Finite Group Databasebeta

← #34100604

Subgroups — #34100604

Order 20096.

338 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#19097750C211yesmaximal, characteristic
2C2×#19097689C212yesmaximal, complemented
3#23757237C211yesmaximal, characteristic
4#23757041C212yesmaximal, complemented
5C22×M16×C157C211yesmaximal, characteristic
6C22×Dic314C2211yescharacteristic
7C2×#14578938C2212yes—
8#19097750—21no—
9#19097750—21no—
10#19097750—21no—
11#19097750—21no—
12C2×C4⋊Dic157C2211yescharacteristic
13C2×#14578933C412yes—
14C2×C22⋊Dic157C412yes—
15#19097689—24no—
16C22⋊#11929434—21no—
17C22⋊#11929434—21no—
18C23×C4×C157C2211yescharacteristic
19C2×M16×C157—21no—
20C2×M16×C157—24no—
21C2×M16×C157C2212yes—
22C22×C8×C157—21no—
23C2×Dic314—22no—
24C2×Dic314—21no—
25C2×Dic314—21no—
26C2×Dic314—22no—
27C22×Dic157C2×C412yes—
28C22×Dic157C2×C412yes—
29#14578938—22no—
30#14578938—22no—
31C2×#11929434—22no—
32C4⋊Dic157—21no—
33C4⋊Dic157—21no—
34#14578933—22no—
35#14578933—22no—
36C22⋊Dic157—22no—
37C22⋊Dic157—22no—
38C24×C157C2×C411yescharacteristic
39C22×C4×C157—22no—
40C22×C4×C157—22no—
41C22×C4×C157C2×C411yescharacteristic
42C22×C4×C157D411yescharacteristic
43C22×C4×C157D411yescharacteristic
44C22×C4×C157D411yescharacteristic
45C22×C4×C157D411yescharacteristic
46C22×C4×C157C2311yescharacteristic
47M16×C157—24no—
48M16×C157—22no—
49M16×C157D414yes—
50C2×C8×C157—24no—
51C2×C8×C157—22no—
52Dic314—41no—
53Dic314—42no—
54Dic314—41no—
55C2×Dic157—22no—
56C2×Dic157—22no—
57C2×Dic157—24no—
58C2×Dic157—42no—
59#11929434—42no—
60C23×C157—22no—
61C23×C157—22no—
62C23×C157—41no—
63C23×C157C22⋊C411yescharacteristic
64C23×C157C22⋊C411yescharacteristic
65C23×C157C22×C411yescharacteristic
66C2×C4×C157—22no—
67C2×C4×C157—21no—
68C2×C4×C157—21no—
69C2×C4×C157—22no—
70C2×C4×C157—22no—
71C2×C4×C157—21no—
72C2×C4×C157—21no—
73C2×C4×C157C22⋊C411yescharacteristic
74C2×C4×C157C22⋊C411yescharacteristic
75C2×C4×C157C2×D412yes—
76C2×C4×C157C2×D411yescharacteristic
77C2×C4×C157C2×D411yescharacteristic
78C2×C4×C157#4112yes—
79C8×C157—24no—
80C22×C157—21no—
81C22×C157—21no—
82C22×C157—21no—
83C22×C157—21no—
84C22×C157—21no—
85C22×C157—21no—
86C22×C157—21no—
87C22×C157—44no—
88C22×C157#10112yescomplemented
89C22×C157C2×C22⋊C411yescharacteristic
90C22×C157C4×D412yes—
91C4×C157—21no—
92C4×C157—21no—
93C4×C157#12011yescharacteristic
94C4×C157#12112yes—
95C4×C157#12311yescharacteristic
96Dic157—42no—
97Dic157—42no—
98C2×C157—22no—
99C2×C157—22no—
100C2×C157—41no—
101C2×C157#38611yescharacteristic
102C2×C157C2×#10111yescharacteristic
103C2×C157#41311yescharacteristic
104C157#190111yescharacteristic, complemented, Sylow
105#1901—1571nomaximal, Sylow
106C22×M16D15711yescharacteristic
107C2×C22⋊Q8—1571no—
108C2×#101—1572no—
109#407—1571no—
110#343—1572no—
111C22×Q8—1571no—
112C23×C4C2×D15711yescharacteristic
113C2×M16—21no—
114C2×M16—24no—
115C2×M16—1572no—
116C2×M16C2×D15712yes—
117C22×C8—21no—
118C22⋊Q8—3141no—
119C22⋊Q8—3141no—
120C22⋊Q8—3141no—
121C22⋊Q8—3141no—
122C2×C4⋊C4—1572no—
123C2×C4⋊C4—1571no—
124C2×C22⋊C4—1572no—
125#101—3144no—
126C22⋊C8—3141no—
127C22⋊C8—3141no—
128C24C4×D15711yescharacteristic
129C2×Q8—3142no—
130C2×Q8—3141no—
131C2×Q8—3141no—
132C2×Q8—3142no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—1572no—
136C22×C4—1572no—
137C22×C4C4×D15711yescharacteristic
138C22×C4C22⋊D15711yescharacteristic
139C22×C4C22⋊D15711yescharacteristic
140C22×C4D62811yescharacteristic
141C22×C4D62811yescharacteristic
142C22×C4C22×D15711yescharacteristic
143M16—24no—
144M16—22no—
145M16—3142no—
146M16—3142no—
147M16C22⋊D15714yes—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—3142no—
151C4⋊C4—3141no—
152C4⋊C4—3141no—
153C4⋊C4—3142no—
154C4⋊C4—3142no—
155C22⋊C4—3142no—
156C22⋊C4—3142no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23#1457893111yescharacteristic
161C23#1457893111yescharacteristic
162C23C2×C4×D15711yescharacteristic
163Q8—6281no—
164Q8—6282no—
165Q8—6281no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3142no—
174C2×C4—3142no—
175C2×C4—3144no—
176C2×C4—6282no—
177C2×C4#1457893111yescharacteristic
178C2×C4#1457893111yescharacteristic
179C2×C4C2×C22⋊D15712yes—
180C2×C4C2×C22⋊D15711yescharacteristic
181C2×C4C2×D62811yescharacteristic
182C2×C4#1457895412yes—
183C8—24no—
184C8—6282no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1909768912yescomplemented
194C22C2×#1457893111yescharacteristic
195C22C4×C22⋊D15712yes—
196C4—21no—
197C4—21no—
198C4—6282no—
199C4—6282no—
200C4#1909774011yescharacteristic
201C4#1909774512yes—
202C4#1909775411yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2375715211yescharacteristic
207C2C2×#1909768911yescharacteristic
208C2#2375725511yescharacteristic
subgroupG/H
centerC22C2×#14578931
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
FittingC22×M16×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC24×C157C2×C4
cosocleC22×C4×C157C23
hypercenterC22×M16D157
hypocenterC157#1901
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#19097689C2semidirect
4. #23757041C2semidirect
88. C22×C157#101semidirect
104. C157#1901semidirect
193. C22#19097689semidirect

Maximal subgroups

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

subgroupindex
C2×#190977502
C2×#190976892
#237572372
#237570412
C22×M16×C1572
#1901157