Subgroups — #27139617

Order 12416.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690721	C₂	1	2	yes	maximal
2	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	—	2	2	no	—
4	#15690721	—	2	2	no	—
5	#15690721	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	#95×C₉₇	—	2	2	no	—
9	C₂×#11715895	—	2	8	no	—
10	C₂×#11715895	—	2	4	no	—
11	C₂²×C₄×C₉₇	—	2	2	no	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₉₇	D₄	1	2	yes	—
14	C₄⋊C₄×C₉₇	—	4	2	no	—
15	C₄²×C₉₇	C₂×C₄	1	2	yes	—
16	C₄²×C₉₇	D₄	1	1	yes	characteristic
17	C₄²×C₉₇	Q₈	1	1	yes	characteristic
18	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₉₇	—	2	1	no	—
20	C₂×C₄×C₉₇	—	2	1	no	—
21	C₂×C₄×C₉₇	—	2	4	no	—
22	C₂×C₄×C₉₇	—	4	2	no	—
23	C₂×C₄×C₉₇	—	4	4	no	—
24	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
28	#11715895	—	2	8	no	—
29	C₂²×C₉₇	#95	1	1	yes	characteristic
30	C₂²×C₉₇	#100	1	1	yes	characteristic
31	C₂²×C₉₇	#101	1	1	yes	characteristic
32	C₂²×C₉₇	#104	1	4	yes	—
33	C₄×C₉₇	—	2	2	no	—
34	C₄×C₉₇	—	2	2	no	—
35	C₄×C₉₇	—	2	1	no	—
36	C₄×C₉₇	—	2	1	no	—
37	C₄×C₉₇	—	8	2	no	—
38	C₂×C₉₇	#329	1	2	yes	—
39	C₂×C₉₇	#330	1	2	yes	—
40	C₂×C₉₇	#339	1	2	yes	—
41	C₂×C₉₇	#343	1	1	yes	characteristic
42	#1295	—	97	1	no	maximal, Sylow
43	C₉₇	#1295	1	1	yes	characteristic, complemented, Sylow
44	C₂×#97	—	97	2	no	—
45	#382	D₉₇	1	1	yes	characteristic
46	C₂²×C₈	—	194	2	no	—
47	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
48	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
49	#97	—	97	4	no	—
50	#97	—	194	2	no	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
55	C₂×C₈	—	194	8	no	—
56	C₂×C₈	—	194	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	Dic₁₉₄	1	1	yes	characteristic
59	C₄²	C₄×D₉₇	1	2	yes	—
60	C₄²	D₃₈₈	1	1	yes	characteristic
61	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
68	C₂×C₄	#13095488	1	2	yes	—
69	C₂×C₄	#13095490	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
71	C₈	—	194	8	no	—
72	C₂²	#15690717	1	1	yes	characteristic
73	C₂²	#15690727	1	1	yes	characteristic
74	C₂²	#15690729	1	1	yes	characteristic
75	C₂²	#15690737	1	4	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	8	2	no	—
81	C₂	#20260475	1	2	yes	—
82	C₂	#20260477	1	2	yes	—
83	C₂	#20260494	1	2	yes	—
84	C₂	#20260504	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₉₇
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	#382×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	#382	D₉₇
hypocenter	C₉₇	#1295
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
5. #15690721	C₄	semidirect
43. C₉₇	#1295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690721	2
#382×C₉₇	2
#1295	97