Finite Group Databasebeta

← #34099330

Subgroups — #34099330

Order 20096.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#23757943C211yesmaximal, characteristic
2#23757619C211yesmaximal, characteristic, complemented
3#23757554C211yesmaximal, characteristic, complemented
4#23757484C211yesmaximal, characteristic, complemented
5#23757003C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic, complemented
7#331×C157C211yesmaximal, characteristic, complemented
8C2×#14578953—21no—
9C2×Dic628—21no—
10#19097771C2211yescharacteristic
11#19097767—21no—
12#19097743—22no—
13#19097742—21no—
14C4×Dic314C2211yescharacteristic
15C4×D628C2211yescharacteristic
16D4×Dic157—21no—
17#19097705—21no—
18#19097696—21no—
19#19097693—21no—
20#19097692—22no—
21C4×#11929434C2211yescharacteristic
22#127×C157C2211yescharacteristic
23C4⋊C8×C157C2211yescharacteristic
24C4⋊C8×C157C2211yescharacteristic
25#14578953—42no—
26#14578953—42no—
27C2×Dic314—21no—
28C2×Dic314D411yescharacteristic
29C2×D628D411yescharacteristic
30C2×C22⋊D157—22no—
31C2×C22⋊D157—22no—
32C22×Dic157—22no—
33C2×C4×D157—21no—
34Dic628—42no—
35C2×#11929434—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157—21no—
38C4⋊Dic157D411yescharacteristic
39C4⋊Dic157D411yescharacteristic
40#14578933—21no—
41C22⋊Dic157—22no—
42#14578931—21no—
43C4×Dic157—21no—
44C2×D4×C157—21no—
45C2×D4×C157—21no—
46C2×D4×C157D412yes—
47C2×C8×C157—21no—
48C2×C8×C157—21no—
49C42×C157C2311yescharacteristic
50C22×D157—21no—
51Dic314—21no—
52Dic314—41no—
53Dic314—41no—
54Dic314D812yes—
55D628—21no—
56D628—21no—
57C22⋊D157—44no—
58C22⋊D157—82no—
59C2×Dic157—21no—
60C2×Dic157—21no—
61C2×Dic157—21no—
62C2×Dic157—44no—
63C2×Dic157—41no—
64C4×D157—41no—
65C4×D157—41no—
66#11929434—41no—
67C23×C157—22no—
68D4×C157—42no—
69D4×C157—41no—
70D4×C157—41no—
71D4×C157—41no—
72D4×C157—41no—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×D411yescharacteristic
76C8×C157—41no—
77C8×C157—41no—
78C22×C157—42no—
79C22×C157—42no—
80C22×C157—42no—
81C22×C157#12011yescharacteristic
82C2×D157—41no—
83C2×D157—41no—
84C2×D157—41no—
85C4×C157—21no—
86C4×C157C2×D811yescharacteristic
87C4×C157#13511yescharacteristic
88C4×C157AGL(1,ℤ/8)11yescharacteristic
89C4×C157#13711yescharacteristic
90Dic157—22no—
91Dic157—41no—
92Dic157—41no—
93Dic157—81no—
94C2×C157—82no—
95C2×C157#44711yescharacteristic
96C2×C157#45211yescharacteristic
97C2×C157#45511yescharacteristic
98D157—81no—
99C157#164111yescharacteristic, complemented, Sylow
100#1641—1571nomaximal, Sylow
101#550—1571no—
102#486—1571no—
103#474—1571no—
104#462—1571no—
105#331D15711yescharacteristic, complemented
106Q8⋊C8—1571no—
107D4⋊C8—1571no—
108C2×#41—3141no—
109C2×Q16—3141no—
110C4⋊Q8—1571no—
111#127—3141no—
112#127C2×D15711yescharacteristic
113#121—3142no—
114#121—3141no—
115C4×Q8—1571no—
116C4×D4—3141no—
117C4×D4—1571no—
118#107—3141no—
119C4⋊C8C2×D15711yescharacteristic
120C4⋊C8C2×D15711yescharacteristic
121#103—3141no—
122#102—3141no—
123#102—3142no—
124C4×C8—1571no—
125#41—6282no—
126#41—6282no—
127C2×Q8—3141no—
128C2×Q8—1571no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—3142no—
132C2×D4—3142no—
133C2×D4—1571no—
134C2×D4D62812yes—
135C22×C4—3142no—
136C22×C4—3141no—
137Q16—6282no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—3141no—
141C4⋊C4—3141no—
142C4⋊C4—3141no—
143C4⋊C4—3141no—
144C4⋊C4—1571no—
145C4⋊C4—1571no—
146C22⋊C4—3142no—
147C22⋊C4—3141no—
148C42—3141no—
149C42C22×D15711yescharacteristic
150C23—22no—
151C23—3141no—
152Q8—3141no—
153Q8—6281no—
154Q8—6281no—
155Q8—1572no—
156D4—42no—
157D4—41no—
158D4—41no—
159D4—41no—
160D4—41no—
161D4—3141no—
162D4—3141no—
163D4—6284no—
164D4—12562no—
165C2×C4—3141no—
166C2×C4—3141no—
167C2×C4—3141no—
168C2×C4—6284no—
169C2×C4—6281no—
170C2×C4—6281no—
171C2×C4—6281no—
172C2×C4D4×D15711yescharacteristic
173C2×C4D4×D15711yescharacteristic
174C2×C4C2×D62811yescharacteristic
175C8—41no—
176C8—41no—
177C8—6281no—
178C22—42no—
179C22—42no—
180C22—42no—
181C22—6281no—
182C22—6281no—
183C22—6281no—
184C22#1909774111yescharacteristic
185C4—21no—
186C4—3142no—
187C4—6281no—
188C4—6281no—
189C4—12561no—
190C4D8×D15711yescharacteristic
191C4#1909779011yescharacteristic
192C4#1909779811yescharacteristic
193C4#1909780011yescharacteristic
194C2—82no—
195C2—12561no—
196C2#2375740811yescharacteristic
197C2#2375743611yescharacteristic
198C2#2375745311yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#331×C157C2
socleC22×C157#120
prime radical#23757943C2
hypercenter#331D157
hypocenterC157#1641
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #23757619C2semidirect
3. #23757554C2semidirect
4. #23757484C2semidirect
5. #23757003C2semidirect
6. #23757000C2semidirect
7. #331×C157C2semidirect
99. C157#1641semidirect
105. #331D157semidirect

Maximal subgroups

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

subgroupindex
#237579432
#237576192
#237575542
#237574842
#237570032
#237570002
#331×C1572
#1641157