Group #20268271 — subgroups

Subgroups — #20268271

Order 6592.

294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13110105	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₀₆	C₂	1	2	yes	maximal, complemented
3	C₄×D₄₁₂	C₂	1	2	yes	maximal, complemented
4	(C₂²⋊D₁₀₃)⋊C₄	C₂	1	4	yes	maximal, complemented
5	#15692979	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Dic₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×D₁₀₃	C₂	1	2	yes	maximal, complemented
8	#117×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13110105	C₄	1	8	yes	complemented
10	C₂×Dic₂₀₆	C₂²	1	1	yes	characteristic
11	C₂×D₄₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₀₃	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₃	C₂²	1	4	yes	—
15	C₂×C₄×D₁₀₃	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
17	C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
18	Dic₁₀₃⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₁₀₃	C₂²	1	4	yes	—
21	C₂²×C₄×C₁₀₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₃	C₂²	1	1	yes	characteristic
24	C₂²⋊C₄×C₁₀₃	C₂²	1	2	yes	—
25	C₄²×C₁₀₃	C₂²	1	2	yes	—
26	C₂³×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₃	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₃	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
30	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
31	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₃	C₂×C₄	1	8	yes	complemented
34	C₂×Dic₁₀₃	—	4	2	no	—
35	C₂×Dic₁₀₃	C₂³	1	4	yes	—
36	C₂×Dic₁₀₃	C₂³	1	2	yes	—
37	D₄₁₂	C₂×C₄	1	4	yes	complemented
38	C₄×D₁₀₃	—	4	4	no	—
39	C₄×D₁₀₃	C₂×C₄	1	8	yes	complemented
40	Dic₂₀₆	C₂×C₄	1	4	yes	complemented
41	C₂²×C₁₀₃	—	2	1	no	—
42	C₂²×C₁₀₃	—	2	1	no	—
43	C₂²×C₁₀₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₀₃	—	2	2	no	—
46	C₂×D₁₀₃	—	2	2	no	—
47	C₂×D₁₀₃	C₂²×C₄	1	4	yes	—
48	C₄×C₁₀₃	—	2	2	no	—
49	C₄×C₁₀₃	—	2	2	no	—
50	C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
51	C₄×C₁₀₃	C₂²×C₄	1	2	yes	—
52	Dic₁₀₃	—	2	4	no	—
53	Dic₁₀₃	C₂²×C₄	1	4	yes	—
54	C₂×C₁₀₃	—	2	2	no	—
55	C₂×C₁₀₃	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₃	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₀₃	—	2	4	no	—
59	C₁₀₃	#520	1	1	yes	characteristic, complemented, Sylow
60	#520	—	103	1	no	maximal, Sylow
61	C₂×#41	—	103	1	no	—
62	C₄×Q₈	—	103	2	no	—
63	C₄×D₄	—	103	4	no	—
64	C₄×D₄	—	103	2	no	—
65	#117	—	103	2	no	—
66	#117	D₁₀₃	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄	—	103	1	no	—
68	C₂×C₄⋊C₄	—	103	2	no	—
69	#41	—	103	8	no	—
70	C₂×Q₈	—	103	1	no	—
71	C₂×D₄	—	103	2	no	—
72	C₂×D₄	—	103	1	no	—
73	C₂²×C₄	—	103	2	no	—
74	C₂²×C₄	—	103	2	no	—
75	C₂²×C₄	—	103	4	no	—
76	C₂²×C₄	C₂×D₁₀₃	1	1	yes	characteristic
77	C₄⋊C₄	—	103	4	no	—
78	C₄⋊C₄	—	103	2	no	—
79	C₄⋊C₄	—	103	2	no	—
80	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
81	C₄⋊C₄	C₂×D₁₀₃	1	1	yes	characteristic
82	C₂²⋊C₄	—	103	4	no	—
83	C₂²⋊C₄	C₂×D₁₀₃	1	2	yes	—
84	C₄²	—	103	4	no	—
85	C₄²	C₂×D₁₀₃	1	2	yes	—
86	C₂³	—	103	2	no	—
87	C₂³	C₂²×D₁₀₃	1	1	yes	characteristic
88	Q₈	—	103	4	no	—
89	D₄	—	103	8	no	—
90	D₄	—	103	4	no	—
91	C₂×C₄	—	103	2	no	—
92	C₂×C₄	—	103	8	no	—
93	C₂×C₄	—	103	4	no	—
94	C₂×C₄	—	412	4	no	—
95	C₂×C₄	—	412	2	no	—
96	C₂×C₄	C₄×D₁₀₃	1	4	yes	complemented
97	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
98	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
99	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	103	4	no	—
104	C₂²	—	206	2	no	—
105	C₂²	—	206	2	no	—
106	C₂²	C₂×C₄×D₁₀₃	1	2	yes	—
107	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	2	no	—
110	C₄	—	103	4	no	—
111	C₄	—	206	4	no	—
112	C₄	C₂×C₄×D₁₀₃	1	2	yes	—
113	C₄	C₂×C₄×D₁₀₃	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	—	206	4	no	—
116	C₂	C₂²×C₄×D₁₀₃	1	1	yes	characteristic
117	C₂	#15693067	1	1	yes	characteristic
118	C₂	#15693068	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13110105

C₂

yes

maximal, characteristic

C₄×Dic₂₀₆

C₂

yes

maximal, complemented

C₄×D₄₁₂

C₂

yes

maximal, complemented

(C₂²⋊D₁₀₃)⋊C₄

C₂

yes

maximal, complemented

#15692979

C₂

yes

maximal, complemented

C₂×C₄⋊Dic₁₀₃

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₀₃

C₂

yes

maximal, complemented

#117×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#13110105

C₄

yes

complemented

C₂×Dic₂₀₆

C₂²

yes

characteristic

C₂×D₄₁₂

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₃

C₂²

yes

—

C₂²×Dic₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₂×C₄×D₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

