Finite Group Databasebeta

← #34101311

Subgroups — #34101311

Order 20096.

244 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 63 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#23757826C211yesmaximal, characteristic, complemented
2#23757289C211yesmaximal, characteristic
3C2×#19097697C211yesmaximal, characteristic
4#23757246C211yesmaximal, characteristic
5#23757034C211yesmaximal, characteristic, complemented
6#23757030C211yesmaximal, characteristic, complemented
7#344×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097770—21no—
10#19097762—21no—
11#19097748—22no—
12C4×Dic314—22no—
13#19097730C2211yescharacteristic
14#19097730C2211yescharacteristic
15C2×C4×Dic157C2211yescharacteristic
16#19097697—21no—
17#19097697—21no—
18#19097697—22no—
19#19097685—22no—
20C2×M16×C157C2211yescharacteristic
21C2×M16×C157C2211yescharacteristic
22#117×C157C2211yescharacteristic
23#14578954—21no—
24#14578954—21no—
25#14578954—21no—
26#14578954—21no—
27C2×Dic314—22no—
28C2×Dic314Q811yescharacteristic, complemented
29C2×D628Q811yescharacteristic, complemented
30C2×C22⋊D157—21no—
31C22×Dic157—21no—
32C2×C4×D157—21no—
33C4⋊Dic157—22no—
34C4⋊Dic157—21no—
35C4⋊Dic157—21no—
36#14578933—22no—
37#14578933—22no—
38C22⋊Dic157—21no—
39C22⋊Dic157—21no—
40C4×Dic157—21no—
41C4×Dic157—21no—
42C4×Dic157D411yescharacteristic
43C4×Dic157D411yescharacteristic
44C4×Dic157D412yes—
45C22×C4×C157C2311yescharacteristic
46M16×C157—21no—
47M16×C157—21no—
48M16×C157—21no—
49M16×C157—21no—
50C2×C8×C157—21no—
51C2×C8×C157—21no—
52C4⋊C4×C157—21no—
53C22⋊C4×C157D412yes—
54C42×C157—21no—
55C22×D157—21no—
56Dic314—21no—
57Dic314—21no—
58Dic314—44no—
59D628—21no—
60D628—21no—
61C22⋊D157—41no—
62C22⋊D157—41no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—22no—
66C2×Dic157—21no—
67C2×Dic157—42no—
68C2×Dic157—41no—
69C4×D157—41no—
70C4×D157—41no—
71C23×C157C2×D411yescharacteristic
72C2×C4×C157—22no—
73C2×C4×C157C2×D411yescharacteristic
74C2×C4×C157C2×D411yescharacteristic
75C2×C4×C157C2×Q811yescharacteristic
76C2×C4×C157#4111yescharacteristic
77C2×C4×C157#4111yescharacteristic
78C2×C4×C157#4111yescharacteristic
79C8×C157—41no—
80C8×C157—41no—
81C22×C157—41no—
82C22×C157#12011yescharacteristic
83C22×C157C22⋊Q811yescharacteristic
84C22×C157#12411yescharacteristic
85C2×D157—41no—
86C2×D157—81no—
87C4×C157—42no—
88C4×C157C22⋊Q811yescharacteristic
89C4×C157C22⋊Q811yescharacteristic
90C4×C157#12411yescharacteristic
91C4×C157#12411yescharacteristic
92Dic157—22no—
93Dic157—41no—
94Dic157—42no—
95Dic157—41no—
96Dic157—81no—
97C2×C157—21no—
98C2×C157—21no—
99C2×C157—21no—
100C2×C157#39311yescharacteristic
101D157—81no—
102C157#203111yescharacteristic, complemented, Sylow
103#2031—1571nomaximal, Sylow
104#533—1571no—
105#421—1571no—
106C2×#104—1571no—
107#410—1571no—
108#344D15711yescharacteristic, complemented
109#339—1571no—
110H(3,ℤ/4)—1571no—
111C2×#41—1571no—
112C2×M16C2×D15711yescharacteristic
113C2×M16C2×D15711yescharacteristic
114C4⋊Q8—3141no—
115#125—3141no—
116C22⋊Q8—3142no—
117C4×Q8—3142no—
118#117—1571no—
119#117—1571no—
120#117C2×D15711yescharacteristic
121C2×C42—1571no—
122#104—3141no—
123#104—3141no—
124#104—3142no—
125#99—3142no—
126#41—3141no—
127#41—3141no—
128#41—3141no—
129#41—3141no—
130C2×Q8—3142no—
131C2×Q8—1571no—
132C2×D4—3141no—
133C2×D4—1571no—
134C22×C4—3141no—
135C22×C4—3141no—
136C22×C4C22×D15711yescharacteristic
137M16—21no—
138M16—21no—
139M16—21no—
140M16—21no—
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—3142no—
145C4⋊C4—3142no—
146C4⋊C4—3142no—
147C4⋊C4—3141no—
148C4⋊C4—3141no—
149C22⋊C4—3141no—
150C22⋊C4—3141no—
151C22⋊C4D62812yes—
152C42—21no—
153C42—3141no—
154C42—3141no—
155C42—1571no—
156C42—1571no—
157C42—1572no—
158C23—3141no—
159C23C2×D62811yescharacteristic
160Q8—3141no—
161Q8—3141no—
162Q8—6284no—
163D4—3141no—
164D4—3141no—
165D4—6281no—
166D4—6281no—
167C2×C4—22no—
168C2×C4—3141no—
169C2×C4—3141no—
170C2×C4—3142no—
171C2×C4—3141no—
172C2×C4—6282no—
173C2×C4—6281no—
174C2×C4—6281no—
175C2×C4—6281no—
176C2×C4D4×D15711yescharacteristic
177C2×C4D4×D15711yescharacteristic
178C2×C4Q8×D15711yescharacteristic
179C2×C4#1457895311yescharacteristic
180C2×C4#1457895411yescharacteristic
181C2×C4#1457895411yescharacteristic
182C8—41no—
183C8—41no—
184C22—41no—
185C22—6281no—
186C22—12561no—
187C22#1909774111yescharacteristic
188C22#1909775111yescharacteristic
189C22#1909775911yescharacteristic
190C4—42no—
191C4—3142no—
192C4—6281no—
193C4—6282no—
194C4—6281no—
195C4—12561no—
196C4#1909774911yescharacteristic
197C4#1909774911yescharacteristic
198C4#1909775711yescharacteristic
199C4#1909775711yescharacteristic
200C2—21no—
201C2—21no—
202C2—21no—
203C2—12561no—
204C2#2375718611yescharacteristic
subgroupG/H
centerC4#19097757
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#344×C157C2
socleC2×C157#393
prime radicalC2×#14578954C22
hypercenter#344D157
hypocenterC157#2031
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #23757826C2semidirect
5. #23757034C2semidirect
6. #23757030C2semidirect
7. #344×C157C2semidirect
28. C2×Dic314Q8semidirect
29. C2×D628Q8semidirect
102. C157#2031semidirect
108. #344D157semidirect

Maximal subgroups

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

subgroupindex
#237578262
#237572892
C2×#190976972
#237572462
#237570342
#237570302
#344×C1572
#2031157