Group #28366254 — subgroups

Subgroups — C₂×#20294950 #28366254

Order 13952.

908 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 245 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×#13182008	C₂	1	3	yes	maximal, complemented
2	C₂²×#13182006	C₂	1	3	yes	maximal
3	#20294950	C₂	1	8	yes	maximal, direct product factor
4	C₂×#95×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₀₉	C₂²	1	3	yes	—
7	C₂×#13182008	—	2	12	no	—
8	C₂×#13182008	C₂²	1	12	yes	complemented
9	C₂×#13182006	—	2	12	no	—
10	C₂×#13182006	C₂²	1	12	yes	—
11	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
12	#95×C₁₀₉	C₂²	1	4	yes	complemented
13	C₂³×D₁₀₉	—	2	6	no	—
14	C₂³×D₁₀₉	—	2	6	no	—
15	C₂³×D₁₀₉	Q₈	1	4	yes	—
16	C₂³×D₁₀₉	C₂³	1	2	yes	—
17	C₂²×Dic₁₀₉	—	2	6	no	—
18	C₂²×Dic₁₀₉	—	2	6	no	—
19	C₂²×Dic₁₀₉	D₄	1	12	yes	—
20	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
21	#13182008	—	2	24	no	—
22	#13182006	—	2	24	no	—
23	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₉	—	2	6	no	—
25	C₂²×C₄×C₁₀₉	—	2	6	no	—
26	C₂²×C₄×C₁₀₉	—	2	6	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
28	C₂³×C₁₀₉	C₂×D₄	1	6	yes	—
29	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
30	C₂³×C₁₀₉	#41	1	6	yes	—
31	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
32	C₂²×D₁₀₉	—	2	24	no	—
33	C₂²×D₁₀₉	—	2	6	no	—
34	C₂²×D₁₀₉	—	2	6	no	—
35	C₂²×D₁₀₉	—	4	12	no	—
36	C₂²×D₁₀₉	—	4	4	no	—
37	C₂²×D₁₀₉	C₂×Q₈	1	4	yes	—
38	C₂×C₄×C₁₀₉	—	2	6	no	—
39	C₂×C₄×C₁₀₉	—	2	6	no	—
40	C₂×C₄×C₁₀₉	—	2	6	no	—
41	C₂×C₄×C₁₀₉	—	4	12	no	—
42	C₂×Dic₁₀₉	—	2	24	no	—
43	C₂×Dic₁₀₉	—	2	6	no	—
44	C₂×Dic₁₀₉	—	2	6	no	—
45	C₂×Dic₁₀₉	C₂×D₄	1	12	yes	—
46	C₂²×C₁₀₉	#121	1	12	yes	—
47	C₂²×C₁₀₉	C₂²⋊Q₈	1	12	yes	—
48	C₂²×C₁₀₉	#126	1	4	yes	—
49	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
50	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₂×#41	1	3	yes	—
52	C₂×D₁₀₉	—	2	12	no	—
53	C₂×D₁₀₉	—	4	6	no	—
54	C₂×D₁₀₉	—	4	2	no	—
55	C₂×D₁₀₉	—	4	16	no	—
56	C₄×C₁₀₉	—	4	6	no	—
57	Dic₁₀₉	—	2	12	no	—
58	C₂×C₁₀₉	#396	1	8	yes	complemented
59	C₂×C₁₀₉	C₂×#121	1	3	yes	—
60	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	3	yes	—
61	C₂×C₁₀₉	C₂×#126	1	1	yes	characteristic
62	D₁₀₉	—	4	4	no	—
63	C₂×#396	—	109	1	no	maximal, Sylow
64	C₁₀₉	C₂×#396	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₄⋊C₄	—	109	3	no	—
66	C₂²×C₂²⋊C₄	—	109	3	no	—
67	#396	—	109	8	no	—
68	C₂×#95	D₁₀₉	1	1	yes	characteristic, complemented
69	C₂⁵	—	109	1	no	—
70	C₂³×C₄	—	109	3	no	—
71	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
72	C₂×C₄⋊C₄	—	109	12	no	—
73	C₂×C₄⋊C₄	—	218	12	no	—
74	C₂×C₂²⋊C₄	—	109	12	no	—
75	C₂×C₂²⋊C₄	—	218	12	no	—
76	#95	C₂×D₁₀₉	1	4	yes	complemented
77	C₂⁴	—	109	4	no	—
78	C₂⁴	—	109	2	no	—
79	C₂⁴	—	218	6	no	—
80	C₂⁴	—	218	6	no	—
81	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	—	2	6	no	—
84	C₂²×C₄	—	2	6	no	—
85	C₂²×C₄	—	109	12	no	—
86	C₂²×C₄	—	109	6	no	—
87	C₂²×C₄	—	218	6	no	—
88	C₂²×C₄	—	218	6	no	—
89	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
90	C₄⋊C₄	—	218	24	no	—
91	C₂²⋊C₄	—	218	24	no	—
92	C₂³	—	109	4	no	—
93	C₂³	—	436	12	no	—
94	C₂³	—	436	4	no	—
95	C₂³	—	218	24	no	—
96	C₂³	—	218	6	no	—
97	C₂³	—	218	6	no	—
98	C₂³	D₄×D₁₀₉	1	6	yes	—
99	C₂³	Q₈×D₁₀₉	1	2	yes	—
100	C₂³	#13182029	1	6	yes	—
101	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	4	12	no	—
106	C₂×C₄	—	109	12	no	—
107	C₂×C₄	—	218	24	no	—
108	C₂×C₄	—	218	6	no	—
109	C₂×C₄	—	218	6	no	—
110	C₂²	—	436	16	no	—
111	C₂²	—	436	6	no	—
112	C₂²	—	436	2	no	—
113	C₂²	—	218	12	no	—
114	C₂²	#16812672	1	12	yes	—
115	C₂²	#16812677	1	12	yes	—
116	C₂²	#16812693	1	4	yes	—
117	C₂²	C₂×D₄×D₁₀₉	1	3	yes	—
118	C₂²	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
119	C₂²	C₂×#13182029	1	3	yes	—
120	C₄	—	4	6	no	—
121	C₄	—	218	12	no	—
122	C₂	—	436	4	no	—
123	C₂	#20294950	1	8	yes	direct product factor
124	C₂	C₂×#16812672	1	3	yes	—
125	C₂	C₂×#16812677	1	3	yes	—
126	C₂	C₂×#16812693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182008

