Group #38766451 — subgroups

Subgroups — Q₈×SD₁₆×C₁₉₉ #38766451

Order 25472.

370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 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	Q₈²×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#498×C₁₉₉	C₂	1	3	yes	maximal, complemented
4	#476×C₁₉₉	C₂	1	3	yes	maximal
5	#475×C₁₉₉	C₂	1	3	yes	maximal, complemented
6	C₈×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₁₉₉	C₂	1	3	yes	maximal
8	C₂²×Q₈×C₁₉₉	—	2	1	no	—
9	C₂×SD₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₁₉₉	—	2	3	no	—
11	C₄⋊Q₈×C₁₉₉	C₂²	1	3	yes	—
12	C₄⋊Q₈×C₁₉₉	C₂²	1	3	yes	—
13	C₂²⋊Q₈×C₁₉₉	—	2	3	no	—
14	C₄×Q₈×C₁₉₉	—	2	1	no	—
15	C₄×Q₈×C₁₉₉	C₂²	1	3	yes	—
16	C₄×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₉₉	C₂²	1	3	yes	—
18	#106×C₁₉₉	C₂²	1	3	yes	—
19	#106×C₁₉₉	C₂²	1	6	yes	—
20	C₄⋊C₈×C₁₉₉	C₂²	1	3	yes	—
21	#103×C₁₉₉	C₂²	1	3	yes	—
22	#102×C₁₉₉	C₂²	1	3	yes	—
23	C₄×C₈×C₁₉₉	C₂²	1	3	yes	—
24	C₂×Q₈×C₁₉₉	—	2	3	no	—
25	C₂×Q₈×C₁₉₉	—	2	3	no	—
26	C₂×Q₈×C₁₉₉	—	4	4	no	—
27	C₂×Q₈×C₁₉₉	D₄	1	1	yes	characteristic
28	C₂×Q₈×C₁₉₉	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₁₉₉	—	2	3	no	—
31	SD₁₆×C₁₉₉	Q₈	1	4	yes	direct product factor
32	C₂×C₈×C₁₉₉	C₂³	1	3	yes	—
33	C₂×C₈×C₁₉₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₉₉	—	2	3	no	—
35	C₄⋊C₄×C₁₉₉	—	2	3	no	—
36	C₄⋊C₄×C₁₉₉	—	2	3	no	—
37	C₄⋊C₄×C₁₉₉	D₄	1	3	yes	—
38	C₄⋊C₄×C₁₉₉	C₂³	1	3	yes	—
39	C₄⋊C₄×C₁₉₉	C₂³	1	3	yes	—
40	C₂²⋊C₄×C₁₉₉	—	2	3	no	—
41	C₄²×C₁₉₉	—	2	3	no	—
42	C₄²×C₁₉₉	C₂³	1	3	yes	—
43	C₂³×C₁₉₉	—	2	1	no	—
44	Q₈×C₁₉₉	—	2	1	no	—
45	Q₈×C₁₉₉	—	4	6	no	—
46	Q₈×C₁₉₉	—	8	3	no	—
47	Q₈×C₁₉₉	SD₁₆	1	4	yes	direct product factor
48	Q₈×C₁₉₉	C₂×Q₈	1	2	yes	complemented
49	D₄×C₁₉₉	—	2	1	no	—
50	D₄×C₁₉₉	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₉₉	—	2	1	no	—
52	C₂×C₄×C₁₉₉	—	2	3	no	—
53	C₂×C₄×C₁₉₉	—	2	3	no	—
54	C₂×C₄×C₁₉₉	—	4	6	no	—
55	C₂×C₄×C₁₉₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₉₉	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
58	C₈×C₁₉₉	—	2	3	no	—
59	C₈×C₁₉₉	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₁₉₉	—	2	2	no	—
61	C₂²×C₁₉₉	—	4	1	no	—
62	C₂²×C₁₉₉	—	4	1	no	—
63	C₂²×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₁₉₉	—	2	2	no	—
65	C₄×C₁₉₉	—	2	3	no	—
66	C₄×C₁₉₉	—	4	3	no	—
67	C₄×C₁₉₉	—	4	3	no	—
68	C₄×C₁₉₉	C₂×SD₁₆	1	6	yes	—
69	C₄×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
70	C₄×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
71	C₂×C₁₉₉	—	4	2	no	—
