉₇	#1330	1	1	yes	characteristic, complemented, Sylow
50	#436	—	97	1	no	—
51	C₄⋊C₁₆	—	97	1	no	—
52	#346	—	97	1	no	—
53	C₂×M₁₆	—	194	1	no	—
54	C₄×D₄	#87604	1	1	yes	characteristic
55	C₂×C₁₆	—	194	1	no	—
56	C₂×C₁₆	—	194	1	no	—
57	C₄⋊C₈	—	97	1	no	—
58	C₂²⋊C₈	—	194	1	no	—
59	C₄×C₈	—	97	1	no	—
60	C₂×D₄	#11715897	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	388	1	no	—
63	M₁₆	—	388	1	no	—
64	C₂×C₈	—	97	1	no	—
65	C₂×C₈	—	97	1	no	—
66	C₂×C₈	—	194	1	no	—
67	C₄⋊C₄	#11715897	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×#87604	1	1	yes	characteristic
70	C₁₆	—	388	1	no	—
71	C₁₆	—	194	2	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊#87604	1	1	yes	characteristic
78	C₂×C₄	C₂×#11715897	1	1	yes	characteristic
79	C₂×C₄	#13095520	1	1	yes	characteristic
80	C₈	—	97	1	no	—
81	C₈	—	97	1	no	—
82	C₈	—	388	1	no	—
83	C₈	—	194	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊#11715897	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#15690872	1	1	yes	characteristic
92	C₄	#15690874	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊#11715897	1	1	yes	characteristic
95	C₂	#20261631	1	1	yes	characteristic
96	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×D₄×C₉₇	C₄
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂×D₄×C₉₇	C₈
cosocle	C₄×#11715895	C₂²
hypercenter	C₄×D₄	#87604
hypocenter	C₉₇	#1330
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13095514	C₂	semidirect
2. #20261623	C₂	semidirect
49. C₉₇	#1330	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
#20260813	2
#1330	97