Finite Group Databasebeta

← D157⋊(Q8⋊C8) #34115425

Subgroups — D157⋊(Q8⋊C8) #34115425

Order 20096.

280 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D157⋊(C4⋊C8)C211yesmaximal, characteristic
2C4×D157⋊C8C211yesmaximal, characteristic
3#23758123C212yesmaximal, complemented
4Q8⋊#11929441C212yesmaximal, complemented
5C4×Q8×D157C211yesmaximal, characteristic
6C2×D157⋊C8—21no—
7C2×D157⋊C8—21no—
8#19097838—21no—
9C4⋊#11929441C2212yes—
10C4×#11929441—21no—
11C4×#11929441C2212yes—
12C2×Q8×D157C411yescharacteristic
13Q8×Dic157C2211yescharacteristic
14C4×Dic314—21no—
15C4×Dic314C411yescharacteristic
16#19097736—21no—
17#19097736C2211yescharacteristic
18C4⋊C4×D157—21no—
19C4⋊C4×D157C411yescharacteristic
20C42×D157—21no—
21C42×D157C2211yescharacteristic
22C4×Q8×C157C411yescharacteristic
23D157⋊C8—24no—
24D157⋊C8—42no—
25C2×#11929441—22no—
26C2×#11929441—22no—
27C2×Dic314—21no—
28C2×Dic314C2×C411yescharacteristic
29Q8×D157—22no—
30Q8×D157C814yescomplemented
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157C2×C411yescharacteristic
34C2×C4×D157D411yescharacteristic
35C2×C4×D157D411yescharacteristic
36C4⋊Dic157—21no—
37C4⋊Dic157C2×C411yescharacteristic
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41C4×Dic157—21no—
42C4×Dic157—21no—
43C4×Dic157—21no—
44C4×Dic157D411yescharacteristic
45C4×Dic157D411yescharacteristic
46C4×Dic157C2311yescharacteristic
47C2×Q8×C157C2×C411yescharacteristic
48C4⋊C4×C157—21no—
49C4⋊C4×C157C2×C411yescharacteristic
50C42×C157—21no—
51C42×C157C2×C411yescharacteristic
52#11929441—24no—
53#11929441—42no—
54C22×D157C22⋊C411yescharacteristic
55Dic314—21no—
56Dic314—24no—
57Dic314C2×C812yescomplemented
58C2×Dic157—21no—
59C2×Dic157—21no—
60C2×Dic157C22⋊C411yescharacteristic
61C2×Dic157C2×D411yescharacteristic
62C2×Dic157C2×D411yescharacteristic
63C4×D157—24no—
64C4×D157—22no—
65C4×D157—42no—
66C4×D157C2×C812yes—
67C4×D157M1612yes—
68C4×D157SD1612yes—
69C4×D157Q1612yes—
70Q8×C157—21no—
71Q8×C157C2×C812yescomplemented
72C2×C4×C157—21no—
73C2×C4×C157—21no—
74C2×C4×C157C22⋊C411yescharacteristic
75C2×C4×C157C22⋊C411yescharacteristic
76C2×C4×C157C22×C411yescharacteristic
77C22×C157C2×C22⋊C411yescharacteristic
78C2×D157C22⋊C812yes—
79C2×D157#10312yes—
80C2×D157#10412yes—
81C4×C157—22no—
82C4×C157—21no—
83C4×C157—41no—
84C4×C157#10312yes—
85C4×C157C22×C811yescharacteristic
86C4×C157C2×M1611yescharacteristic
87Dic157—22no—
88Dic157—21no—
89Dic157—41no—
90Dic157C22⋊C812yes—
91Dic157C2×SD1611yescharacteristic
92Dic157C2×Q1611yescharacteristic
93C2×C157C2×C22⋊C811yescharacteristic
94C2×C157C2×#10311yescharacteristic
95C2×C157C2×#10411yescharacteristic
