Group #27139619 — subgroups

Subgroups — #27139619

Order 12416.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄⋊#11715895	C₂	1	2	yes	maximal
2	#384×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	—	2	2	no	—
4	C₄⋊#11715895	—	2	2	no	—
5	C₄⋊#11715895	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
7	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
8	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×#11715895	—	2	4	no	—
10	C₂×#11715895	—	4	2	no	—
11	C₂×#11715895	—	4	2	no	—
12	C₂²×C₄×C₉₇	—	2	2	no	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₉₇	D₄	1	2	yes	—
15	C₄⋊C₄×C₉₇	—	4	2	no	—
16	C₄⋊C₄×C₉₇	—	4	2	no	—
17	C₄⋊C₄×C₉₇	—	4	2	no	—
18	C₄²×C₉₇	C₂×C₄	1	2	yes	—
19	C₄²×C₉₇	D₄	1	1	yes	characteristic
20	C₄²×C₉₇	Q₈	1	1	yes	characteristic
21	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₉₇	—	2	1	no	—
23	C₂×C₄×C₉₇	—	2	1	no	—
24	C₂×C₄×C₉₇	—	4	2	no	—
25	C₂×C₄×C₉₇	—	4	4	no	—
26	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	D₈	1	2	yes	—
31	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
32	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
33	C₂×C₄×C₉₇	Q₁₆	1	2	yes	—
34	#11715895	—	4	4	no	—
35	C₂²×C₉₇	#95	1	1	yes	characteristic
36	C₂²×C₉₇	#100	1	1	yes	characteristic
37	C₂²×C₉₇	#101	1	1	yes	characteristic
38	C₂²×C₉₇	#102	1	2	yes	—
39	C₂²×C₉₇	#103	1	2	yes	—
40	C₄×C₉₇	—	2	1	no	—
41	C₄×C₉₇	—	2	1	no	—
42	C₄×C₉₇	—	8	2	no	—
43	C₄×C₉₇	#102	1	2	yes	—
44	C₄×C₉₇	#103	1	2	yes	—
45	C₄×C₉₇	#106	1	2	yes	—
46	C₄×C₉₇	#107	1	2	yes	—
47	C₂×C₉₇	#331	1	1	yes	characteristic
48	C₂×C₉₇	#332	1	2	yes	—
49	C₂×C₉₇	#333	1	1	yes	characteristic
50	C₂×C₉₇	#340	1	2	yes	—
51	C₂×C₉₇	#343	1	1	yes	characteristic
52	#1296	—	97	1	no	maximal, Sylow
53	C₉₇	#1296	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄⋊C₈	—	97	2	no	—
55	#384	D₉₇	1	1	yes	characteristic
56	C₂²×C₈	—	194	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
59	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
60	C₄⋊C₈	—	97	4	no	—
61	C₄⋊C₈	—	194	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
65	C₂×C₈	—	388	2	no	—
66	C₂×C₈	—	388	2	no	—
67	C₂×C₈	—	194	4	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₁₉₄	1	1	yes	characteristic
72	C₄²	C₄×D₉₇	1	2	yes	—
73	C₄²	D₃₈₈	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
80	C₂×C₄	#13095488	1	2	yes	—
81	C₂×C₄	#13095490	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
83	C₂×C₄	#13095496	1	2	yes	—
84	C₂×C₄	#13095498	1	2	yes	—
85	C₂×C₄	#13095499	1	2	yes	—
86	C₂×C₄	#13095501	1	2	yes	—
87	C₈	—	388	4	no	—
88	C₂²	#15690717	1	1	yes	characteristic
89	C₂²	#15690727	1	1	yes	characteristic
90	C₂²	#15690729	1	1	yes	characteristic
91	C₂²	#15690731	1	2	yes	—
92	C₂²	#15690734	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#15690730	1	2	yes	—
97	C₄	#15690733	1	2	yes	—
98	C₄	#15690741	1	2	yes	—
99	C₄	#15690743	1	2	yes	—
100	C₂	#20260479	1	1	yes	characteristic
101	C₂	#20260481	1	2	yes	—
102	C₂	#20260484	1	1	yes	characteristic
103	C₂	#20260497	1	2	yes	—
104	C₂	#20260504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal

#384×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

—

C₄⋊#11715895

—

C₄⋊#11715895

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

Q₈

yes

characteristic

C₂³×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

D₈

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

—

#11715895

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

#103

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#102

yes

—

C₄×C₉₇

#103

yes

—

C₄×C₉₇

#106

yes

—

C₄×C₉₇

#107

yes

—

C₂×C₉₇

#331

yes

characteristic

C₂×C₉₇

#332

yes

—

C₂×C₉₇

#333

yes

characteristic

C₂×C₉₇

#340

yes

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

#384

D₉₇

yes

characteristic

C₂²×C₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂×C₈

—

388

—

C₂×C₈

—

388

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₁₉₄

yes

characteristic

C₄²

C₄×D₉₇

yes

—

C₄²

D₃₈₈

yes

characteristic

C₂³

C₂²⋊Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

characteristic

C₂×C₄

#13095488

yes

—

C₂×C₄

#13095490

yes

—

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

#13095496

yes

—

C₂×C₄

#13095498

yes

—

C₂×C₄

#13095499

yes

—

C₂×C₄

#13095501

yes

—

C₈

—

388

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#15690734

yes

—

C₄

—

C₄

—

C₄

—

C₄

#15690730

yes

—

C₄

#15690733

yes

—

C₄

#15690741

yes

—

C₄

#15690743

yes

—

100

C₂

#20260479

yes

characteristic

101

C₂

#20260481

yes

—

102

C₂

#20260484

yes

characteristic

103

C₂

#20260497

yes

—

104

C₂

#20260504

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₉₇
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	#384×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	#384	D₉₇
hypocenter	C₉₇	#1296
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₉₇

commutator

C₂×C₄×C₉₇

C₄²

Frattini

C₂×C₄²

C₂×D₉₇

Fitting

#384×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂²⋊C₄

cosocle

C₂×C₄²×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊#11715895	C₄	semidirect
53. C₉₇	#1296	semidirect

H ≅ G/N

type

5. C₄⋊#11715895

C₄

semidirect

53. C₉₇

#1296

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
#384×C₉₇	2
#1296	97

subgroup

index

C₂×C₄⋊#11715895

#384×C₉₇

#1296

Subgroups — #27139619

Conjugacy classes of subgroups

Complements

Maximal subgroups