Group #27194287 — subgroups

Subgroups — #27194287

Order 13184.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20268507	C₂	1	1	yes	maximal, characteristic
2	#20268078	C₂	1	2	yes	maximal, complemented
3	#20267540	C₂	1	1	yes	maximal, characteristic, complemented
4	#20267520	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15693010	—	2	2	no	—
7	#15692996	—	2	2	no	—
8	C₄×D₄₁₂	C₂²	1	2	yes	—
9	#15692979	—	2	2	no	—
10	#15692952	—	2	2	no	—
11	C₄⋊(C₁₀₃⋊C₈)	C₂²	1	2	yes	—
12	#15692941	—	2	2	no	—
13	C₂×Q₁₆×C₁₀₃	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
17	C₂×D₄₁₂	—	2	2	no	—
18	C₂×D₄₁₂	D₄	1	2	yes	—
19	C₂×C₄×D₁₀₃	—	2	2	no	—
20	C₂×C₁₀₃⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₃	D₄	1	2	yes	—
22	(C₂×D₁₀₃)⋊C₄	—	2	4	no	—
23	(C₂×D₁₀₃)⋊C₄	—	2	2	no	—
24	C₄×Dic₁₀₃	—	2	2	no	—
25	C₂×Q₈×C₁₀₃	—	2	1	no	—
26	C₂×Q₈×C₁₀₃	D₄	1	2	yes	—
27	Q₁₆×C₁₀₃	—	4	4	no	—
28	C₂×C₈×C₁₀₃	—	2	1	no	—
29	C₄⋊C₄×C₁₀₃	—	2	2	no	—
30	C₄⋊C₄×C₁₀₃	—	2	2	no	—
31	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₃	—	2	2	no	—
33	Q₈×C₁₀₃	—	4	2	no	—
34	Q₈×C₁₀₃	—	4	2	no	—
35	C₂×C₄×C₁₀₃	—	2	2	no	—
36	C₂×C₄×C₁₀₃	—	4	1	no	—
37	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₀₃	—	2	2	no	—
40	D₄₁₂	—	2	2	no	—
41	D₄₁₂	—	2	2	no	—
42	D₄₁₂	—	8	2	no	—
43	C₄×D₁₀₃	—	4	2	no	—
44	C₄×D₁₀₃	—	4	2	no	—
45	C₈×C₁₀₃	—	2	2	no	—
46	C₁₀₃⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₃	#120	1	1	yes	characteristic
48	C₂×D₁₀₃	—	4	2	no	—
49	C₂×D₁₀₃	—	4	2	no	—
50	C₂×D₁₀₃	—	4	2	no	—
51	C₄×C₁₀₃	—	2	1	no	—
52	C₄×C₁₀₃	—	4	2	no	—
53	C₄×C₁₀₃	—	8	1	no	—
54	C₄×C₁₀₃	#135	1	2	yes	—
55	C₄×C₁₀₃	#137	1	2	yes	—
56	Dic₁₀₃	—	4	2	no	—
57	C₂×C₁₀₃	#452	1	2	yes	—
58	C₂×C₁₀₃	#453	1	1	yes	characteristic
59	D₁₀₃	—	8	2	no	—
60	#1634	—	103	1	no	maximal, Sylow
61	C₁₀₃	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	—	103	1	no	—
63	#494	D₁₀₃	1	1	yes	characteristic, complemented
64	#476	—	103	2	no	—
65	#333	—	103	1	no	—
66	D₄⋊C₈	—	103	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₀₃	1	1	yes	characteristic
70	#124	—	206	2	no	—
71	#121	—	206	2	no	—
72	C₄×D₄	—	103	2	no	—
73	#117	—	206	2	no	—
74	#106	—	206	2	no	—
75	C₄⋊C₈	—	103	2	no	—
76	#102	—	206	2	no	—
77	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₀₃	1	2	yes	—
80	C₂×D₄	—	103	2	no	—
81	C₂×D₄	—	206	2	no	—
82	C₂²×C₄	—	206	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	206	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	103	2	no	—
89	C₂²⋊C₄	—	206	4	no	—
90	C₂²⋊C₄	—	206	2	no	—
91	C₄²	—	206	2	no	—
92	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
93	C₂³	—	206	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	824	2	no	—
97	D₄	—	206	2	no	—
98	D₄	—	206	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	412	2	no	—
102	C₂×C₄	—	412	2	no	—
103	C₂×C₄	—	206	2	no	—
104	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	412	2	no	—
108	C₂²	—	412	2	no	—
109	C₂²	—	412	2	no	—
110	C₂²	—	412	2	no	—
111	C₂²	#15692989	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	412	2	no	—
116	C₄	#15693044	1	2	yes	—
117	C₄	#15693053	1	2	yes	—
118	C₂	—	824	2	no	—
119	C₂	#20267952	1	2	yes	—
120	C₂	#20267958	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268507

