Subgroups — #27158781

Order 12416.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261838	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261829	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261769	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095523	—	2	1	no	—
5	C₂×#13095523	—	2	1	no	—
6	C₂×#13095521	—	2	1	no	—
7	C₄⋊#11715897	C₄	1	1	yes	characteristic
8	#15690867	—	2	1	no	—
9	C₄×#11715897	C₂²	1	1	yes	characteristic
10	#15690772	C₄	1	1	yes	characteristic
11	#13095523	—	2	2	no	—
12	#13095523	—	4	1	no	—
13	#13095521	—	4	1	no	—
14	#13095521	—	4	1	no	—
15	C₂×#11715897	—	2	1	no	—
16	C₂×#11715897	D₄	1	1	yes	characteristic
17	C₂×#11715897	D₄	1	1	yes	characteristic
18	C₂×D₃₈₈	C₈	1	1	yes	characteristic
19	C₂×C₄×D₉₇	—	2	1	no	—
20	#13095488	—	2	1	no	—
21	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₉₇	C₈	1	1	yes	characteristic
23	C₂²×D₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	—	2	1	no	—
25	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
26	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
28	D₃₈₈	—	4	1	no	—
29	C₄×D₉₇	—	4	1	no	—
30	C₄×D₉₇	—	4	1	no	—
31	#11715897	—	2	1	no	—
32	#11715897	—	4	1	no	—
33	#11715897	D₈	1	1	yes	characteristic
34	#11715897	SD₁₆	1	1	yes	characteristic
35	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×D₉₇	—	4	1	no	—
37	C₂×D₉₇	—	4	1	no	—
38	C₂×D₉₇	—	4	1	no	—
39	C₄×C₉₇	—	2	1	no	—
40	C₄×C₉₇	—	4	1	no	—
41	Dic₉₇	—	2	1	no	—
42	Dic₉₇	#102	1	1	yes	characteristic
43	Dic₉₇	#104	1	1	yes	characteristic
44	C₂×C₉₇	D₄⋊C₈	1	1	yes	characteristic
45	C₂×C₉₇	#349	1	1	yes	characteristic
46	C₂×C₉₇	#350	1	1	yes	characteristic
47	D₉₇	—	8	1	no	—
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	—	97	1	no	—
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	—	97	1	no	—
61	C₂²×C₄	—	194	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	#11715906	1	1	yes	characteristic
68	C₂²⋊C₄	—	194	1	no	—
69	C₄²	—	97	1	no	—
70	C₁₆	—	388	1	no	—
71	C₁₆	—	194	2	no	—
72	C₂³	—	194	1	no	—
73	D₄	—	388	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	97	1	no	—
76	C₂×C₄	—	97	1	no	—
77	C₂×C₄	—	388	1	no	—
78	C₂×C₄	—	388	1	no	—
79	C₂×C₄	C₂×#11715906	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	—	388	1	no	—
85	C₂²	—	388	1	no	—
86	C₂²	—	388	1	no	—
87	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	4	1	no	—
90	C₄	—	97	1	no	—
91	C₄	—	97	1	no	—
92	C₄	—	194	1	no	—
93	C₂	—	776	1	no	—
94	C₂	D₄⋊#11715906	1	1	yes	characteristic
95	C₂	#20261834	1	1	yes	characteristic
96	C₂	#20261836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11715906
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₂×C₄	C₂×#11715906
Fitting	C₄⋊C₄×C₉₇	C₈
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂×D₃₈₈	C₈
cosocle	C₄×#11715897	C₂²
hypercenter	C₄⋊C₄	#11715906
hypocenter	C₉₇	#1330
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #20261838	C₂	semidirect
2. #20261829	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
#20261838	2
#20261829	2
#20261769	2
#1330	97