Finite Group Databasebeta

← #34098581

Subgroups — #34098581

Order 20096.

252 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×Q8×D157C211yesmaximal, characteristic
2#23757329C211yesmaximal, characteristic
3C4×#14578937C211yesmaximal, characteristic
4#23757005C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757003C211yesmaximal, characteristic, complemented
7Q8⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157C411yescharacteristic
9C2×#14578937—21no—
10Q8×Dic157C411yescharacteristic
11C4×Dic314—21no—
12C4×Dic314C2211yescharacteristic
13#19097736—21no—
14#19097736C411yescharacteristic
15C4⋊C4×D157—21no—
16C4⋊C4×D157C411yescharacteristic
17C42×D157—21no—
18C42×D157C2211yescharacteristic
19C4⋊#11929434C2211yescharacteristic
20#19097683—21no—
21#19097681—21no—
22#19097681—21no—
23C4×#11929434C2211yescharacteristic
24C4×Q8×C157C2211yescharacteristic
25C4⋊C8×C157C2211yescharacteristic
26C4×C8×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314C2×C411yescharacteristic
29Q8×D157—22no—
30Q8×D157—22no—
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157C2×C411yescharacteristic
34C2×C4×D157D411yescharacteristic
35C2×C4×D157D411yescharacteristic
36#14578937—24no—
37C2×#11929434—21no—
38C2×#11929434—21no—
39C4⋊Dic157—21no—
40C4⋊Dic157C2×C411yescharacteristic
41#14578933—21no—
42#14578933—21no—
43#14578933—21no—
44C4×Dic157—21no—
45C4×Dic157—21no—
46C4×Dic157—21no—
47C4×Dic157C2×C411yescharacteristic
48C4×Dic157D411yescharacteristic
49C4×Dic157D411yescharacteristic
50C2×Q8×C157C2×C411yescharacteristic
51C2×C8×C157—21no—
52C2×C8×C157—21no—
53C4⋊C4×C157—21no—
54C4⋊C4×C157C2×C411yescharacteristic
55C42×C157—21no—
56C42×C157C2311yescharacteristic
57C22×D157C22⋊C411yescharacteristic
58Dic314—24no—
59Dic314—21no—
60Dic314M1612yescomplemented
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157C22⋊C411yescharacteristic
64C2×Dic157C22⋊C411yescharacteristic
65C2×Dic157C22⋊C411yescharacteristic
66C4×D157—24no—
67C4×D157—22no—
68C4×D157—21no—
69C4×D157—21no—
70C4×D157—21no—
71C4×D157—21no—
72C4×D157—42no—
73#11929434—22no—
74#11929434—41no—
75Q8×C157—21no—
76Q8×C157M1612yescomplemented
77C2×C4×C157—21no—
78C2×C4×C157—21no—
79C2×C4×C157C22×C411yescharacteristic
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C8×C157—22no—
83C8×C157—41no—
84C22×C157C2×C22⋊C411yescharacteristic
85C2×D157—21no—
86C2×D157—21no—
87C2×D157#10412yes—
88C4×C157—22no—
89C4×C157—21no—
90C4×C157—41no—
91C4×C157C2×M1611yescharacteristic
92C4×C157C2×M1611yescharacteristic
93C4×C157#13711yescharacteristic
94C4×C157#13711yescharacteristic
95Dic157—22no—
96Dic157—21no—
97Dic157—21no—
98Dic157—21no—
99Dic157—41no—
100C2×C157#40711yescharacteristic
101C2×C157#41911yescharacteristic
102C2×C157C2×#10411yescharacteristic
103D157—22no—
104C157#149111yescharacteristic, complemented, Sylow
105#1491—1571nomaximal, Sylow
106C2×C4×Q8—1571no—
107#432—1571no—
108C4×M16—1571no—
109Q8⋊C8—1571no—
110Q8⋊C8—1571no—
111Q8⋊C8—1571no—
112Q8⋊C8D15711yescharacteristic, complemented
113C22×Q8—1571no—
114C2×M16—3141no—
115C4×Q8—3141no—
116C4×Q8—3141no—
117C4×Q8—1571no—
118C4×Q8—1571no—
119C4×Q8—1571no—
120C4×Q8C2×D15711yescharacteristic
121C2×C4⋊C4—3141no—
122C2×C4⋊C4—1571no—
123C2×C42—3141no—
124C2×C42—1571no—
125C4⋊C8—1571no—
126C4⋊C8C2×D15711yescharacteristic
127C22⋊C8—3141no—
128#97—3141no—
129#97—3141no—
130C4×C8—1571no—
131C4×C8C2×D15711yescharacteristic
132C2×Q8—3141no—
133C2×Q8—3142no—
134C2×Q8—3142no—
135C2×Q8—1571no—
136C2×Q8C4×D15711yescharacteristic
137C22×C4—3141no—
138C22×C4—3141no—
139C22×C4—1571no—
140C22×C4—1571no—
141C22×C4—1571no—
142M16—3144no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—3141no—
146C2×C8—3141no—
147C4⋊C4—21no—
148C4⋊C4—3141no—
149C4⋊C4—3141no—
150C4⋊C4—3141no—
151C4⋊C4—3141no—
152C4⋊C4—1571no—
153C4⋊C4C4×D15711yescharacteristic
154C42—21no—
155C42—3141no—
156C42—3141no—
157C42—3141no—
158C42—1571no—
159C42—1571no—
160C42—1571no—
161C42C22×D15711yescharacteristic
162C23—1571no—
163Q8—21no—
164Q8—3144no—
165Q8—3141no—
166Q8—1572no—
167Q8#1457893712yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—3144no—
171C2×C4—3142no—
172C2×C4—3141no—
173C2×C4—3141no—
174C2×C4—3141no—
175C2×C4—3141no—
176C2×C4—3141no—
177C2×C4—3141no—
178C2×C4—6282no—
179C2×C4—1571no—
180C2×C4—1571no—
181C2×C4—1571no—
182C2×C4C2×C4×D15711yescharacteristic
183C2×C4D4×D15711yescharacteristic
184C2×C4D4×D15711yescharacteristic
185C8—22no—
186C8—41no—
187C8—3142no—
188C8—6281no—
189C22—3141no—
190C22—3141no—
191C22—1572no—
192C22C22⋊C4×D15711yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—3142no—
197C4—3141no—
198C4—3141no—
199C4—3141no—
200C4—6281no—
201C4M16×D15711yescharacteristic
202C4C2×#1457893711yescharacteristic
203C4#1909780011yescharacteristic
204C4#1909780311yescharacteristic
205C2—3142no—
206C2#2375723511yescharacteristic
207C2#2375728011yescharacteristic
208C2#104×D15711yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
FittingQ8⋊C8×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC22×D157C22⋊C4
cosocleC42×C157C23
hypercenterQ8⋊C8D157
hypocenterC157#1491
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #23757005C2semidirect
5. Q8⋊#11929434C2semidirect
6. #23757003C2semidirect
7. Q8⋊C8×C157C2semidirect
60. Dic314M16semidirect
76. Q8×C157M16semidirect
104. C157#1491semidirect
112. Q8⋊C8D157semidirect
167. Q8#14578937semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D1572
#237573292
C4×#145789372
#237570052
Q8⋊#119294342
#237570032
Q8⋊C8×C1572
#1491157