C₂

yes

maximal, characteristic

#20268078

C₂

yes

maximal, complemented

#20267540

C₂

yes

maximal, characteristic, complemented

#20267520

C₂

yes

maximal, complemented

#494×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#15693010

—

#15692996

—

C₄×D₄₁₂

C₂²

yes

—

#15692979

—

#15692952

—

C₄⋊(C₁₀₃⋊C₈)

C₂²

yes

—

#15692941

—

C₂×Q₁₆×C₁₀₃

—

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×D₄₁₂

—

C₂×D₄₁₂

D₄

yes

—

C₂×C₄×D₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

D₄

yes

—

(C₂×D₁₀₃)⋊C₄

—

(C₂×D₁₀₃)⋊C₄

—

C₄×Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

D₄

yes

—

Q₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

—

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

#120

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#135

yes

—

C₄×C₁₀₃

#137

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#452

yes

—

C₂×C₁₀₃

#453

yes

characteristic

D₁₀₃

—

#1634

—

103

maximal, Sylow

C₁₀₃

#1634

yes

characteristic, complemented, Sylow

#555

—

103

—

#494

D₁₀₃

yes

characteristic, complemented

#476

—

103

—

#333

—

103

—

D₄⋊C₈

—

103

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₃

yes

characteristic

#124

—

206

—

#121

—

206

—

C₄×D₄

—

103

—

#117

—

206

—

#106

—

206

—

C₄⋊C₈

—

103

—

#102

—

206

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₃

yes

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

C₂²×C₄

—

206

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

103

—

C₂²⋊C₄

—

206

—

C₂²⋊C₄

—

206

—

C₄²

—

206

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

206

—

Q₈

—

Q₈

—

D₄

—

824

—

D₄

—

206

—

D₄

—

206

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

412

—

102

C₂×C₄

—

412

—

103

C₂×C₄

—

206

—

104

C₂×C₄

D₄×D₁₀₃

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

106

C₈

—

107

C₈

—

412

—

108

C₂²

—

412

—

109

C₂²

—

412

—

110

C₂²

—

412

—

111

C₂²

#15692989

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

412

—

116

C₄

#15693044

yes

—

117

C₄

#15693053

yes

—

118

C₂

—

824

—

119

C₂

#20267952

yes

—

120

C₂

#20267958

yes

characteristic

	subgroup	G/H
center	C₂²	#15692989
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#494×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#120
prime radical	#20268507	C₂
hypercenter	#494	D₁₀₃
hypocenter	C₁₀₃	#1634
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#15692989

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#494×C₁₀₃

C₂

C₂²×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #20268078	C₂	semidirect
3. #20267540	C₂	semidirect
4. #20267520	C₂	semidirect
5. #494×C₁₀₃	C₂	semidirect
61. C₁₀₃	#1634	semidirect
63. #494	D₁₀₃	semidirect

H ≅ G/N

type

2. #20268078

C₂

semidirect

3. #20267540

C₂

semidirect

4. #20267520

C₂

semidirect

5. #494×C₁₀₃

C₂

semidirect

61. C₁₀₃

#1634

semidirect

63. #494

D₁₀₃

semidirect

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

subgroup	index
#20268507	2
#20268078	2
#20267540	2
#20267520	2
#494×C₁₀₃	2
#1634	103

subgroup

index

#494×C₁₀₃

103

Subgroups — #27194287

Conjugacy classes of subgroups

Complements

Maximal subgroups