Finite Group Databasebeta

← #34101039

Subgroups — #34101039

Order 20096.

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14578943C211yesmaximal, characteristic
2C2×#19097707C211yesmaximal, characteristic, complemented
3#23757281C211yesmaximal, characteristic, complemented
4#23757277C211yesmaximal, characteristic
5C2×#19097689C211yesmaximal, characteristic, complemented
6C2×#19097687C211yesmaximal, characteristic
7#405×C157C211yesmaximal, characteristic, complemented
8C22×Dic314C2211yescharacteristic
9C22×D628C2211yescharacteristic
10C2×#14578943—21no—
11C2×#14578943—21no—
12C2×#14578943—22no—
13C2×#14578943C414yescomplemented
14C2×#14578938C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16#19097707—22no—
17#19097694—22no—
18#19097691—22no—
19#19097689—22no—
20#19097687—22no—
21C2×M16×C157C2211yescharacteristic
22C22×C8×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#97×C157—21no—
25C4×C8×C157—21no—
26C23×D157—21no—
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×Dic314—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314C2×C412yes—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×D628—21no—
37C2×D628—21no—
38C2×D628C2×C412yes—
39C22×Dic157—21no—
40#14578943—24no—
41#14578943—24no—
42#14578938—21no—
43#14578938—21no—
44#14578938—21no—
45#14578938—21no—
46C2×#11929434—21no—
47C2×#11929434—21no—
48C22×C4×C157C2311yescharacteristic
49M16×C157—21no—
50M16×C157D412yes—
51C2×C8×C157—21no—
52C2×C8×C157C2×C412yes—
53C2×C8×C157D411yescharacteristic
54C2×C8×C157D411yescharacteristic
55C2×C8×C157D412yes—
56C4⋊C4×C157D412yes—
57C22⋊C4×C157—21no—
58C42×C157—21no—
59C22×D157—22no—
60C22×D157—41no—
61C22×D157—41no—
62C22×D157—82no—
63Dic314—22no—
64Dic314—22no—
65Dic314—41no—
66Dic314—41no—
67D628—22no—
68D628—22no—
69D628—41no—
70D628—41no—
71C2×Dic157—22no—
72C2×Dic157—41no—
73C2×Dic157—41no—
74#11929434—41no—
75#11929434—41no—
76C23×C157#4111yescharacteristic
77C2×C4×C157—22no—
78C2×C4×C157C22×C411yescharacteristic
79C2×C4×C157C2×D411yescharacteristic
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157#4111yescharacteristic
84C8×C157—22no—
85C8×C157C22⋊C412yes—
86C8×C157C22⋊C412yes—
87C22×C157—22no—
88C22×C157C4×D411yescharacteristic
89C22×C157#12111yescharacteristic
90C22×C157#12411yescharacteristic
91C2×D157—42no—
92C2×D157—82no—
93C2×D157—82no—
94C2×D157—82no—
95C4×C157—42no—
96C4×C157C2×C22⋊C411yescharacteristic
97C4×C157C4×D411yescharacteristic
98C4×C157#12111yescharacteristic
99C4×C157#12711yescharacteristic
100Dic157—42no—
101C2×C157—21no—
102C2×C157—21no—
103C2×C157#39011yescharacteristic
104C2×C157#47112yes—
105D157—82no—
106C157#197711yescharacteristic, complemented, Sylow
107#1977—1571nomaximal, Sylow
108C22×SD16—1571no—
109C2×#108—1571no—
110#419—1571no—
111#418—1571no—
112C2×#101—1571no—
113C2×#100—1571no—
114#405D15711yescharacteristic, complemented
115C22×Q8—1571no—
116C22×D4—1571no—
117C2×SD16—3141no—
118C2×SD16—3141no—
119C2×SD16—3142no—
120C2×SD16—1574no—
121C2×M16—1571no—
122C2×M16—1571no—
123C2×M16C2×D15711yescharacteristic
124C22×C8C2×D15711yescharacteristic
125#117C2×D15711yescharacteristic
126#108—3142no—
127#103—3142no—
128#102—3142no—
129#101—3142no—
130#100—3142no—
131#97—21no—
132C4×C8—21no—
133C24—3141no—
134C2×Q8—3141no—
135C2×Q8—3141no—
136C2×Q8—3141no—
137C2×Q8—3141no—
138C2×Q8—3141no—
139C2×Q8—1572no—
140C2×D4—3141no—
141C2×D4—3141no—
142C2×D4—3141no—
143C2×D4—3141no—
144C2×D4—3141no—
145C2×D4—1572no—
146C22×C4—3141no—
147C22×C4C22×D15711yescharacteristic
148SD16—3144no—
149SD16—3144no—
150M16—21no—
151M16—3141no—
152M16—3141no—
153M16—3141no—
154M16—3141no—
155M16D62812yes—
156C2×C8—21no—
157C2×C8—3141no—
158C2×C8—3141no—
159C2×C8C4×D15712yes—
160C2×C8C22⋊D15711yescharacteristic
161C2×C8C22⋊D15711yescharacteristic
162C2×C8D62812yes—
163C4⋊C4D62812yes—
164C22⋊C4—21no—
165C42—21no—
166C23—3142no—
167C23—6281no—
168C23—6281no—
169C23—12562no—
170C23#1457895411yescharacteristic
171Q8—3142no—
172Q8—3142no—
173Q8—6281no—
174Q8—6281no—
175D4—3142no—
176D4—3142no—
177D4—6281no—
178D4—6281no—
179C2×C4—22no—
180C2×C4—3142no—
181C2×C4—6281no—
182C2×C4—6281no—
183C2×C4C2×C4×D15711yescharacteristic
184C2×C4C2×C22⋊D15711yescharacteristic
185C2×C4C2×D62811yescharacteristic
186C2×C4C2×D62811yescharacteristic
187C2×C4C2×D62811yescharacteristic
188C2×C4#1457895411yescharacteristic
189C8—22no—
190C8—6281no—
191C8—6281no—
192C8#1457893112yes—
193C8#1457893112yes—
194C22—22no—
195C22—6282no—
196C22—12562no—
197C22—12562no—
198C22—12562no—
199C22C4×D62811yescharacteristic
200C22#1909774511yescharacteristic
201C22#1909775911yescharacteristic
202C4—42no—
203C4—6282no—
204C4C2×#1457893111yescharacteristic
205C4C4×D62811yescharacteristic
206C4#1909774511yescharacteristic
207C4#1909776811yescharacteristic
208C2—21no—
209C2—21no—
210C2—12562no—
211C2#2375717511yescharacteristic
212C2#2375753612yes—
subgroupG/H
centerC22#19097745
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#405×C157C2
socleC22×C157#121
prime radicalC22×D628C22
cosocleC22×C4×C157C23
hypercenter#405D157
hypocenterC157#1977
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405

Complements

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

NH ≅ G/Ntype
2. C2×#19097707C2semidirect
3. #23757281C2semidirect
5. C2×#19097689C2semidirect
7. #405×C157C2semidirect
13. C2×#14578943C4semidirect
106. C157#1977semidirect
114. #405D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789432
C2×#190977072
#237572812
#237572772
C2×#190976892
C2×#190976872
#405×C1572
#1977157