Group #27135177 — subgroups

Subgroups — #27135177

Order 12352.

238 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂²×D₇₇₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690534	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690532	C₂	1	2	yes	maximal, complemented
4	C₂×#15690532	C₂	1	2	yes	maximal, complemented
5	#95×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₉₃	C₂²	1	2	yes	—
7	C₂×D₇₇₂	—	2	4	no	—
8	C₂×D₇₇₂	—	2	2	no	—
9	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₉₃	C₂²	1	1	yes	characteristic
11	#15690534	—	2	2	no	—
12	#15690532	—	2	4	no	—
13	#15690532	—	2	4	no	—
14	C₂²×C₄×C₁₉₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₁₉₃	C₂²	1	1	yes	characteristic
16	C₂²×D₁₉₃	—	2	2	no	—
17	C₂²×D₁₉₃	—	2	2	no	—
18	C₂²×D₁₉₃	—	2	2	no	—
19	C₂²×D₁₉₃	—	2	2	no	—
20	C₂²×D₁₉₃	—	2	2	no	—
21	C₂²×D₁₉₃	—	2	2	no	—
22	C₂²×D₁₉₃	D₄	1	4	yes	—
23	D₇₇₂	—	2	8	no	—
24	C₂×Dic₁₉₃	—	2	1	no	—
25	C₂×Dic₁₉₃	—	2	1	no	—
26	C₂×Dic₁₉₃	—	2	2	no	—
27	C₂×Dic₁₉₃	—	2	1	no	—
28	C₂×Dic₁₉₃	D₄	1	2	yes	complemented
29	C₂³×C₁₉₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₉₃	—	2	2	no	—
31	C₂×C₄×C₁₉₃	—	2	2	no	—
32	C₂×C₄×C₁₉₃	—	2	2	no	—
33	C₂×C₄×C₁₉₃	—	2	2	no	—
34	C₂×C₄×C₁₉₃	D₄	1	2	yes	—
35	C₂²×C₁₉₃	C₂×D₄	1	2	yes	—
36	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₉₃	#41	1	2	yes	—
39	C₂²×C₁₉₃	#41	1	1	yes	characteristic
40	C₂×D₁₉₃	—	2	4	no	—
41	C₂×D₁₉₃	—	2	4	no	—
42	C₂×D₁₉₃	—	2	4	no	—
43	C₂×D₁₉₃	—	4	2	no	—
44	C₂×D₁₉₃	—	4	2	no	—
45	C₂×D₁₉₃	—	4	2	no	—
46	C₂×D₁₉₃	—	4	2	no	—
47	C₄×C₁₉₃	—	2	2	no	—
48	C₄×C₁₉₃	—	4	2	no	—
49	Dic₁₉₃	—	2	2	no	—
50	Dic₁₉₃	—	4	1	no	—
51	C₂×C₁₉₃	#120	1	1	yes	characteristic
