Subgroups — #27140030

Order 12416.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260786	C₂	1	2	yes	maximal
2	C₄×#13095494	C₂	1	1	yes	maximal, characteristic
3	#20260457	C₂	1	2	yes	maximal, complemented
4	#20260456	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈²×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095494	C₄	1	2	yes	—
7	C₄²×D₉₇	C₂²	1	1	yes	characteristic
8	#15690739	C₄	1	4	yes	—
9	#15690722	C₄	1	4	yes	—
10	#15690720	C₄	1	2	yes	—
11	C₄×#11715895	C₂²	1	1	yes	characteristic
12	C₄×#11715895	C₂²	1	2	yes	—
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₉₇	C₂×C₄	1	2	yes	—
17	#13095494	—	2	4	no	—
18	C₂×#11715895	C₂×C₄	1	2	yes	—
19	C₂×#11715895	C₂×C₄	1	4	yes	—
20	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₉₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
24	C₄²×C₉₇	C₂³	1	1	yes	characteristic
25	C₂²×D₉₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
28	C₂×Dic₉₇	C₄²	1	1	yes	characteristic
29	C₂×Dic₉₇	C₄²	1	2	yes	—
30	C₄×D₉₇	—	2	2	no	—
31	C₄×D₉₇	—	2	4	no	—
32	C₈×C₉₇	—	2	4	no	—
33	C₈×C₉₇	M₁₆	1	4	yes	—
34	#11715895	—	2	2	no	—
35	#11715895	—	2	4	no	—
36	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₉₇	—	2	1	no	—
38	C₂×D₉₇	—	2	2	no	—
39	C₄×C₉₇	C₂×M₁₆	1	2	yes	—
40	C₄×C₉₇	#131	1	4	yes	—
41	Dic₉₇	—	2	1	no	—
42	Dic₉₇	—	2	2	no	—
43	C₂×C₉₇	C₄×M₁₆	1	1	yes	characteristic
44	C₂×C₉₇	#405	1	2	yes	—
45	D₉₇	—	4	1	no	—
46	#1453	—	97	1	no	maximal, Sylow
47	C₉₇	#1453	1	1	yes	characteristic, complemented, Sylow
48	#431	—	97	2	no	—
49	C₄×M₁₆	—	97	1	no	—
50	#322	—	97	1	no	—
51	#322	—	97	2	no	—
52	C₈²	D₉₇	1	1	yes	characteristic, complemented
53	C₂×M₁₆	—	97	2	no	—
54	C₂×C₄²	—	97	1	no	—
55	C₄⋊C₈	—	97	4	no	—
56	C₂²⋊C₈	—	97	4	no	—
57	#97	—	97	2	no	—
58	C₄×C₈	—	97	1	no	—
59	C₄×C₈	—	97	2	no	—
60	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
61	C₄×C₈	C₂×D₉₇	1	2	yes	—
62	C₂²×C₄	—	97	2	no	—
63	C₂²×C₄	—	97	1	no	—
64	M₁₆	—	194	4	no	—
65	C₂×C₈	—	97	2	no	—
66	C₂×C₈	—	97	4	no	—
67	C₂×C₈	C₄×D₉₇	1	4	yes	—
68	C₂×C₈	C₄×D₉₇	1	2	yes	—
69	C₄²	—	97	1	no	—
70	C₄²	—	97	2	no	—
71	C₄²	C₂²×D₉₇	1	1	yes	characteristic
72	C₂³	—	97	1	no	—
73	C₂×C₄	—	97	1	no	—
74	C₂×C₄	—	97	2	no	—
75	C₂×C₄	—	194	2	no	—
76	C₂×C₄	—	194	4	no	—
77	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	—	194	2	no	—
81	C₈	—	194	4	no	—
82	C₈	#13095494	1	4	yes	—
83	C₂²	—	194	1	no	—
84	C₂²	—	194	2	no	—
85	C₂²	C₄²×D₉₇	1	1	yes	characteristic
86	C₄	—	194	1	no	—
87	C₄	—	194	2	no	—
88	C₄	C₂×#13095494	1	2	yes	—
89	C₄	#15690816	1	4	yes	—
90	C₂	—	388	1	no	—
91	C₂	C₄×#13095494	1	1	yes	characteristic
92	C₂	#20260689	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₉₇
commutator, socle	C₂²×C₉₇	C₂×C₄²
Fitting	C₈²×C₉₇	C₂
prime radical	C₂²×D₉₇	C₄²
cosocle	C₄²×C₉₇	C₂³
hypercenter	C₈²	D₉₇
hypocenter	C₉₇	#1453
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
3. #20260457	C₂	semidirect
4. #20260456	C₂	semidirect
5. C₈²×C₉₇	C₂	semidirect
47. C₉₇	#1453	semidirect
52. C₈²	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20260786	2
C₄×#13095494	2
#20260457	2
#20260456	2
C₈²×C₉₇	2
#1453	97