C₂

yes

maximal, complemented

C₂²×#13182006

C₂

yes

maximal

#20294950

C₂

yes

maximal, direct product factor

C₂×#95×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₀₉

C₂²

yes

characteristic

C₂³×Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

C₂×#13182008

C₂²

yes

complemented

C₂×#13182006

—

C₂×#13182006

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

—

#95×C₁₀₉

C₂²

yes

complemented

C₂³×D₁₀₉

—

C₂³×D₁₀₉

—

C₂³×D₁₀₉

Q₈

yes

—

C₂³×D₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182006

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

#121

yes

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

#126

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₉

C₂×#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#396

yes

complemented

C₂×C₁₀₉

C₂×#121

yes

—

C₂×C₁₀₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₀₉

C₂×#126

yes

characteristic

D₁₀₉

—

C₂×#396

—

109

maximal, Sylow

C₁₀₉

C₂×#396

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

—

109

—

C₂²×C₂²⋊C₄

—

109

—

#396

—

109

—

C₂×#95

D₁₀₉

yes

characteristic, complemented

C₂⁵

—

109

—

C₂³×C₄

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

218

—

#95

C₂×D₁₀₉

yes

complemented

C₂⁴

—

109

—

C₂⁴

—

109

—

C₂⁴

—

218

—

C₂⁴

—

218

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

D₄×D₁₀₉

yes

—

C₂³

Q₈×D₁₀₉

yes

—

100

C₂³

#13182029

yes

—

101

C₂³

C₂³×D₁₀₉

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

218

—

108

C₂×C₄

—

218

—

109

C₂×C₄

—

218

—

110

C₂²

—

436

—

111

C₂²

—

436

—

112

C₂²

—

436

—

113

C₂²

—

218

—

114

C₂²

#16812672

yes

—

115

C₂²

#16812677

yes

—

116

C₂²

#16812693

yes

—

117

C₂²

C₂×D₄×D₁₀₉

yes

—

118

C₂²

C₂×Q₈×D₁₀₉

yes

characteristic

119

C₂²

C₂×#13182029

yes

—

120

C₄

—

121

C₄

—

218

—

122

C₂

—

436

—

123

C₂

#20294950

yes

direct product factor

124

C₂

C₂×#16812672

yes

—

125

C₂

C₂×#16812677

yes

—

126

C₂

C₂×#16812693

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator, cosocle	C₂³×C₁₀₉	C₂⁴
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂×#95×C₁₀₉	C₂
socle	C₂⁴×C₁₀₉	C₂³
prime radical	C₂⁴×D₁₀₉	C₂²
hypercenter	C₂×#95	D₁₀₉
hypocenter	C₁₀₉	C₂×#396
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂×#95

subgroup

G/H

center

C₂⁴

C₂²×D₁₀₉

commutator, cosocle

C₂³×C₁₀₉

C₂⁴

Frattini

C₂³

C₂³×D₁₀₉

Fitting

C₂×#95×C₁₀₉

C₂

socle

C₂⁴×C₁₀₉

C₂³

prime radical

C₂⁴×D₁₀₉

C₂²

hypercenter

C₂×#95

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂⁴ ◁ C₂×#95

N	H ≅ G/N	type
1. C₂²×#13182008	C₂	semidirect
3. #20294950	C₂	direct (N × H)
4. C₂×#95×C₁₀₉	C₂	semidirect
8. C₂×#13182008	C₂²	semidirect
12. #95×C₁₀₉	C₂²	semidirect
58. C₂×C₁₀₉	#396	semidirect
64. C₁₀₉	C₂×#396	semidirect
68. C₂×#95	D₁₀₉	semidirect
76. #95	C₂×D₁₀₉	semidirect
123. C₂	#20294950	direct (N × H)

H ≅ G/N

type

1. C₂²×#13182008

C₂

semidirect

3. #20294950

C₂

direct (N × H)

4. C₂×#95×C₁₀₉

C₂

semidirect

8. C₂×#13182008

C₂²

semidirect

12. #95×C₁₀₉

C₂²

semidirect

58. C₂×C₁₀₉

#396

semidirect

64. C₁₀₉

C₂×#396

semidirect

68. C₂×#95

D₁₀₉

semidirect

76. #95

C₂×D₁₀₉

semidirect

123. C₂

#20294950

direct (N × H)

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

subgroup	index
C₂²×#13182008	2
C₂²×#13182006	2
#20294950	2
C₂×#95×C₁₀₉	2
C₂×#396	109

subgroup

index

C₂×#95×C₁₀₉

109

Subgroups — C₂×#20294950 #28366254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20294950 #28366254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294950 #28366254