72	C₂×C₁₉₉	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₁₉₉	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₁₉₉	#577	1	1	yes	characteristic
75	C₁₉₉	Q₈×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
76	Q₈×SD₁₆	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	Q₈²	C₂×C₁₉₉	1	1	yes	characteristic, complemented
78	D₄×Q₈	C₂×C₁₉₉	1	1	yes	characteristic
79	#498	C₂×C₁₉₉	1	3	yes	complemented
80	#476	C₂×C₁₉₉	1	3	yes	—
81	#475	C₂×C₁₉₉	1	3	yes	complemented
82	C₈×Q₈	C₂×C₁₉₉	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	C₂×C₁₉₉	1	3	yes	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂×SD₁₆	C₂²×C₁₉₉	1	1	yes	characteristic
86	C₄⋊Q₈	—	2	3	no	—
87	C₄⋊Q₈	C₂²×C₁₉₉	1	3	yes	—
88	C₄⋊Q₈	C₂²×C₁₉₉	1	3	yes	—
89	C₂²⋊Q₈	—	2	3	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂²×C₁₉₉	1	3	yes	—
92	C₄×Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
93	C₄×D₄	C₂²×C₁₉₉	1	3	yes	—
94	#106	C₂²×C₁₉₉	1	3	yes	—
95	#106	C₂²×C₁₉₉	1	6	yes	—
96	C₄⋊C₈	C₂²×C₁₉₉	1	3	yes	—
97	#103	C₂²×C₁₉₉	1	3	yes	—
98	#102	C₂²×C₁₉₉	1	3	yes	—
99	C₄×C₈	C₂²×C₁₉₉	1	3	yes	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	2	3	no	—
102	C₂×Q₈	—	4	4	no	—
103	C₂×Q₈	D₄×C₁₉₉	1	1	yes	characteristic
104	C₂×Q₈	C₂³×C₁₉₉	1	1	yes	characteristic
105	C₂×D₄	C₂³×C₁₉₉	1	1	yes	characteristic
106	C₂²×C₄	—	2	3	no	—
107	SD₁₆	Q₈×C₁₉₉	1	4	yes	direct product factor
108	C₂×C₈	C₂³×C₁₉₉	1	3	yes	—
109	C₂×C₈	C₂³×C₁₉₉	1	1	yes	characteristic
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	2	3	no	—
113	C₄⋊C₄	D₄×C₁₉₉	1	3	yes	—
114	C₄⋊C₄	C₂³×C₁₉₉	1	3	yes	—
115	C₄⋊C₄	C₂³×C₁₉₉	1	3	yes	—
116	C₂²⋊C₄	—	2	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂³×C₁₉₉	1	3	yes	—
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	4	6	no	—
122	Q₈	—	8	3	no	—
123	Q₈	SD₁₆×C₁₉₉	1	4	yes	direct product factor
124	Q₈	C₂×Q₈×C₁₉₉	1	2	yes	complemented
125	D₄	—	2	1	no	—
126	D₄	C₂×Q₈×C₁₉₉	1	2	yes	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	C₂×D₄×C₁₉₉	1	3	yes	—
132	C₂×C₄	C₂×D₄×C₁₉₉	1	3	yes	—
133	C₂×C₄	C₂⁴×C₁₉₉	1	1	yes	characteristic
134	C₈	—	2	3	no	—
135	C₈	C₂×Q₈×C₁₉₉	1	2	yes	complemented
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂²×D₄×C₁₉₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	3	no	—
142	C₄	—	4	3	no	—
143	C₄	—	4	3	no	—
144	C₄	C₂×SD₁₆×C₁₉₉	1	6	yes	—
145	C₄	C₂²×Q₈×C₁₉₉	1	1	yes	characteristic
146	C₄	2₋¹⁺⁴×C₁₉₉	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	D₄×Q₈×C₁₉₉	1	1	yes	characteristic
149	C₂	C₂²×SD₁₆×C₁₉₉	1	1	yes	characteristic
150	C₂	#577×C₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₉₉

