Finite Group Databasebeta

← #34099323

Subgroups — #34099323

Order 20096.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#23758009C211yesmaximal, characteristic
2#23757619C211yesmaximal, characteristic
3#23757559C211yesmaximal, characteristic, complemented
4#23757484C211yesmaximal, characteristic, complemented
5#23757003C211yesmaximal, characteristic, complemented
6#23757003C211yesmaximal, characteristic, complemented
7#331×C157C211yesmaximal, characteristic
8C2×Dic628—21no—
9#19097771C2211yescharacteristic
10#19097758—21no—
11#19097758—21no—
12#19097752—22no—
13C4×Dic314C2211yescharacteristic
14C4×Dic314C2211yescharacteristic
15D4×Dic157—21no—
16D4×Dic157—21no—
17#19097703—21no—
18#19097696—21no—
19#19097696—21no—
20#19097692—22no—
21C4×#11929434C2211yescharacteristic
22#127×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25C2×Dic314—21no—
26C2×Dic314D411yescharacteristic
27C2×Dic314D411yescharacteristic
28C22×Dic157—22no—
29Dic628—42no—
30C2×#11929434—21no—
31C4⋊Dic157—21no—
32C4⋊Dic157—21no—
33C4⋊Dic157D411yescharacteristic
34C4⋊Dic157D411yescharacteristic
35#14578933—21no—
36#14578933—21no—
37C22⋊Dic157—22no—
38C22⋊Dic157—22no—
39C22⋊Dic157—22no—
40C4×Dic157—21no—
41C4×Dic157—21no—
42C2×D4×C157—21no—
43C2×D4×C157—21no—
44C2×D4×C157D412yes—
45C2×C8×C157—21no—
46C2×C8×C157—21no—
47C42×C157C2311yescharacteristic
48Dic314—21no—
49Dic314—21no—
50Dic314—21no—
51Dic314—21no—
52Dic314—41no—
53Dic314—41no—
54C2×Dic157—21no—
55C2×Dic157—21no—
56C2×Dic157—21no—
57C2×Dic157—21no—
58C2×Dic157—41no—
59C2×Dic157—82no—
60#11929434—41no—
61C23×C157—22no—
62D4×C157—42no—
63D4×C157—41no—
64D4×C157—41no—
65D4×C157—41no—
66D4×C157—41no—
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C8×C157—41no—
71C8×C157—41no—
72C22×C157—42no—
73C22×C157—42no—
74C22×C157—42no—
75C22×C157#12011yescharacteristic
76C4×C157—21no—
77C4×C157#13511yescharacteristic
78C4×C157#13511yescharacteristic
79C4×C157AGL(1,ℤ/8)11yescharacteristic
80C4×C157#13711yescharacteristic
81Dic157—41no—
82Dic157—41no—
83Dic157—41no—
84Dic157—41no—
85Dic157—81no—
86C2×C157—82no—
87C2×C157#44911yescharacteristic
88C2×C157#45211yescharacteristic
89C2×C157#45511yescharacteristic
90C157#164011yescharacteristic, complemented, Sylow
91#1640—1571nomaximal, Sylow
92#559—1571no—
93#486—1571no—
94#475—1571no—
95#462—1571no—
96#331D15711yescharacteristic
97Q8⋊C8—1571no—
98Q8⋊C8—1571no—
99C2×Q16—3141no—
100C4⋊Q8—1571no—
101#127C2×D15711yescharacteristic
102#124—3141no—
103#124—3141no—
104#123—3142no—
105C4×Q8—1571no—
106C4×Q8—1571no—
107C4×D4—3141no—
108C4×D4—3141no—
109#106—3141no—
110C4⋊C8C2×D15711yescharacteristic
111C4⋊C8C2×D15711yescharacteristic
112#103—3141no—
113#103—3141no—
114#102—3142no—
115C4×C8—1571no—
116C2×Q8—3141no—
117C2×Q8—1571no—
118C2×Q8—1571no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D62812yes—
122C22×C4—3142no—
123Q16—6282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—3141no—
127C4⋊C4—3141no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4—3141no—
131C4⋊C4—1571no—
132C4⋊C4—1571no—
133C22⋊C4—3142no—
134C22⋊C4—3142no—
135C22⋊C4—3142no—
136C42—3141no—
137C42—3141no—
138C42C22×D15711yescharacteristic
139C23—22no—
140Q8—3141no—
141Q8—3141no—
142Q8—3141no—
143Q8—3141no—
144Q8—6281no—
145Q8—6281no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—3141no—
152C2×C4—3141no—
153C2×C4—3141no—
154C2×C4—3141no—
155C2×C4—6281no—
156C2×C4—12562no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×D62811yescharacteristic
160C8—41no—
161C8—41no—
162C8—6281no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#1909774111yescharacteristic
167C4—21no—
168C4—6281no—
169C4—6281no—
170C4—6281no—
171C4—6281no—
172C4—12561no—
173C4#1909779011yescharacteristic
174C4#1909779211yescharacteristic
175C4#1909779511yescharacteristic
176C4#1909780011yescharacteristic
177C2—82no—
178C2#2375741911yescharacteristic
179C2#2375743611yescharacteristic
180C2#2375745311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157#120
prime radical#127×C157C22
hypercenter#331D157
hypocenterC157#1640
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
3. #23757559C2semidirect
4. #23757484C2semidirect
5. #23757003C2semidirect
6. #23757003C2semidirect
90. C157#1640semidirect

Maximal subgroups

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

subgroupindex
#237580092
#237576192
#237575592
#237574842
#237570032
#331×C1572
#1640157