96D157Q8⋊C814yescomplemented
97C157C2×Q8⋊C811yescharacteristic, complemented, Sylow
98C2×Q8⋊C8—1571nomaximal, Sylow
99C2×C4×Q8—1571no—
100C2×C4⋊C8—1571no—
101C2×C4×C8—1571no—
102Q8⋊C8—1572no—
103Q8⋊C8—1572no—
104C22×Q8—1571no—
105C22×C8—3141no—
106C22×C8—3141no—
107C4×Q8—3141no—
108C4×Q8—3141no—
109C4×Q8—1571no—
110C4×Q8—1571no—
111C4×Q8—1571no—
112C4×Q8#1059784511yescharacteristic
113C2×C4⋊C4—3141no—
114C2×C4⋊C4—1571no—
115C2×C42—3141no—
116C2×C42—1571no—
117C4⋊C8—3141no—
118C4⋊C8—1572no—
119C4×C8—3141no—
120C4×C8—1572no—
121C2×Q8—3141no—
122C2×Q8—3142no—
123C2×Q8—1574no—
124C2×Q8—1571no—
125C2×Q8C2×#1059784511yescharacteristic
126C22×C4—3141no—
127C22×C4—3141no—
128C22×C4—1571no—
129C22×C4—1571no—
130C22×C4—1571no—
131C2×C8—3144no—
132C2×C8—3142no—
133C2×C8—3142no—
134C2×C8—6282no—
135C4⋊C4—21no—
136C4⋊C4—3141no—
137C4⋊C4—3141no—
138C4⋊C4—3141no—
139C4⋊C4—3141no—
140C4⋊C4—1571no—
141C4⋊C4C2×#1059784511yescharacteristic
142C42—21no—
143C42—3141no—
144C42—3141no—
145C42—3141no—
146C42—1571no—
147C42—1571no—
148C42—1571no—
149C42C2×#1059784511yescharacteristic
150C23—1571no—
151Q8—21no—
152Q8—3144no—
153Q8—3141no—
154Q8—1572no—
155Q8D157⋊C812yescomplemented
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—3142no—
159C2×C4—3144no—
160C2×C4—3141no—
161C2×C4—3141no—
162C2×C4—6282no—
163C2×C4—1571no—
164C2×C4—1571no—
165C2×C4—1571no—
166C2×C4—1572no—
167C2×C4—1572no—
168C2×C4—1572no—
169C2×C4—1572no—
170C2×C4C22⋊#1059784511yescharacteristic
171C2×C4C22⋊#1059784511yescharacteristic
172C2×C4C22×#1059784511yescharacteristic
173C8—3144no—
174C8—6282no—
175C22—1572no—
176C22—1572no—
177C22—1572no—
178C22C2×C22⋊#1059784511yescharacteristic
179C4—22no—
180C4—21no—
181C4—41no—
182C4—3142no—
183C4—3141no—
184C4—6281no—
185C4—1572no—
186C4—1571no—
187C4—1571no—
188C4#1909783412yes—
189C4C2×D157⋊C811yescharacteristic
190C4D157⋊M1611yescharacteristic
191C2—1574no—
192C2D157⋊(C22⋊C8)11yescharacteristic
193C2C2×#1909783411yescharacteristic
194C2D157⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#10597845
commutatorC4×C157C22×C8
FrattiniC2×C4C22×#10597845
FittingC4×Q8×C157C4
socleC22×C157C2×C22⋊C4
prime radicalC22×D157C22⋊C4
cosocleC4×Dic157C23
hypercenterC4×Q8#10597845
hypocenterC157C2×Q8⋊C8
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #23758123C2semidirect
4. Q8⋊#11929441C2semidirect
30. Q8×D157C8semidirect
57. Dic314C2×C8semidirect
71. Q8×C157C2×C8semidirect
96. D157Q8⋊C8semidirect
97. C157C2×Q8⋊C8semidirect
155. Q8D157⋊C8semidirect

Maximal subgroups

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

subgroupindex
D157⋊(C4⋊C8)2
C4×D157⋊C82
#237581232
Q8⋊#119294412
C4×Q8×D1572
C2×Q8⋊C8157