Finite Group Databasebeta

← #34098708

Subgroups — #34098708

Order 20096.

288 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C22⋊#11929434C211yesmaximal, characteristic
2#23757133C211yesmaximal, characteristic
3#23757010C214yesmaximal, complemented
4C2×C22⋊Q8×C157C211yesmaximal, characteristic
5C22×#11929434—21no—
6C2×C4⋊Dic157—21no—
7C2×C22⋊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—
21C2×#11929434—22no—
22C2×#11929434—42no—
23C4⋊Dic157—42no—
24C22⋊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—42no—
45C2×Dic157—42no—
46C2×Dic157—42no—
47#11929434—42no—
48C23×C157—41no—
49C23×C157—41no—
50C23×C157C22⋊C411yescharacteristic
51C23×C157C22⋊C411yescharacteristic
52C23×C157SD1612yes—
53C23×C157Q1612yes—
54C23×C157C2×D411yescharacteristic
55Q8×C157—42no—
56Q8×C157—42no—
57C2×C4×C157—22no—
58C2×C4×C157—22no—
59C2×C4×C157—22no—
60C2×C4×C157—22no—
61C2×C4×C157—22no—
62C2×C4×C157—41no—
63C2×C4×C157—41no—
64C2×C4×C157—41no—
65C2×C4×C157—42no—
66C2×C4×C157—42no—
67C2×C4×C157C22⋊C411yescharacteristic
68C2×C4×C157C22⋊C411yescharacteristic
69C2×C4×C157C22×C411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C22×C157—21no—
72C22×C157—21no—
73C22×C157—21no—
74C22×C157—21no—
75C22×C157—21no—
76C22×C157—21no—
77C22×C157—42no—
78C22×C157—42no—
79C22×C157#10312yes—
80C22×C157#10312yes—
81C22×C157C2×C22⋊C411yescharacteristic
82C22×C157C2×SD1611yescharacteristic
83C22×C157C2×Q1611yescharacteristic
84C4×C157—21no—
85C4×C157—21no—
86C4×C157—21no—
87C4×C157—42no—
88C4×C157—42no—
89C4×C157#9912yes—
90Dic157—82no—
91C2×C157—21no—
92C2×C157—22no—
93C2×C157—22no—
94C2×C157—21no—
95C2×C157C2×#9911yescharacteristic
96C2×C157C2×#10311yescharacteristic
97C2×C157#42111yescharacteristic
98C157#152011yescharacteristic, complemented, Sylow
99#1520—1571nomaximal, Sylow
100C2×C22⋊Q8D15711yescharacteristic
101C2×C22⋊C8—1571no—
102#380—1571no—
103#328—1574no—
104C22×Q8Dic15711yescharacteristic, complemented
105C23×C4C2×D15711yescharacteristic
106C22×C8—3141no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8Dic15712yescomplemented
110C22⋊Q8C2×D15712yes—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—3141no—
113C2×C4⋊C4Dic15711yescharacteristic, complemented
114C2×C22⋊C4—21no—
115C2×C22⋊C4—3141no—
116C22⋊C8—3141no—
117C22⋊C8—1572no—
118#95—1572no—
119C24C22⋊D15711yescharacteristic
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×Dic15712yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
126C22×C4—3142no—
127C22×C4C2×Dic15711yescharacteristic
128C22×C4C2×Dic15711yescharacteristic
129C22×C4C22⋊D15711yescharacteristic
130C22×C4C22⋊D15711yescharacteristic
131C22×C4C22⋊D15711yescharacteristic
132C22×C4C22×D15711yescharacteristic
133C2×C8—3142no—
134C2×C8—6282no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—6282no—
139C4⋊C4C2×Dic15712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—6282no—
143C23—41no—
144C23—41no—
145C23C22⋊Dic15711yescharacteristic
146C23C22⋊Dic15711yescharacteristic
147C23#1457894112yes—
148C23#1457894412yes—
149C23C2×C22⋊D15711yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—6282no—
163C2×C4—6282no—
164C2×C4—6282no—
165C2×C4C22⋊Dic15711yescharacteristic
166C2×C4C22⋊Dic15711yescharacteristic
167C2×C4C22×Dic15711yescharacteristic
168C2×C4C2×C22⋊D15711yescharacteristic
169C8—6282no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22#1909769512yes—
179C22#1909769512yes—
180C22C2×C22⋊Dic15711yescharacteristic
181C22C2×#1457894111yescharacteristic
182C22C2×#1457894411yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—12562no—
189C4#1909768612yes—
190C2—21no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2C2×#1909768611yescharacteristic
195C2C2×#1909769511yescharacteristic
196C2#2375729011yescharacteristic
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#1520
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
98. C157#1520semidirect
104. C22×Q8Dic157semidirect
109. C22⋊Q8Dic157semidirect
113. C2×C4⋊C4Dic157semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊#119294342
#237571332
#237570102
C2×C22⋊Q8×C1572
#1520157