Subgroups — #27156949

Order 12416.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20261751	C₂	1	2	yes	maximal
2	#20261577	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11715897	C₄	1	4	yes	—
5	C₂²⋊#11715897	C₄	1	4	yes	—
6	C₄×#11715897	C₂²	1	4	yes	—
7	C₂²×#11715895	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	C₈×Dic₉₇	C₄	1	2	yes	—
10	C₈×Dic₉₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₉₇	C₄	1	1	yes	characteristic
12	C₂×#11715897	C₂×C₄	1	8	yes	—
13	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
14	C₂×#11715895	—	2	2	no	—
15	C₂×#11715895	C₂×C₄	1	2	yes	—
16	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
17	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
18	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₉₇	—	2	2	no	—
21	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
22	C₂³×C₉₇	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
25	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
26	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×Dic₉₇	—	2	4	no	—
28	C₂×Dic₉₇	C₄²	1	2	yes	—
29	C₂×Dic₉₇	C₂²×C₄	1	2	yes	—
30	#11715897	—	2	8	no	—
31	C₈×C₉₇	—	2	2	no	—
32	#11715895	—	2	2	no	—
33	C₂²×C₉₇	—	2	2	no	—
34	C₂²×C₉₇	#97	1	2	yes	—
35	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
36	C₄×C₉₇	#97	1	2	yes	—
37	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
38	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
39	Dic₉₇	—	2	2	no	—
40	Dic₉₇	#131	1	4	yes	—
41	C₂×C₉₇	—	2	2	no	—
42	C₂×C₉₇	C₂×#97	1	1	yes	characteristic
43	C₂×C₉₇	#405	1	2	yes	—
44	#1452	—	97	1	no	maximal, Sylow
45	C₉₇	#1452	1	1	yes	characteristic, complemented, Sylow
46	#431	—	97	2	no	—
47	C₂×C₄×C₈	—	97	1	no	—
48	#322	—	97	4	no	—
49	C₂²×C₈	—	97	1	no	—
50	C₂²×C₈	#87604	1	1	yes	characteristic
51	C₂×C₄²	—	97	1	no	—
52	C₄⋊C₈	—	97	4	no	—
53	C₂²⋊C₈	—	97	4	no	—
54	C₄×C₈	—	97	4	no	—
55	C₄×C₈	—	97	2	no	—
56	C₄×C₈	—	97	2	no	—
57	C₂²×C₄	—	97	2	no	—
58	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	97	8	no	—
61	C₂×C₈	—	97	2	no	—
62	C₂×C₈	—	194	2	no	—
63	C₂×C₈	C₂×#87604	1	2	yes	—
64	C₄²	—	97	2	no	—
65	C₄²	—	97	1	no	—
66	C₄²	—	97	1	no	—
67	C₂³	C₄×#87604	1	1	yes	characteristic
68	C₂×C₄	—	97	2	no	—
69	C₂×C₄	—	97	2	no	—
70	C₂×C₄	—	194	4	no	—
71	C₂×C₄	C₄×#87604	1	1	yes	characteristic
72	C₂×C₄	#13095520	1	2	yes	—
73	C₂×C₄	#13095521	1	2	yes	—
74	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	194	8	no	—
77	C₈	—	194	2	no	—
78	C₂²	—	2	2	no	—
79	C₂²	#15690865	1	2	yes	—
80	C₂²	C₂×C₄×#87604	1	1	yes	characteristic
81	C₄	—	97	4	no	—
82	C₄	—	194	2	no	—
83	C₄	#15690865	1	2	yes	—
84	C₄	C₂×#13095520	1	1	yes	characteristic
85	C₄	C₂×#13095521	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	C₂×#15690865	1	1	yes	characteristic
88	C₂	#20261698	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₂²×C₈×C₉₇	C₄
prime radical	C₂³×C₉₇	C₄²
cosocle	C₄×Dic₉₇	C₂³
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	#1452
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #20261577	C₂	semidirect
45. C₉₇	#1452	semidirect

Maximal subgroups

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

subgroup	index
#20261751	2
#20261577	2
C₂×C₈×Dic₉₇	2
#1452	97