C₂

yes

maximal, characteristic, complemented

D₄×Q₈×C₁₉₉

C₂

yes

maximal, characteristic

#498×C₁₉₉

C₂

yes

maximal, complemented

#476×C₁₉₉

C₂

yes

maximal

#475×C₁₉₉

C₂

yes

maximal, complemented

C₈×Q₈×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₁₉₉

C₂

yes

maximal

C₂²×Q₈×C₁₉₉

—

C₂×SD₁₆×C₁₉₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

—

C₂²⋊Q₈×C₁₉₉

—

C₄×Q₈×C₁₉₉

—

C₄×Q₈×C₁₉₉

C₂²

yes

—

C₄×Q₈×C₁₉₉

C₂²

yes

characteristic

C₄×D₄×C₁₉₉

C₂²

yes

—

#106×C₁₉₉

C₂²

yes

—

#106×C₁₉₉

C₂²

yes

—

C₄⋊C₈×C₁₉₉

C₂²

yes

—

#103×C₁₉₉

C₂²

yes

—

#102×C₁₉₉

C₂²

yes

—

C₄×C₈×C₁₉₉

C₂²

yes

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

characteristic

C₂×Q₈×C₁₉₉

C₂³

yes

characteristic

C₂×D₄×C₁₉₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉₉

—

SD₁₆×C₁₉₉

Q₈

yes

direct product factor

C₂×C₈×C₁₉₉

C₂³

yes

—

C₂×C₈×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

C₂³

yes

—

C₄⋊C₄×C₁₉₉

C₂³

yes

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

—

C₂³×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

SD₁₆

yes

direct product factor

Q₈×C₁₉₉

C₂×Q₈

yes

complemented

D₄×C₁₉₉

—

D₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂⁴

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

C₂×Q₈

yes

complemented

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂²×D₄

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×SD₁₆

yes

—

C₄×C₁₉₉

C₂²×Q₈

yes

characteristic

C₄×C₁₉₉

2₋¹⁺⁴

yes

characteristic

C₂×C₁₉₉

—

C₂×C₁₉₉

D₄×Q₈

yes

characteristic

C₂×C₁₉₉

C₂²×SD₁₆

yes

characteristic

C₂×C₁₉₉

#577

yes

characteristic

C₁₉₉

Q₈×SD₁₆

yes

characteristic, direct product factor, Sylow

Q₈×SD₁₆

C₁₉₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₁₉₉

yes

characteristic, complemented

D₄×Q₈

C₂×C₁₉₉

yes

characteristic

#498

C₂×C₁₉₉

yes

complemented

#476

C₂×C₁₉₉

yes

—

#475

C₂×C₁₉₉

yes

complemented

C₈×Q₈

C₂×C₁₉₉

yes

characteristic, complemented

C₄×SD₁₆

C₂×C₁₉₉

yes

—

C₂²×Q₈

—

C₂×SD₁₆

C₂²×C₁₉₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₉₉

yes

—

C₄⋊Q₈

C₂²×C₁₉₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₉₉

yes

—

C₄×Q₈

C₂²×C₁₉₉

yes

characteristic

C₄×D₄

C₂²×C₁₉₉

yes

—

#106

C₂²×C₁₉₉

yes

—

#106

C₂²×C₁₉₉

yes

—

C₄⋊C₈

C₂²×C₁₉₉

yes

—

#103

C₂²×C₁₉₉

yes

—

#102

C₂²×C₁₉₉

yes

—

C₄×C₈

C₂²×C₁₉₉

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

D₄×C₁₉₉

yes

characteristic

104

C₂×Q₈

C₂³×C₁₉₉

yes

characteristic

105

C₂×D₄

C₂³×C₁₉₉

yes

characteristic

106

C₂²×C₄

—

107

SD₁₆

Q₈×C₁₉₉

yes

direct product factor

108

C₂×C₈

C₂³×C₁₉₉

yes

—

109

C₂×C₈

C₂³×C₁₉₉

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₄×C₁₉₉

yes

—

114

C₄⋊C₄

C₂³×C₁₉₉

yes

—

115

C₄⋊C₄

C₂³×C₁₉₉

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂³×C₁₉₉

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

SD₁₆×C₁₉₉

yes

direct product factor

124

Q₈

C₂×Q₈×C₁₉₉

yes

complemented

125

D₄

—

126

D₄

C₂×Q₈×C₁₉₉

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

132

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

133

C₂×C₄

C₂⁴×C₁₉₉

yes

characteristic

134

C₈

—

135

C₈

C₂×Q₈×C₁₉₉

yes

complemented

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂²×D₄×C₁₉₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

