Finite Group Databasebeta

← #34099301

Subgroups — #34099301

Order 20096.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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#23757625C211yesmaximal, characteristic, complemented
2#23757568C211yesmaximal, characteristic, complemented
3#23757563C211yesmaximal, characteristic
4#23757019C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic
7D4×Q8×C157C211yesmaximal, characteristic
8#19097771C2211yescharacteristic
9#19097704—21no—
10#19097702—21no—
11C4⋊#11929434C2211yescharacteristic
12C4⋊#11929434C2211yescharacteristic
13#19097695—21no—
14#19097694—22no—
15#19097692—21no—
16C4×#11929434C2211yescharacteristic
17C22×Q8×C157—21no—
18C4⋊Q8×C157—21no—
19C4⋊Q8×C157C2211yescharacteristic
20C22⋊Q8×C157—21no—
21C22⋊Q8×C157—22no—
22C4×Q8×C157C2211yescharacteristic
23C4×D4×C157—21no—
24C4×D4×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×#11929434—21no—
27C2×#11929434—21no—
28C2×#11929434—21no—
29C4⋊Dic157—21no—
30C4⋊Dic157—21no—
31C2×Q8×C157—22no—
32C2×Q8×C157—21no—
33C2×Q8×C157—42no—
34C2×Q8×C157—42no—
35C2×Q8×C157D411yescharacteristic
36C2×D4×C157D411yescharacteristic
37C22×C4×C157—22no—
38C22×C4×C157—21no—
39C4⋊C4×C157—22no—
40C4⋊C4×C157—21no—
41C4⋊C4×C157—21no—
42C4⋊C4×C157D411yescharacteristic
43C4⋊C4×C157D411yescharacteristic
44C4⋊C4×C157D412yes—
45C22⋊C4×C157—22no—
46C22⋊C4×C157—21no—
47C42×C157—21no—
48C42×C157C2311yescharacteristic
49Dic314—42no—
50C2×Dic157—41no—
51#11929434—41no—
52#11929434—41no—
53#11929434—41no—
54C23×C157—21no—
55Q8×C157—21no—
56Q8×C157—44no—
57Q8×C157—41no—
58Q8×C157—41no—
59Q8×C157SD1612yes—
60D4×C157—21no—
61D4×C157Q1612yes—
62C2×C4×C157—21no—
63C2×C4×C157—21no—
64C2×C4×C157—22no—
65C2×C4×C157—21no—
66C2×C4×C157—44no—
67C2×C4×C157—41no—
68C2×C4×C157—41no—
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C2×C4×C157C2×D411yescharacteristic
72C22×C157—22no—
73C22×C157—41no—
74C22×C157—41no—
75C22×C157#12011yescharacteristic
76C4×C157—22no—
77C4×C157—21no—
78C4×C157—42no—
79C4×C157—41no—
80C4×C157—41no—
81C4×C157C2×SD1611yescharacteristic
82C4×C157C2×Q1611yescharacteristic
83C4×C157AGL(1,ℤ/8)11yescharacteristic
84C4×C157#13711yescharacteristic
85Dic157—81no—
86C2×C157—42no—
87C2×C157#45011yescharacteristic
88C2×C157#45111yescharacteristic
89C2×C157#45411yescharacteristic
90C157#163711yescharacteristic, complemented, Sylow
91#1637—1571nomaximal, Sylow
92D4×Q8D15711yescharacteristic
93#487—1571no—
94#477—1571no—
95#476—1571no—
96#332—1571no—
97Q8⋊C8—1571no—
98D4⋊C8—1571no—
99C22×Q8—21no—
100C4⋊Q8—21no—
101C4⋊Q8—1571no—
102C4⋊Q8C2×D15711yescharacteristic
103C22⋊Q8—21no—
104C22⋊Q8—22no—
105C4×Q8C2×D15711yescharacteristic
106C4×D4—21no—
107C4×D4C2×D15711yescharacteristic
108#107—3141no—
109#106—3141no—
110C4⋊C8—1571no—
111C4⋊C8—1571no—
112#103—3141no—
113#103—3142no—
114#102—3141no—
115C4×C8—1571no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—3141no—
121C2×Q8C22⋊D15711yescharacteristic
122C2×D4C22⋊D15711yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C2×C8—3141no—
126C2×C8—3141no—
127C2×C8—3141no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3141no—
132C4⋊C4—3141no—
133C4⋊C4C22⋊D15712yes—
134C4⋊C4C22⋊D15711yescharacteristic
135C4⋊C4C22⋊D15711yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—21no—
138C42—21no—
139C42C22×D15711yescharacteristic
140C23—21no—
141Q8—21no—
142Q8—44no—
143Q8—41no—
144Q8—41no—
145Q8—6282no—
146Q8#1457894212yes—
147D4—21no—
148D4#1457894412yes—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—44no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—6281no—
157C2×C4C2×C22⋊D15711yescharacteristic
158C2×C4C2×C22⋊D15711yescharacteristic
159C2×C4C2×C22⋊D15711yescharacteristic
160C8—6281no—
161C8—6281no—
162C8—6281no—
163C22—22no—
164C22—41no—
165C22—41no—
166C22#1909774011yescharacteristic
167C4—22no—
168C4—21no—
169C4—42no—
170C4—41no—
171C4—41no—
172C4—12561no—
173C4C2×#1457894211yescharacteristic
174C4C2×#1457894411yescharacteristic
175C4#1909779711yescharacteristic
176C4#1909780211yescharacteristic
177C2—42no—
178C2#2375742411yescharacteristic
179C2#2375742911yescharacteristic
180C2#2375744611yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
FittingD4×Q8×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenterD4×Q8D157
hypocenterC157#1637
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23757625C2semidirect
2. #23757568C2semidirect
4. #23757019C2semidirect
5. Q8⋊#11929434C2semidirect
90. C157#1637semidirect

Maximal subgroups

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

subgroupindex
#237576252
#237575682
#237575632
#237570192
Q8⋊#119294342
D4⋊#119294342
D4×Q8×C1572
#1637157