Group #27141892 — subgroups

Subgroups — #27141892

Order 12416.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#15690741	C₂	1	4	yes	maximal
2	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
3	#20260605	C₂	1	1	yes	maximal, characteristic
4	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	—	2	2	no	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	#15690741	—	2	8	no	—
9	C₄×#11715895	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₉₇	—	2	2	no	—
14	C₂×#11715895	C₂×C₄	1	4	yes	—
15	C₂×#11715895	D₄	1	4	yes	—
16	C₂×#11715895	Q₈	1	4	yes	—
17	C₄⋊Dic₉₇	—	2	4	no	—
18	C₄⋊Dic₉₇	—	4	4	no	—
19	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₉₇	—	2	4	no	—
23	C₄⋊C₄×C₉₇	—	4	4	no	—
24	C₄²×C₉₇	C₂×C₄	1	2	yes	—
25	C₄²×C₉₇	Q₈	1	2	yes	—
26	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	—	4	2	no	—
28	C₂×C₄×C₉₇	—	4	4	no	—
29	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
31	C₂×C₄×C₉₇	SD₁₆	1	4	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×Dic₉₇	—	4	2	no	—
37	C₂×Dic₉₇	—	4	4	no	—
38	#11715895	C₄⋊C₄	1	8	yes	complemented
39	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₉₇	#127	1	1	yes	characteristic
41	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
43	C₂²×C₉₇	C₂×SD₁₆	1	2	yes	—
44	C₄×C₉₇	—	8	2	no	—
45	C₄×C₉₇	#106	1	8	yes	complemented
46	C₄×C₉₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₄×C₉₇	C₄⋊Q₈	1	2	yes	—
48	Dic₉₇	—	8	2	no	—
49	C₂×C₉₇	#384	1	1	yes	characteristic
50	C₂×C₉₇	C₂×#106	1	2	yes	—
51	C₂×C₉₇	#492	1	1	yes	characteristic
52	C₂×C₉₇	#492	1	1	yes	characteristic
53	C₂×C₉₇	#498	1	2	yes	—
54	#1844	—	97	1	no	maximal, Sylow
55	C₉₇	#1844	1	1	yes	characteristic, complemented, Sylow
56	C₂×#106	—	97	4	no	—
57	C₂×C₄×C₈	—	97	1	no	—
58	#384	—	97	1	no	—
59	#384	D₉₇	1	1	yes	characteristic
60	C₂²×C₈	—	97	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	97	2	no	—
63	C₂×C₄⋊C₄	—	194	2	no	—
64	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
65	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
66	#106	—	194	8	no	—
67	C₄×C₈	—	97	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	194	2	no	—
70	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
71	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
72	C₂×C₈	—	97	4	no	—
73	C₂×C₈	—	97	4	no	—
74	C₂×C₈	—	97	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	388	4	no	—
78	C₄⋊C₄	—	194	4	no	—
79	C₄²	Dic₁₉₄	1	2	yes	—
80	C₄²	C₄×D₉₇	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	388	2	no	—
85	C₂×C₄	—	388	4	no	—
86	C₂×C₄	#13095490	1	4	yes	—
87	C₂×C₄	#13095498	1	4	yes	—
88	C₂×C₄	#13095499	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
90	C₂×C₄	D₄×D₉₇	1	2	yes	—
91	C₂×C₄	Q₈×D₉₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
93	C₈	—	97	8	no	—
94	C₂²	C₂×#13095490	1	1	yes	characteristic
95	C₂²	#15690806	1	1	yes	characteristic
96	C₂²	#15690808	1	1	yes	characteristic
97	C₂²	C₂×#13095498	1	2	yes	—
98	C₂²	C₂×#13095499	1	2	yes	—
99	C₄	—	8	2	no	—
100	C₄	—	776	2	no	—
101	C₄	#15690741	1	8	yes	complemented
102	C₄	C₄⋊C₄×D₉₇	1	2	yes	—
103	C₄	#15690809	1	2	yes	—
104	C₂	#20260604	1	1	yes	characteristic
105	C₂	C₂×#15690741	1	2	yes	—
106	C₂	#20261107	1	1	yes	characteristic
107	C₂	#20261109	1	1	yes	characteristic
108	C₂	#20261131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690741

C₂

yes

maximal

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260605

C₂

yes

maximal, characteristic

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690741

—

C₄×#11715895

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

D₄

yes

—

C₂×#11715895

Q₈

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

C₄⋊C₄

yes

complemented

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

#127

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₂²×C₉₇

C₂×SD₁₆

yes

—

C₄×C₉₇

—

C₄×C₉₇

#106

yes

complemented

C₄×C₉₇

C₂×C₄⋊C₄

yes

—

C₄×C₉₇

C₄⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

#384

yes

characteristic

C₂×C₉₇

C₂×#106

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#1844

—

maximal, Sylow

C₉₇

#1844

yes

characteristic, complemented, Sylow

C₂×#106

—

C₂×C₄×C₈

—

#384

—

#384

D₉₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#106

—

194

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₄×D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

#13095490

yes

—

C₂×C₄

#13095498

yes

—

C₂×C₄

#13095499

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

D₄×D₉₇

yes

—

C₂×C₄

Q₈×D₉₇

yes

—

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#13095499

yes

—

C₄

—

100

C₄

—

776

—

101

C₄

#15690741

yes

complemented

102

C₄

C₄⋊C₄×D₉₇

yes

—

103

C₄

#15690809

yes

—

104

C₂

#20260604

yes

characteristic

105

C₂

C₂×#15690741

yes

—

106

C₂

#20261107

yes

characteristic

107

C₂

#20261109

yes

characteristic

108

C₂

#20261131

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#384	D₉₇
hypocenter	C₉₇	#1844
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11715895	C₄	semidirect
38. #11715895	C₄⋊C₄	semidirect
45. C₄×C₉₇	#106	semidirect
55. C₉₇	#1844	semidirect
101. C₄	#15690741	semidirect

H ≅ G/N

type

9. C₄×#11715895

C₄

semidirect

38. #11715895

C₄⋊C₄

semidirect

45. C₄×C₉₇

#106

semidirect

55. C₉₇

#1844

semidirect

101. C₄

#15690741

semidirect

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

subgroup	index
C₂×#15690741	2
C₂×C₄×#11715895	2
#20260605	2
#384×C₉₇	2
#1844	97

subgroup

index

#384×C₉₇

Subgroups — #27141892

Conjugacy classes of subgroups

Complements

Maximal subgroups