C₂×SD₁₆×C₁₉₉

yes

—

145

C₄

C₂²×Q₈×C₁₉₉

yes

characteristic

146

C₄

2₋¹⁺⁴×C₁₉₉

yes

characteristic

147

C₂

—

148

C₂

D₄×Q₈×C₁₉₉

yes

characteristic

149

C₂

C₂²×SD₁₆×C₁₉₉

yes

characteristic

150

C₂

#577×C₁₉₉

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉₉
prime radical	C₂×D₄×C₁₉₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ C₄×Q₈×C₁₉₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₉₉

prime radical

C₂×D₄×C₁₉₉

C₂³

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₉ ◁ C₄×Q₈×C₁₉₉ ◁ G

N	H ≅ G/N	type
1. Q₈²×C₁₉₉	C₂	semidirect
3. #498×C₁₉₉	C₂	semidirect
5. #475×C₁₉₉	C₂	semidirect
6. C₈×Q₈×C₁₉₉	C₂	semidirect
31. SD₁₆×C₁₉₉	Q₈	direct (N × H)
47. Q₈×C₁₉₉	SD₁₆	direct (N × H)
48. Q₈×C₁₉₉	C₂×Q₈	semidirect
59. C₈×C₁₉₉	C₂×Q₈	semidirect
75. C₁₉₉	Q₈×SD₁₆	direct (N × H)
76. Q₈×SD₁₆	C₁₉₉	direct (N × H)
77. Q₈²	C₂×C₁₉₉	semidirect
79. #498	C₂×C₁₉₉	semidirect
81. #475	C₂×C₁₉₉	semidirect
82. C₈×Q₈	C₂×C₁₉₉	semidirect
107. SD₁₆	Q₈×C₁₉₉	direct (N × H)
123. Q₈	SD₁₆×C₁₉₉	direct (N × H)
124. Q₈	C₂×Q₈×C₁₉₉	semidirect
135. C₈	C₂×Q₈×C₁₉₉	semidirect

H ≅ G/N

type

1. Q₈²×C₁₉₉

C₂

semidirect

3. #498×C₁₉₉

C₂

semidirect

5. #475×C₁₉₉

C₂

semidirect

6. C₈×Q₈×C₁₉₉

C₂

semidirect

31. SD₁₆×C₁₉₉

Q₈

direct (N × H)

47. Q₈×C₁₉₉

SD₁₆

direct (N × H)

48. Q₈×C₁₉₉

C₂×Q₈

semidirect

59. C₈×C₁₉₉

C₂×Q₈

semidirect

75. C₁₉₉

Q₈×SD₁₆

direct (N × H)

76. Q₈×SD₁₆

C₁₉₉

direct (N × H)

77. Q₈²

C₂×C₁₉₉

semidirect

79. #498

C₂×C₁₉₉

semidirect

81. #475

C₂×C₁₉₉

semidirect

82. C₈×Q₈

C₂×C₁₉₉

semidirect

107. SD₁₆

Q₈×C₁₉₉

direct (N × H)

123. Q₈

SD₁₆×C₁₉₉

direct (N × H)

124. Q₈

C₂×Q₈×C₁₉₉

semidirect

135. C₈

C₂×Q₈×C₁₉₉

semidirect

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

subgroup	index
Q₈²×C₁₉₉	2
D₄×Q₈×C₁₉₉	2
#498×C₁₉₉	2
#476×C₁₉₉	2
#475×C₁₉₉	2
C₈×Q₈×C₁₉₉	2
C₄×SD₁₆×C₁₉₉	2
Q₈×SD₁₆	199

subgroup

index

Q₈²×C₁₉₉

D₄×Q₈×C₁₉₉

#498×C₁₉₉

#476×C₁₉₉

#475×C₁₉₉

C₈×Q₈×C₁₉₉

C₄×SD₁₆×C₁₉₉

Q₈×SD₁₆

199

Subgroups — Q₈×SD₁₆×C₁₉₉ #38766451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×SD16×C199 #38766451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×SD₁₆×C₁₉₉ #38766451