Group #27150005 — subgroups

Subgroups — #27150005

Order 12416.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20260674	C₂	1	3	yes	maximal
2	#20260668	C₂	1	6	yes	maximal
3	#20260655	C₂	1	3	yes	maximal
4	#20260603	C₂	1	2	yes	maximal
5	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₉₇	C₂²	1	3	yes	—
8	C₂×#13095490	C₂²	1	6	yes	—
9	#15690717	C₂²	1	6	yes	—
10	#15690717	C₂²	1	3	yes	—
11	#15690717	C₂²	1	3	yes	—
12	#15690717	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₉₇	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₉₇	C₂²	1	3	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₉₄	—	4	3	no	—
17	C₂²×Dic₉₇	C₂³	1	2	yes	—
18	C₂²×Dic₉₇	C₂³	1	6	yes	—
19	C₄⋊Dic₉₇	—	4	3	no	—
20	#13095490	—	4	6	no	—
21	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₉₇	C₂³	1	3	yes	—
24	C₄⋊C₄×C₉₇	—	4	3	no	—
25	C₄⋊C₄×C₉₇	—	4	3	no	—
26	C₄²×C₉₇	Q₈	1	4	yes	complemented
27	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	3	no	—
29	C₂×C₄×C₉₇	—	2	3	no	—
30	C₂×C₄×C₉₇	—	2	6	no	—
31	C₂×C₄×C₉₇	—	2	3	no	—
32	C₂×C₄×C₉₇	—	2	3	no	—
33	C₂×C₄×C₉₇	C₂×Q₈	1	6	yes	—
34	C₂×Dic₉₇	—	2	6	no	—
35	C₂×Dic₉₇	—	2	6	no	—
36	C₂×Dic₉₇	—	2	12	no	—
37	Dic₁₉₄	—	4	4	no	—
38	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₉₇	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₉₇	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₉₇	—	2	6	no	—
42	C₄×C₉₇	—	4	1	no	—
43	C₄×C₉₇	—	4	3	no	—
44	Dic₉₇	—	4	2	no	—
45	Dic₉₇	—	4	6	no	—
46	C₂×C₉₇	#544	1	3	yes	—
47	C₂×C₉₇	#560	1	1	yes	characteristic
48	C₂×C₉₇	#563	1	3	yes	—
49	#2843	—	97	1	no	maximal, Sylow
50	C₉₇	#2843	1	1	yes	characteristic, complemented, Sylow
51	#400	—	97	3	no	—
52	#398	—	97	6	no	—
53	#395	—	97	3	no	—
54	#384	D₉₇	1	1	yes	characteristic
55	#383	—	97	2	no	—
56	C₂²×Q₈	—	97	1	no	—
57	C₂×C₄⋊C₄	—	97	6	no	—
58	C₂×C₄⋊C₄	—	97	3	no	—
59	C₂×C₄⋊C₄	C₂×D₉₇	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₉₇	1	3	yes	—
61	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
62	#95	—	97	6	no	—
63	#95	—	97	6	no	—
64	#95	—	97	3	no	—
65	#95	—	97	3	no	—
66	C₂×Q₈	—	388	3	no	—
67	C₂²×C₄	—	97	6	no	—
68	C₂²×C₄	—	97	2	no	—
69	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
71	C₂²×C₄	C₂²×D₉₇	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	388	6	no	—
75	C₄⋊C₄	—	388	3	no	—
76	C₄²	Dic₁₉₄	1	4	yes	complemented
77	C₂³	C₂³×D₉₇	1	1	yes	characteristic
78	Q₈	—	388	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	194	6	no	—
85	C₂×C₄	—	194	6	no	—
86	C₂×C₄	—	194	12	no	—
87	C₂×C₄	C₂×Dic₁₉₄	1	6	yes	—
88	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
89	C₂²	#15690855	1	3	yes	—
90	C₂²	#15690858	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	388	2	no	—
95	C₄	—	388	6	no	—
96	C₂	#20261366	1	3	yes	—
97	C₂	#20261472	1	1	yes	characteristic
98	C₂	#20261484	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260674

C₂

yes

maximal

#20260668

C₂

yes

maximal

#20260655

C₂

yes

maximal

#20260603

C₂

yes

maximal

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095490

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

Q₈

yes

complemented

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

C₂²×C₉₇

C₂²×Q₈

yes

characteristic

C₂²×C₉₇

H(5,𝔽₂)

yes

—

C₂²×C₉₇

2₋¹⁺⁴

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#544

yes

—

C₂×C₉₇

#560

yes

characteristic

C₂×C₉₇

#563

yes

—

#2843

—

maximal, Sylow

C₉₇

#2843

yes

characteristic, complemented, Sylow

—

—

—

D₉₇

yes

characteristic

#383

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

—

—

—

—

C₂×Q₈

—

388

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

388

—

C₄²

Dic₁₉₄

yes

complemented

C₂³

C₂³×D₉₇

yes

characteristic

Q₈

—

388

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂×Dic₁₉₄

yes

—

C₂²

C₂²×Dic₁₉₄

yes

characteristic

C₂²

#15690855

yes

—

C₂²

#15690858

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

388

—

C₄

—

388

—

C₂

#20261366

yes

—

C₂

#20261472

yes

characteristic

C₂

#20261484

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#384×C₉₇	C₂
hypercenter	#384	D₉₇
hypocenter	C₉₇	#2843
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, prime radical, cosocle

C₂³×C₉₇

C₂⁴

Fitting

#384×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₉₇	Q₈	semidirect
50. C₉₇	#2843	semidirect
76. C₄²	Dic₁₉₄	semidirect

H ≅ G/N

type

26. C₄²×C₉₇

Q₈

semidirect

50. C₉₇

#2843

semidirect

76. C₄²

Dic₁₉₄

semidirect

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

subgroup	index
#20260674	2
#20260668	2
#20260655	2
#20260603	2
#384×C₉₇	2
#2843	97

subgroup

index

#384×C₉₇

Subgroups — #27150005

Conjugacy classes of subgroups

Complements

Maximal subgroups