Group #27141365 — subgroups

Subgroups — #27141365

Order 12416.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20261112	C₂	1	1	yes	maximal, characteristic
2	#20261109	C₂	1	2	yes	maximal, complemented
3	#20261083	C₂	1	2	yes	maximal, complemented
4	#20260456	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095498	—	2	4	no	—
7	C₂×D₇₇₆	—	2	2	no	—
8	#15690807	C₂²	1	1	yes	characteristic
9	#15690807	C₂²	1	1	yes	characteristic
10	#15690730	—	2	4	no	—
11	C₄×#11715895	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
13	#106×C₉₇	—	2	2	no	—
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×D₃₈₈	—	2	1	no	—
16	C₂×D₃₈₈	—	2	1	no	—
17	C₂×D₃₈₈	—	2	2	no	—
18	C₂×D₃₈₈	—	2	2	no	—
19	#13095498	—	4	8	no	—
20	D₇₇₆	—	4	4	no	—
21	C₂×#11715895	D₄	1	4	yes	—
22	C₂×Q₈×C₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	D₄	1	2	yes	—
24	C₄⋊C₄×C₉₇	—	2	2	no	—
25	C₄⋊C₄×C₉₇	—	2	2	no	—
26	C₄²×C₉₇	C₂³	1	1	yes	characteristic
27	C₂²×D₉₇	—	4	1	no	—
28	C₂²×D₉₇	—	4	1	no	—
29	Q₈×C₉₇	—	4	4	no	—
30	C₂×C₄×C₉₇	—	4	2	no	—
31	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
33	D₃₈₈	—	4	2	no	—
34	D₃₈₈	—	4	2	no	—
35	D₃₈₈	—	4	4	no	—
36	D₃₈₈	—	4	4	no	—
37	C₈×C₉₇	SD₁₆	1	4	yes	complemented
38	#11715895	—	2	4	no	—
39	C₂²×C₉₇	#127	1	1	yes	characteristic
40	C₂×D₉₇	—	8	1	no	—
41	C₂×D₉₇	—	8	1	no	—
42	C₂×D₉₇	—	8	2	no	—
43	C₂×D₉₇	—	8	2	no	—
44	C₄×C₉₇	—	8	2	no	—
45	C₄×C₉₇	C₂×SD₁₆	1	2	yes	—
46	C₄×C₉₇	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₉₇	#492	1	1	yes	characteristic
48	C₂×C₉₇	#496	1	2	yes	—
49	D₉₇	—	16	1	no	—
50	D₉₇	—	16	1	no	—
51	#1715	—	97	1	no	maximal, Sylow
52	C₉₇	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	D₉₇	1	1	yes	characteristic, complemented
54	#493	—	97	1	no	—
55	#492	—	97	2	no	—
56	#486	—	97	2	no	—
57	#322	—	97	1	no	—
58	C₂×SD₁₆	—	194	4	no	—
59	C₂×D₈	—	194	2	no	—
60	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
61	#127	—	97	1	no	—
62	#127	—	97	1	no	—
63	#106	—	2	2	no	—
64	#102	—	194	4	no	—
65	C₄×C₈	—	97	2	no	—
66	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	194	2	no	—
69	C₂×D₄	—	194	2	no	—
70	C₂×D₄	—	194	1	no	—
71	C₂×D₄	—	194	1	no	—
72	SD₁₆	—	388	8	no	—
73	D₈	—	388	4	no	—
74	C₂×C₈	—	97	4	no	—
75	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₉₇	1	1	yes	characteristic
79	C₂³	—	388	1	no	—
80	C₂³	—	388	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	388	2	no	—
83	D₄	—	388	2	no	—
84	D₄	—	388	4	no	—
85	D₄	—	388	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₉₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
89	C₈	—	194	4	no	—
90	C₈	#13095498	1	4	yes	complemented
91	C₂²	—	776	2	no	—
92	C₂²	—	776	2	no	—
93	C₂²	—	776	1	no	—
94	C₂²	—	776	1	no	—
95	C₂²	#15690806	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#13095498	1	2	yes	—
98	C₄	#15690837	1	4	yes	—
99	C₂	—	1552	1	no	—
100	C₂	—	1552	1	no	—
101	C₂	#20261109	1	1	yes	characteristic
102	C₂	#20261121	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261112