52	C₂×C₁₉₃	#121	1	2	yes	—
53	C₂×C₁₉₃	#121	1	1	yes	characteristic
54	C₂×C₁₉₃	#123	1	2	yes	—
55	C₂×C₁₉₃	#124	1	1	yes	characteristic
56	D₁₉₃	—	4	4	no	—
57	C₁₉₃	#394	1	1	yes	characteristic, complemented, Sylow
58	#394	—	193	1	no	maximal, Sylow
59	C₂²×D₄	—	193	1	no	—
60	C₂×C₄⋊C₄	—	193	1	no	—
61	C₂×C₂²⋊C₄	—	193	2	no	—
62	C₂×C₂²⋊C₄	—	193	2	no	—
63	#95	D₁₉₃	1	1	yes	characteristic, complemented
64	C₂⁴	—	193	2	no	—
65	C₂×D₄	—	386	4	no	—
66	C₂×D₄	—	386	2	no	—
67	C₂²×C₄	—	193	1	no	—
68	C₂²×C₄	—	193	1	no	—
69	C₂²×C₄	C₂×D₁₉₃	1	2	yes	—
70	C₂²×C₄	C₂×D₁₉₃	1	1	yes	characteristic
71	C₄⋊C₄	—	386	2	no	—
72	C₂²⋊C₄	—	386	4	no	—
73	C₂²⋊C₄	—	386	4	no	—
74	C₂³	—	386	2	no	—
75	C₂³	—	386	2	no	—
76	C₂³	—	386	2	no	—
77	C₂³	—	386	2	no	—
78	C₂³	—	386	2	no	—
79	C₂³	—	386	2	no	—
80	C₂³	—	193	4	no	—
81	C₂³	C₂²×D₁₉₃	1	1	yes	characteristic
82	D₄	—	386	8	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	386	2	no	—
88	C₂×C₄	—	386	1	no	—
89	C₂×C₄	—	386	1	no	—
90	C₂×C₄	—	386	1	no	—
91	C₂×C₄	—	193	2	no	—
92	C₂×C₄	D₇₇₂	1	2	yes	—
93	C₂²	—	772	2	no	—
94	C₂²	—	772	2	no	—
95	C₂²	—	772	2	no	—
96	C₂²	—	772	2	no	—
97	C₂²	—	386	4	no	—
98	C₂²	—	386	4	no	—
99	C₂²	—	386	4	no	—
100	C₂²	D₄×D₁₉₃	1	2	yes	—
101	C₂²	D₄×D₁₉₃	1	1	yes	characteristic
102	C₂²	C₂×D₇₇₂	1	1	yes	characteristic
103	C₂²	#15690555	1	2	yes	—
104	C₂²	#15690556	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	4	2	no	—
107	C₄	—	772	1	no	—
108	C₄	—	386	2	no	—
109	C₂	—	772	4	no	—
110	C₂	#20259869	1	1	yes	characteristic
111	C₂	#20259872	1	2	yes	—
112	C₂	#20259874	1	1	yes	characteristic
113	C₂	#20259884	1	2	yes	—
114	C₂	#20259887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₇₇₂