C₄⋊Dic₁₀₃

C₂²

yes

—

Dic₁₀₃⋊C₄

C₂²

yes

—

(C₂×D₁₀₃)⋊C₄

C₂²

yes

—

C₄×Dic₁₀₃

C₂²

yes

—

C₂²×C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₃

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₃

C₂²

yes

—

C₄²×C₁₀₃

C₂²

yes

—

C₂³×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₄×C₁₀₃

C₂³

yes

characteristic

C₂²⋊D₁₀₃

C₂×C₄

yes

complemented

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

C₂³

yes

—

C₂×Dic₁₀₃

C₂³

yes

—

D₄₁₂

C₂×C₄

yes

complemented

C₄×D₁₀₃

—

C₄×D₁₀₃

C₂×C₄

yes

complemented

Dic₂₀₆

C₂×C₄

yes

complemented

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂²×C₄

yes

—

C₂²×C₁₀₃

C₂⁴

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

C₂²×C₄

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²×C₄

yes

—

C₄×C₁₀₃

C₂²×C₄

yes

—

Dic₁₀₃

—

Dic₁₀₃

C₂²×C₄

yes

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂³×C₄

yes

characteristic

C₂×C₁₀₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₃

2₋¹⁺⁴

yes

characteristic

D₁₀₃

—

C₁₀₃

#520

yes

characteristic, complemented, Sylow

#520

—

103

maximal, Sylow

C₂×#41

—

103

—

C₄×Q₈

—

103

—

C₄×D₄

—

103

—

C₄×D₄

—

103

—

#117

—

103

—

#117

D₁₀₃

yes

characteristic, complemented

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

#41

—

103

—

C₂×Q₈

—

103

—

C₂×D₄

—

103

—

C₂×D₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

—

103

—

C₂²×C₄

C₂×D₁₀₃

yes

characteristic

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

103

—

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₄⋊C₄

C₂×D₁₀₃

yes

characteristic

C₂²⋊C₄

—

103

—

C₂²⋊C₄

C₂×D₁₀₃

yes

—

C₄²

—

103

—

C₄²

C₂×D₁₀₃

yes

—

C₂³

—

103

—

C₂³

C₂²×D₁₀₃

yes

characteristic

Q₈

—

103

—

D₄

—

103

—

D₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

103

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

C₄×D₁₀₃

yes

complemented

C₂×C₄

C₂²×D₁₀₃

yes

—

C₂×C₄

C₂²×D₁₀₃

yes

—

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

100

C₂×C₄

C₂²×D₁₀₃

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

103

—

104

C₂²

—

206

—

105

C₂²

—

206

—

106

C₂²

C₂×C₄×D₁₀₃

yes

—

107

C₂²

C₂³×D₁₀₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

103

—

111

C₄

—

206

—

112

C₄

C₂×C₄×D₁₀₃

yes

—

113

C₄

C₂×C₄×D₁₀₃

yes

—

114

C₂

—

115

C₂

—

206

—

116

C₂

C₂²×C₄×D₁₀₃

yes

characteristic

117

C₂

#15693067

yes

characteristic

118

C₂

#15693068

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator	C₂×C₁₀₃	C₂³×C₄
Fitting	#117×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂⁴
hypercenter	#117	D₁₀₃
hypocenter	C₁₀₃	#520
LCS	G ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₃

commutator

C₂×C₁₀₃

C₂³×C₄

Fitting

#117×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂⁴

D₁₀₃

C₁₀₃

G ▷ C₂×C₁₀₃ ▷ C₁₀₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
2. C₄×Dic₂₀₆	C₂	semidirect
3. C₄×D₄₁₂	C₂	semidirect
4. (C₂²⋊D₁₀₃)⋊C₄	C₂	semidirect
5. #15692979	C₂	semidirect
6. C₂×C₄⋊Dic₁₀₃	C₂	semidirect
7. C₄⋊C₄×D₁₀₃	C₂	semidirect
8. #117×C₁₀₃	C₂	semidirect
9. #13110105	C₄	semidirect
28. C₂×C₄×C₁₀₃	C₂×C₄	semidirect
33. C₂²⋊D₁₀₃	C₂×C₄	semidirect
37. D₄₁₂	C₂×C₄	semidirect
39. C₄×D₁₀₃	C₂×C₄	semidirect
40. Dic₂₀₆	C₂×C₄	semidirect
59. C₁₀₃	#520	semidirect
66. #117	D₁₀₃	semidirect
96. C₂×C₄	C₄×D₁₀₃	semidirect

H ≅ G/N

type

2. C₄×Dic₂₀₆

C₂

semidirect

3. C₄×D₄₁₂

C₂

semidirect

4. (C₂²⋊D₁₀₃)⋊C₄

C₂

semidirect

5. #15692979

C₂

semidirect

6. C₂×C₄⋊Dic₁₀₃

C₂

semidirect

7. C₄⋊C₄×D₁₀₃

C₂

semidirect

8. #117×C₁₀₃

C₂

semidirect

9. #13110105

C₄

semidirect

28. C₂×C₄×C₁₀₃

C₂×C₄

semidirect

33. C₂²⋊D₁₀₃

C₂×C₄

semidirect

37. D₄₁₂

C₂×C₄

semidirect

39. C₄×D₁₀₃

C₂×C₄

semidirect

40. Dic₂₀₆

C₂×C₄

semidirect

59. C₁₀₃

#520

semidirect

66. #117

D₁₀₃

semidirect

96. C₂×C₄

C₄×D₁₀₃

semidirect

Subgroups — #20268271

Conjugacy classes of subgroups

Complements