Finite Group Databasebeta

← #34098656

Subgroups — #34098656

Order 20096.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#23757237C211yesmaximal, characteristic
2C4×C22⋊Dic157C211yesmaximal, characteristic
3#23757010C214yesmaximal, complemented
4C2×C22⋊Q8×C157C211yesmaximal, characteristic
5C2×#14578938—21no—
6C2×C22⋊Dic157—21no—
7C2×C4×Dic157—21no—
8C22⋊#11929434—21no—
9C22⋊#11929434C2212yes—
10#19097678C2212yes—
11C22×Q8×C157C411yescharacteristic, complemented
12C23×C4×C157C2211yescharacteristic
13C22⋊Q8×C157—21no—
14C22⋊Q8×C157—21no—
15C22⋊Q8×C157C412yescomplemented
16C22⋊Q8×C157C2212yes—
17C2×C4⋊C4×C157—21no—
18C2×C4⋊C4×C157C411yescharacteristic, complemented
19C2×C22⋊C4×C157—21no—
20C22×Dic157—22no—
21#14578938—42no—
22C2×#11929434—22no—
23C22⋊Dic157—24no—
24C4×Dic157—42no—
25C24×C157D411yescharacteristic
26C2×Q8×C157—21no—
27C2×Q8×C157—42no—
28C2×Q8×C157C2×C412yes—
29C22×C4×C157—21no—
30C22×C4×C157—21no—
31C22×C4×C157—42no—
32C22×C4×C157C2×C411yescharacteristic
33C22×C4×C157C2×C411yescharacteristic
34C22×C4×C157D411yescharacteristic
35C22×C4×C157D411yescharacteristic
36C22×C4×C157D411yescharacteristic
37C22×C4×C157C2311yescharacteristic
38C4⋊C4×C157—22no—
39C4⋊C4×C157—21no—
40C4⋊C4×C157—22no—
41C4⋊C4×C157C2×C412yes—
42C22⋊C4×C157—22no—
43C22⋊C4×C157—22no—
44C2×Dic157—24no—
45C2×Dic157—42no—
46C2×Dic157—42no—
47#11929434—42no—
48C23×C157—22no—
49C23×C157—22no—
50C23×C157—41no—
51C23×C157C22⋊C411yescharacteristic
52C23×C157C22⋊C411yescharacteristic
53C23×C157C2×D411yescharacteristic
54Q8×C157—42no—
55Q8×C157—42no—
56C2×C4×C157—22no—
57C2×C4×C157—22no—
58C2×C4×C157—22no—
59C2×C4×C157—22no—
60C2×C4×C157—22no—
61C2×C4×C157—41no—
62C2×C4×C157—41no—
63C2×C4×C157—41no—
64C2×C4×C157—42no—
65C2×C4×C157—42no—
66C2×C4×C157C22⋊C411yescharacteristic
67C2×C4×C157C22⋊C411yescharacteristic
68C2×C4×C157C22×C411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C22×C157—21no—
71C22×C157—21no—
72C22×C157—21no—
73C22×C157—21no—
74C22×C157—21no—
75C22×C157—21no—
76C22×C157—21no—
77C22×C157—44no—
78C22×C157#10412yes—
79C22×C157C2×C22⋊C411yescharacteristic
80C22×C157#13712yes—
81C4×C157—21no—
82C4×C157—21no—
83C4×C157—21no—
84C4×C157—42no—
85C4×C157—42no—
86C4×C157#9912yes—
87Dic157—44no—
88C2×C157—22no—
89C2×C157—22no—
90C2×C157—41no—
91C2×C157C2×#9911yescharacteristic
92C2×C157#41911yescharacteristic
93C2×C157C2×#10411yescharacteristic
94C157#150811yescharacteristic, complemented, Sylow
95#1508—1571nomaximal, Sylow
96C2×C22⋊Q8D15711yescharacteristic
97#407—1571no—
98C4×C22⋊C4—1571no—
99#328—1574no—
100C22×Q8Dic15711yescharacteristic, complemented
101C23×C4C2×D15711yescharacteristic
102C2×M16—3141no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8Dic15712yescomplemented
106C22⋊Q8C2×D15712yes—
107C2×C4⋊C4—21no—
108C2×C4⋊C4Dic15711yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4—3141no—
111C2×C42—3141no—
112C22⋊C8—3141no—
113C22⋊C8—1572no—
114#95—1572no—
115C24C22⋊D15711yescharacteristic
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×Dic15712yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—42no—
122C22×C4—3142no—
123C22×C4C2×Dic15711yescharacteristic
124C22×C4C2×Dic15711yescharacteristic
125C22×C4C22⋊D15711yescharacteristic
126C22×C4C22⋊D15711yescharacteristic
127C22×C4C22⋊D15711yescharacteristic
128C22×C4C22×D15711yescharacteristic
129M16—6282no—
130C2×C8—3142no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4C2×Dic15712yes—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—3144no—
138C42—6282no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23C22⋊Dic15711yescharacteristic
143C23C22⋊Dic15711yescharacteristic
144C23C2×C22⋊D15711yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—3144no—
158C2×C4—6282no—
159C2×C4—6282no—
160C2×C4C22⋊Dic15711yescharacteristic
161C2×C4C22⋊Dic15711yescharacteristic
162C2×C4C22×Dic15711yescharacteristic
163C2×C4C2×C22⋊D15711yescharacteristic
164C8—6282no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22#1909769812yes—
174C22C2×C22⋊Dic15711yescharacteristic
175C22#1909780212yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—42no—
181C4—6284no—
182C4#1909768612yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×#1909768611yescharacteristic
187C2#2375728211yescharacteristic
188C2C2×#1909769811yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×C22⋊Q8×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC24×C157D4
cosocleC22×C4×C157C23
hypercenterC2×C22⋊Q8D157
hypocenterC157#1508
LCSG ▷ C2×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #23757010C2semidirect
11. C22×Q8×C157C4semidirect
15. C22⋊Q8×C157C4semidirect
18. C2×C4⋊C4×C157C4semidirect
94. C157#1508semidirect
100. C22×Q8Dic157semidirect
105. C22⋊Q8Dic157semidirect
108. C2×C4⋊C4Dic157semidirect

Maximal subgroups

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

subgroupindex
#237572372
C4×C22⋊Dic1572
#237570102
C2×C22⋊Q8×C1572
#1508157