Group #44576177 — subgroups

Subgroups — C₁₂₇×#12504 #44576177

Order 32512.

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₁₂₇×#2252	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2247	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2241	C₂	1	2	yes	maximal
4	C₁₂₇×#1707	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₂₇	—	2	2	no	—
7	#500×C₁₂₇	C₂²	1	2	yes	—
8	#498×C₁₂₇	—	2	2	no	—
9	#494×C₁₂₇	C₂²	1	2	yes	—
10	#487×C₁₂₇	—	2	2	no	—
11	#366×C₁₂₇	—	2	2	no	—
12	#358×C₁₂₇	—	2	4	no	—
13	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
17	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
18	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
19	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
20	Q₃₂×C₁₂₇	—	4	4	no	—
21	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
22	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
23	#107×C₁₂₇	—	2	2	no	—
24	#107×C₁₂₇	—	2	2	no	—
25	#106×C₁₂₇	—	4	4	no	—
26	#103×C₁₂₇	—	4	4	no	—
27	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₁₂₇	—	4	2	no	—
31	C₂×Q₈×C₁₂₇	—	4	2	no	—
32	Q₁₆×C₁₂₇	—	4	4	no	—
33	Q₁₆×C₁₂₇	—	4	4	no	—
34	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
35	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
36	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₁₂₇	—	4	2	no	—
39	C₄⋊C₄×C₁₂₇	—	4	2	no	—
40	C₄⋊C₄×C₁₂₇	—	4	2	no	—
41	C₄⋊C₄×C₁₂₇	—	4	2	no	—
42	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₂₇	—	2	2	no	—
44	C₁₆×C₁₂₇	SD₁₆	1	4	yes	—
45	Q₈×C₁₂₇	—	8	2	no	—
46	Q₈×C₁₂₇	—	8	2	no	—
47	Q₈×C₁₂₇	—	8	4	no	—
48	C₂×C₄×C₁₂₇	—	8	2	no	—
49	C₂×C₄×C₁₂₇	—	8	2	no	—
50	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
53	C₈×C₁₂₇	—	2	2	no	—
54	C₈×C₁₂₇	Q₃₂	1	4	yes	complemented
55	C₈×C₁₂₇	C₂×SD₁₆	1	2	yes	—
56	C₈×C₁₂₇	#135	1	2	yes	—
57	C₂²×C₁₂₇	#493	1	1	yes	characteristic
58	C₄×C₁₂₇	—	16	2	no	—
59	C₄×C₁₂₇	—	16	2	no	—
60	C₄×C₁₂₇	#492	1	1	yes	characteristic
61	C₄×C₁₂₇	#495	1	1	yes	characteristic
62	C₄×C₁₂₇	C₂×Q₃₂	1	2	yes	—
63	C₄×C₁₂₇	#508	1	2	yes	—
64	#12504	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₁₂₇	#1706	1	1	yes	characteristic
66	C₂×C₁₂₇	#2247	1	1	yes	characteristic
67	C₂×C₁₂₇	#2249	1	1	yes	characteristic
68	#2252	C₂×C₁₂₇	1	1	yes	characteristic
69	#2247	C₂×C₁₂₇	1	1	yes	characteristic
70	#2241	C₂×C₁₂₇	1	2	yes	—
71	#1707	C₂×C₁₂₇	1	2	yes	—
72	C₈×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
73	C₁₂₇	#12504	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×Q₃₂	—	2	2	no	—
75	#500	C₂²×C₁₂₇	1	2	yes	—
76	#498	—	2	2	no	—
77	#494	C₂²×C₁₂₇	1	2	yes	—
78	#487	—	2	2	no	—
79	#366	—	2	2	no	—
80	#358	—	2	4	no	—
81	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
82	C₄×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
83	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	Q₃₂	—	4	4	no	—
89	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
90	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	4	4	no	—
94	#103	—	4	4	no	—
95	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
96	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	4	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
103	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
104	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
105	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	SD₁₆×C₁₂₇	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	8	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
119	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
120	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Q₃₂×C₁₂₇	1	4	yes	complemented
123	C₈	C₂×SD₁₆×C₁₂₇	1	2	yes	—
124	C₈	#135×C₁₂₇	1	2	yes	—
125	C₂²	#493×C₁₂₇	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	16	2	no	—
128	C₄	#492×C₁₂₇	1	1	yes	characteristic
129	C₄	#495×C₁₂₇	1	1	yes	characteristic
130	C₄	C₂×Q₃₂×C₁₂₇	1	2	yes	—
131	C₄	#508×C₁₂₇	1	2	yes	—
132	C₂	C₁₂₇×#1706	1	1	yes	characteristic
133	C₂	C₁₂₇×#2247	1	1	yes	characteristic
134	C₂	C₁₂₇×#2249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₂₇×#2252