C₂

yes

maximal, characteristic

C₂×#15690534

C₂

yes

maximal, characteristic, complemented

C₂×#15690532

C₂

yes

maximal, complemented

C₂×#15690532

C₂

yes

maximal, complemented

#95×C₁₉₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₉₃

C₂²

yes

—

C₂×D₇₇₂

—

C₂×D₇₇₂

—

C₂²×Dic₁₉₃

C₂²

yes

characteristic

C₂²×Dic₁₉₃

C₂²

yes

—

—

—

C₂²×C₄×C₁₉₃

C₂²

yes

—

C₂²×C₄×C₁₉₃

C₂²

yes

characteristic

C₂²×D₁₉₃

—

C₂²×D₁₉₃

—

C₂²×D₁₉₃

—

C₂²×D₁₉₃

—

C₂²×D₁₉₃

—

C₂²×D₁₉₃

—

C₂²×D₁₉₃

D₄

yes

—

D₇₇₂

—

C₂×Dic₁₉₃

—

C₂×Dic₁₉₃

—

C₂×Dic₁₉₃

—

C₂×Dic₁₉₃

—

C₂×Dic₁₉₃

D₄

yes

complemented

C₂³×C₁₉₃

C₂³

yes

characteristic

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

D₄

yes

—

C₂²×C₁₉₃

C₂×D₄

yes

—

C₂²×C₁₉₃

C₂×D₄

yes

characteristic

C₂²×C₁₉₃

C₂×D₄

yes

characteristic

C₂²×C₁₉₃

#41

yes

—

C₂²×C₁₉₃

#41

yes

characteristic

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₂×D₁₉₃

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

Dic₁₉₃

—

Dic₁₉₃

—

C₂×C₁₉₃

#120

yes

characteristic

C₂×C₁₉₃

#121

yes

—

C₂×C₁₉₃

#121

yes

characteristic

C₂×C₁₉₃

#123

yes

—

C₂×C₁₉₃

#124

yes

characteristic

D₁₉₃

—

C₁₉₃

#394

yes

characteristic, complemented, Sylow

#394

—

193

maximal, Sylow

C₂²×D₄

—

193

—

C₂×C₄⋊C₄

—

193

—

C₂×C₂²⋊C₄

—

193

—

C₂×C₂²⋊C₄

—

193

—

#95

D₁₉₃

yes

characteristic, complemented

C₂⁴

—

193

—

C₂×D₄

—

386

—

C₂×D₄

—

386

—

C₂²×C₄

—

193

—

C₂²×C₄

—

193

—

C₂²×C₄

C₂×D₁₉₃

yes

—

C₂²×C₄

C₂×D₁₉₃

yes

characteristic

C₄⋊C₄

—

386

—

C₂²⋊C₄

—

386

—

C₂²⋊C₄

—

386

—

C₂³

—

386

—

C₂³

—

386

—

C₂³

—

386

—

C₂³

—

386

—

C₂³

—

386

—

C₂³

—

386

—

C₂³

—

193

—

C₂³

C₂²×D₁₉₃

yes

characteristic

D₄

—

386

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

386

—

C₂×C₄

—

386

—

C₂×C₄

—

386

—

C₂×C₄

—

386

—

C₂×C₄

—

193

—

C₂×C₄

D₇₇₂

yes

—

C₂²

—

772

—

C₂²

—

772

—

C₂²

—

772

—

C₂²

—

772

—

C₂²

—

386

—

C₂²

—

386

—

C₂²

—

386

—

100

C₂²

D₄×D₁₉₃

yes

—

101

C₂²

D₄×D₁₉₃

yes

characteristic

102

C₂²

C₂×D₇₇₂

yes

characteristic

103

C₂²

#15690555

yes

—

104

C₂²

#15690556

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

772

—

108

C₄

—

386

—

109

C₂

—

772

—

110

C₂

#20259869

yes

characteristic

111

C₂

#20259872

yes

—

112

C₂

#20259874

yes

characteristic

113

C₂

#20259884

yes

—

114

C₂

#20259887

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₉₃
commutator, socle, cosocle	C₂³×C₁₉₃	C₂³
Fitting	#95×C₁₉₃	C₂
prime radical	C₂²×D₇₇₂	C₂
hypercenter	#95	D₁₉₃
hypocenter	C₁₉₃	#394
LCS	G ▷ C₂³×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₉₃

commutator, socle, cosocle

C₂³×C₁₉₃

C₂³

Fitting

#95×C₁₉₃

C₂

prime radical

C₂²×D₇₇₂

C₂

D₁₉₃

C₁₉₃

G ▷ C₂³×C₁₉₃ ▷ C₁₉₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×#15690534	C₂	semidirect
3. C₂×#15690532	C₂	semidirect
4. C₂×#15690532	C₂	semidirect
5. #95×C₁₉₃	C₂	semidirect
28. C₂×Dic₁₉₃	D₄	semidirect
57. C₁₉₃	#394	semidirect
63. #95	D₁₉₃	semidirect

H ≅ G/N

type

2. C₂×#15690534

C₂

semidirect

3. C₂×#15690532

C₂

semidirect

4. C₂×#15690532

C₂

semidirect

5. #95×C₁₉₃

C₂

semidirect

28. C₂×Dic₁₉₃

D₄

semidirect

57. C₁₉₃

#394

semidirect

63. #95

D₁₉₃

semidirect

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

subgroup	index
C₂²×D₇₇₂	2
C₂×#15690534	2
C₂×#15690532	2
#95×C₁₉₃	2
#394	193

subgroup

index

C₂²×D₇₇₂

C₂×#15690534

C₂×#15690532

#95×C₁₉₃

#394

193

Subgroups — #27135177

Conjugacy classes of subgroups

Complements

Maximal subgroups