Group #30638620 — subgroups

Subgroups — C₄×#16861140 #30638620

Order 16256.

280 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16861140	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#22549638	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₂₇	—	2	1	no	—
10	#16861140	C₄	1	8	yes	direct product factor
11	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
12	C₁₂₇⋊#95	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₂₇	—	2	1	no	—
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	#95×C₁₂₇	—	2	1	no	—
17	C₂²×Dic₁₂₇	—	2	2	no	—
18	C₂×C₁₂₇⋊C₈	—	2	2	no	—
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₂₇	—	4	2	no	—
25	C₂²×C₄×C₁₂₇	—	2	2	no	—
26	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₂₇	—	2	2	no	—
30	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₂₇	—	4	2	no	—
32	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₄²×C₁₂₇	Q₈	1	2	yes	—
34	C₂×Dic₁₂₇	—	2	4	no	—
35	C₂×Dic₁₂₇	—	4	2	no	—
36	C₂×Dic₁₂₇	—	4	2	no	—
37	C₁₂₇⋊C₈	—	2	2	no	—
38	C₁₂₇⋊C₈	C₄²	1	4	yes	complemented
39	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	—	2	2	no	—
41	C₂×C₄×C₁₂₇	—	2	4	no	—
42	C₂×C₄×C₁₂₇	—	4	2	no	—
43	C₂×C₄×C₁₂₇	—	4	2	no	—
44	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
47	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	#41	1	2	yes	—
52	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
54	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂²×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₂₇	#135	1	2	yes	—
57	C₄×C₁₂₇	—	2	2	no	—
58	C₄×C₁₂₇	—	4	4	no	—
59	C₄×C₁₂₇	#106	1	4	yes	complemented
60	C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
61	C₄×C₁₂₇	#117	1	1	yes	characteristic
62	C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
63	Dic₁₂₇	—	4	4	no	—
64	C₂×C₁₂₇	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₁₂₇	C₂×#106	1	1	yes	characteristic
66	C₂×C₁₂₇	#427	1	1	yes	characteristic
67	C₂×C₁₂₇	C₄×SD₁₆	1	2	yes	—
68	C₂×C₁₂₇	C₄×SD₁₆	1	2	yes	—
69	C₄×#106	—	127	1	no	maximal, Sylow
70	C₁₂₇	C₄×#106	1	1	yes	characteristic, complemented, Sylow
71	C₂×#106	—	127	2	no	—
72	C₂×C₄×C₈	—	127	1	no	—
73	C₄×C₄⋊C₄	—	127	1	no	—
74	C₄×C₄⋊C₄	D₁₂₇	1	1	yes	characteristic
75	#340	—	127	2	no	—
76	C₂²×C₈	—	127	1	no	—
77	C₂²×C₈	—	127	1	no	—
78	C₂×C₄⋊C₄	—	127	2	no	—
79	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	—	254	1	no	—
82	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
83	#106	—	127	8	no	—
84	C₄×C₈	—	127	4	no	—
85	#95	—	2	1	no	—
86	#95	—	254	1	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	254	2	no	—
89	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
91	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₂×C₈	—	127	2	no	—
93	C₂×C₈	—	127	4	no	—
94	C₂×C₈	—	127	2	no	—
95	C₂×C₈	—	254	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	127	4	no	—
98	C₄⋊C₄	—	254	2	no	—
99	C₄⋊C₄	C₄×D₁₂₇	1	4	yes	—
100	C₄²	—	4	2	no	—
101	C₄²	—	508	2	no	—
102	C₄²	C₄×D₁₂₇	1	2	yes	—
103	C₄²	Dic₂₅₄	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	508	2	no	—
110	C₂×C₄	—	508	2	no	—
111	C₂×C₄	—	254	4	no	—
112	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
113	C₂×C₄	Dic₁₂₇⋊C₄	1	2	yes	—
114	C₂×C₄	#14523235	1	2	yes	—
115	C₂×C₄	#14523236	1	2	yes	—
116	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
118	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
119	C₂×C₄	#14523248	1	2	yes	—
120	C₈	—	127	4	no	—
121	C₈	—	254	2	no	—
122	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
123	C₂²	C₄×C₂²⋊D₁₂₇	1	2	yes	—
124	C₂²	C₂×#14523235	1	1	yes	characteristic
125	C₂²	C₂×#14523236	1	1	yes	characteristic
126	C₂²	#16861229	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	508	4	no	—
130	C₄	#16861140	1	4	yes	direct product factor
131	C₄	C₄²×D₁₂₇	1	1	yes	characteristic
132	C₄	#16861166	1	1	yes	characteristic
133	C₄	C₄×Dic₂₅₄	1	2	yes	—
134	C₂	C₄×Dic₁₂₇⋊C₄	1	1	yes	characteristic
135	C₂	C₂×#16861140	1	1	yes	characteristic
136	C₂	#22549912	1	1	yes	characteristic
137	C₂	C₄×#14523236	1	2	yes	—
138	C₂	C₄×#14523235	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16861140

C₂

yes

maximal

C₂×C₄×C₁₂₇⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

#22549638

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂²×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₂₇