C₂

yes

maximal, characteristic

#20261109

C₂

yes

maximal, complemented

#20261083

C₂

yes

maximal, complemented

#20260456

C₂

yes

maximal, characteristic, complemented

#498×C₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#13095498

—

C₂×D₇₇₆

—

#15690807

C₂²

yes

characteristic

#15690807

C₂²

yes

characteristic

#15690730

—

C₄×#11715895

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

#106×C₉₇

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

C₂×D₃₈₈

—

#13095498

—

D₇₇₆

—

C₂×#11715895

D₄

yes

—

C₂×Q₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂²×D₉₇

—

C₂²×D₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

D₃₈₈

—

C₈×C₉₇

SD₁₆

yes

complemented

#11715895

—

C₂²×C₉₇

#127

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₂×D₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×SD₁₆

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₂×C₉₇

#492

yes

characteristic

C₂×C₉₇

#496

yes

—

D₉₇

—

D₉₇

—

#1715

—

maximal, Sylow

C₉₇

#1715

yes

characteristic, complemented, Sylow

#498

D₉₇

yes

characteristic, complemented

—

—

—

—

C₂×SD₁₆

—

194

—

C₂×D₈

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

—

—

—

—

—

194

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

C₂×D₄

—

194

—

SD₁₆

—

388

—

D₈

—

388

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

—

388

—

C₂³

—

388

—

Q₈

—

D₄

—

388

—

D₄

—

388

—

D₄

—

388

—

D₄

—

388

—

C₂×C₄

—

C₂×C₄

D₄×D₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₈

—

194

—

C₈

#13095498

yes

complemented

C₂²

—

776

—

C₂²

—

776

—

C₂²

—

776

—

C₂²

—

776

—

C₂²

#15690806

yes

characteristic

C₄

—

C₄

C₂×#13095498

yes

—

C₄

#15690837

yes

—

C₂

—

1552

—

100

C₂

—

1552

—

101

C₂

#20261109

yes

characteristic

102

C₂

#20261121

yes

—

	subgroup	G/H
center	C₂²	#15690806
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#498×C₉₇	C₂
socle	C₂²×C₉₇	#127
prime radical	#20261112	C₂
hypercenter	#498	D₉₇
hypocenter	C₉₇	#1715
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#15690806

commutator, cosocle

C₄²×C₉₇

C₂³

Frattini

C₄²

C₂²×D₉₇

Fitting

#498×C₉₇

C₂

C₂²×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #20261109	C₂	semidirect
3. #20261083	C₂	semidirect
4. #20260456	C₂	semidirect
5. #498×C₉₇	C₂	semidirect
37. C₈×C₉₇	SD₁₆	semidirect
52. C₉₇	#1715	semidirect
53. #498	D₉₇	semidirect
90. C₈	#13095498	semidirect

H ≅ G/N

type

2. #20261109

C₂

semidirect

3. #20261083

C₂

semidirect

4. #20260456

C₂

semidirect

5. #498×C₉₇

C₂

semidirect

37. C₈×C₉₇

SD₁₆

semidirect

52. C₉₇

#1715

semidirect

53. #498

D₉₇

semidirect

90. C₈

#13095498

semidirect

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

subgroup	index
#20261112	2
#20261109	2
#20261083	2
#20260456	2
#498×C₉₇	2
#1715	97

subgroup

index

#498×C₉₇

Subgroups — #27141365

Conjugacy classes of subgroups

Complements

Maximal subgroups