C₂

yes

maximal, characteristic

C₁₂₇×#2247

C₂

yes

maximal, characteristic

C₁₂₇×#2241

C₂

yes

maximal

C₁₂₇×#1707

C₂

yes

maximal

C₈×C₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₁₂₇

—

#500×C₁₂₇

C₂²

yes

—

#498×C₁₂₇

—

#494×C₁₂₇

C₂²

yes

—

#487×C₁₂₇

—

#366×C₁₂₇

—

#358×C₁₂₇

—

C₄×C₁₆×C₁₂₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₂₇

C₂²

yes

characteristic

C₈²×C₁₂₇

C₂²

yes

characteristic

C₂×Q₁₆×C₁₂₇

—

C₂×Q₁₆×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

C₄⋊Q₈×C₁₂₇

—

Q₃₂×C₁₂₇

—

C₂×C₁₆×C₁₂₇

D₄

yes

—

C₂×C₁₆×C₁₂₇

D₄

yes

—

#107×C₁₂₇

—

#107×C₁₂₇

—

#106×C₁₂₇

—

#103×C₁₂₇

—

C₄×C₈×C₁₂₇

D₄

yes

characteristic

C₄×C₈×C₁₂₇

D₄

yes

characteristic

C₄×C₈×C₁₂₇

C₂³

yes

characteristic

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

—

Q₁₆×C₁₂₇

—

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

D₈

yes

—

C₂×C₈×C₁₂₇

D₈

yes

—

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂×D₄

yes

characteristic

C₁₆×C₁₂₇

—

C₁₆×C₁₂₇

SD₁₆

yes

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

#127

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₂₇

C₂×D₈

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

Q₃₂

yes

complemented

C₈×C₁₂₇

C₂×SD₁₆

yes

—

C₈×C₁₂₇

#135

yes

—

C₂²×C₁₂₇

#493

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

C₂×Q₃₂

yes

—

C₄×C₁₂₇

#508

yes

—

#12504

C₁₂₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

characteristic

#2247

C₂×C₁₂₇

yes

characteristic

#2241

C₂×C₁₂₇

yes

—

#1707

C₂×C₁₂₇

yes

—

C₈×C₁₆

C₂×C₁₂₇

yes

characteristic

C₁₂₇

#12504

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

#500

C₂²×C₁₂₇

yes

—

#498

—

#494

C₂²×C₁₂₇

yes

—

#487

—

#366

—

#358

—

C₄×C₁₆

C₂²×C₁₂₇

yes

characteristic

C₄×C₁₆

C₂²×C₁₂₇

yes

characteristic

C₈²

C₂²×C₁₂₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₁₂₇

yes

—

C₂×C₁₆

D₄×C₁₂₇

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₂₇

yes

characteristic

C₄×C₈

D₄×C₁₂₇

yes

characteristic

C₄×C₈

C₂³×C₁₂₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

D₈×C₁₂₇

yes

—

103

C₂×C₈

D₈×C₁₂₇

yes

—

104

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

105

C₂×C₈

C₂×D₄×C₁₂₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

C₂×D₄×C₁₂₇

yes

characteristic

111

C₁₆

—

112

C₁₆

SD₁₆×C₁₂₇

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#127×C₁₂₇

yes

characteristic

119

C₂×C₄

C₂×D₈×C₁₂₇

yes

characteristic

120

C₂×C₄

C₂×D₈×C₁₂₇

yes

characteristic

121

C₈

—

122

C₈

Q₃₂×C₁₂₇

yes

complemented

123

C₈

C₂×SD₁₆×C₁₂₇

yes

—

124

C₈

#135×C₁₂₇

yes

—

125

C₂²

#493×C₁₂₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#492×C₁₂₇

yes

characteristic

129

C₄

#495×C₁₂₇

yes

characteristic

130

C₄

C₂×Q₃₂×C₁₂₇

yes

—

131

C₄

#508×C₁₂₇

yes

—

132

C₂

C₁₂₇×#1706

yes

characteristic

133

C₂

C₁₂₇×#2247

yes

characteristic

134

C₂

C₁₂₇×#2249

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₂₇	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₂₇

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₂₇

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ C₈²×C₁₂₇ ◁ G

N	H ≅ G/N	type
54. C₈×C₁₂₇	Q₃₂	semidirect
64. #12504	C₁₂₇	direct (N × H)
73. C₁₂₇	#12504	direct (N × H)
122. C₈	Q₃₂×C₁₂₇	semidirect

H ≅ G/N

type

54. C₈×C₁₂₇

Q₃₂

semidirect

64. #12504

C₁₂₇

direct (N × H)

73. C₁₂₇

#12504

direct (N × H)

122. C₈

Q₃₂×C₁₂₇

semidirect

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

subgroup	index
C₁₂₇×#2252	2
C₁₂₇×#2247	2
C₁₂₇×#2241	2
C₁₂₇×#1707	2
C₈×C₁₆×C₁₂₇	2
#12504	127

subgroup

index

C₁₂₇×#2252

C₁₂₇×#2247

C₁₂₇×#2241

C₁₂₇×#1707

C₈×C₁₆×C₁₂₇

#12504

127

Subgroups — C₁₂₇×#12504 #44576177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C127×#12504 #44576177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12504 #44576177