C₂²

yes

—

C₂×C₄×Dic₁₂₇

—

#16861140

C₄

yes

direct product factor

C₄×C₁₂₇⋊C₈

C₄

yes

complemented

C₁₂₇⋊#95

—

C₂×C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂×C₄²×C₁₂₇

—

C₂×C₄²×C₁₂₇

C₂²

yes

characteristic

#95×C₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂×C₄

yes

—

C₄×Dic₁₂₇

—

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

D₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

—

C₄²×C₁₂₇

C₂×C₄

yes

—

C₄²×C₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₄²

yes

complemented

C₂³×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

SD₁₆

yes

—

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

—

C₂×C₄×C₁₂₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₂₇

C₄×D₄

yes

—

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₂₇

#135

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#106

yes

complemented

C₄×C₁₂₇

C₂×C₄²

yes

characteristic

C₄×C₁₂₇

#117

yes

characteristic

C₄×C₁₂₇

C₄×Q₈

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

C₄×C₄⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

C₄×SD₁₆

yes

—

C₂×C₁₂₇

C₄×SD₁₆

yes

—

C₄×#106

—

127

maximal, Sylow

C₁₂₇

C₄×#106

yes

characteristic, complemented, Sylow

C₂×#106

—

127

—

C₂×C₄×C₈

—

127

—

C₄×C₄⋊C₄

—

127

—

C₄×C₄⋊C₄

D₁₂₇

yes

characteristic

#340

—

127

—

C₂²×C₈

—

127

—

C₂²×C₈

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

254

—

C₂×C₄²

C₂×D₁₂₇

yes

characteristic

#106

—

127

—

C₄×C₈

—

127

—

#95

—

#95

—

254

—

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₄⋊C₄

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₄×D₁₂₇

yes

—

100

C₄²

—

101

C₄²

—

508

—

102

C₄²

C₄×D₁₂₇

yes

—

103

C₄²

Dic₂₅₄

yes

—

104

C₂³

C₂×C₂²⋊D₁₂₇

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

508

—

110

C₂×C₄

—

508

—

111

C₂×C₄

—

254

—

112

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

113

C₂×C₄

Dic₁₂₇⋊C₄

yes

—

114

C₂×C₄

#14523235

yes

—

115

C₂×C₄

#14523236

yes

—

116

C₂×C₄

C₂×C₄×D₁₂₇

yes

characteristic

117

C₂×C₄

C₂×C₄×D₁₂₇

yes

—

118

C₂×C₄

C₂×Dic₂₅₄

yes

characteristic

119

C₂×C₄

#14523248

yes

—

120

C₈

—

127

—

121

C₈

—

254

—

122

C₂²

C₂×Dic₁₂₇⋊C₄

yes

characteristic

123

C₂²

C₄×C₂²⋊D₁₂₇

yes

—

124

C₂²

C₂×#14523235

yes

characteristic

125

C₂²

C₂×#14523236

yes

characteristic

126

C₂²

#16861229

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

508

—

130

C₄

#16861140

yes

direct product factor

131

C₄

C₄²×D₁₂₇

yes

characteristic

132

C₄

#16861166

yes

characteristic

133

C₄

C₄×Dic₂₅₄

yes

—

134

C₂

C₄×Dic₁₂₇⋊C₄

yes

characteristic

135

C₂

C₂×#16861140

yes

characteristic

136

C₂

#22549912

yes

characteristic

137

C₂

C₄×#14523236

yes

—

138

C₂

C₄×#14523235

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄×C₄⋊C₄×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₂₇
hypocenter	C₁₂₇	C₄×#106
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

C₄×C₄⋊C₄×C₁₂₇

C₂

socle, prime radical

C₂³×C₁₂₇

C₂×D₄

cosocle

C₂²×C₄×C₁₂₇

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₂₇

hypocenter

C₁₂₇

C₄×#106

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
10. #16861140	C₄	direct (N × H)
11. C₄×C₁₂₇⋊C₈	C₄	semidirect
38. C₁₂₇⋊C₈	C₄²	semidirect
59. C₄×C₁₂₇	#106	semidirect
70. C₁₂₇	C₄×#106	semidirect
130. C₄	#16861140	direct (N × H)

H ≅ G/N

type

10. #16861140

C₄

direct (N × H)

11. C₄×C₁₂₇⋊C₈

C₄

semidirect

38. C₁₂₇⋊C₈

C₄²

semidirect

59. C₄×C₁₂₇

#106

semidirect

70. C₁₂₇

C₄×#106

semidirect

130. C₄

#16861140

direct (N × H)

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

subgroup	index
C₂×#16861140	2
C₂×C₄×C₁₂₇⋊C₈	2
C₄×C₄⋊Dic₁₂₇	2
#22549638	2
C₄×C₄⋊C₄×C₁₂₇	2
C₄×#106	127

subgroup

index

C₂×#16861140

C₂×C₄×C₁₂₇⋊C₈

C₄×C₄⋊Dic₁₂₇

#22549638

C₄×C₄⋊C₄×C₁₂₇

C₄×#106

127

Subgroups — C₄×#16861140 #30638620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#16861140 #30638620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16861140 #30638620