Finite Group Databasebeta

← #34116188

Subgroups — #34116188

Order 20096.

308 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 66 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#23758351C211yesmaximal, characteristic
2#23758260C211yesmaximal, characteristic
3C2×#19097832C212yesmaximal, complemented
4#23758154C212yesmaximal, complemented
5#23757845C211yesmaximal, characteristic
6#19097858—24no—
7D157⋊M16—21no—
8C2×#14578963—21no—
9C2×#14578963C2212yes—
10C2×#14578963C2212yes—
11#19097832—24no—
12#19097832—21no—
13#19097831—21no—
14C2×#14578953C2211yescharacteristic
15C22×Dic314C2211yescharacteristic
16#19097770—21no—
17#19097757—21no—
18#19097755C411yescharacteristic
19#19097754C411yescharacteristic
20#19097752C411yescharacteristic
21#19097749—21no—
22#19097748—21no—
23#19097727C2211yescharacteristic
24#123×C157C411yescharacteristic
25#14578964—22no—
26#14578963—22no—
27#14578963—22no—
28#14578963—22no—
29#14578963—22no—
30#14578963D414yes—
31D157⋊C8—22no—
32C2×#11929441—22no—
33C2×#11929441—22no—
34C2×#11929441—22no—
35#14578953—22no—
36#14578953—22no—
37C2×Dic314—21no—
38C2×Dic314—21no—
39C2×Dic314—22no—
40C2×Dic314—22no—
41C2×Dic314D411yescharacteristic
42C2×C22⋊D157C2×C411yescharacteristic
43C2×C22⋊D157D411yescharacteristic
44C22×Dic157C2×C411yescharacteristic
45C22×Dic157D411yescharacteristic
46C22×Dic157C2311yescharacteristic
47C2×C4×D157D411yescharacteristic
48C4⋊Dic157—21no—
49#14578933—21no—
50#14578933—21no—
51#14578933—21no—
52C22⋊Dic157—21no—
53C22⋊Dic157C2×C411yescharacteristic
54#14578931—21no—
55#14578931—21no—
56C4×Dic157—21no—
57C2×D4×C157C2×C411yescharacteristic
58C22×C4×C157C2×C411yescharacteristic
59C4⋊C4×C157—21no—
60C22⋊C4×C157—21no—
61C22⋊C4×C157C2×C411yescharacteristic
62#11929441—24no—
63#11929441—42no—
64C22×D157C22⋊C411yescharacteristic
65Dic314—22no—
66Dic314—41no—
67Dic314—42no—
68Dic314—41no—
69C22⋊D157—22no—
70C22⋊D157—22no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—22no—
76C2×Dic157C22⋊C411yescharacteristic
77C2×Dic157C2×D412yes—
78C2×Dic157C2×D411yescharacteristic
79C2×Dic157C2×D411yescharacteristic
80C2×Dic157#4112yes—
81C4×D157—21no—
82C4×D157—21no—
83C23×C157C22⋊C411yescharacteristic
84C23×C157C22×C411yescharacteristic
85D4×C157—22no—
86C2×C4×C157—21no—
87C2×C4×C157—21no—
88C2×C4×C157—22no—
89C2×C4×C157C22⋊C411yescharacteristic
90C22×C157—21no—
91C22×C157—41no—
92C22×C157—41no—
93C22×C157C2×C22⋊C411yescharacteristic
94C22×C157C4×D412yes—
95C2×D157—21no—
96C2×D157—41no—
97C4×C157—21no—
98C4×C157—41no—
99C4×C157—41no—
100Dic157—21no—
101Dic157—41no—
102Dic157—41no—
103Dic157#12011yescharacteristic
104Dic157#12112yes—
105Dic157#12311yescharacteristic
106C2×C157—21no—
107C2×C157—22no—
108C2×C157—41no—
109C2×C157#38611yescharacteristic
110D157—41no—
111C157#191511yescharacteristic, complemented, Sylow
112#1915—1571nomaximal, Sylow
113#568—1571no—
114#536—1571no—
115#413—1571no—
116C2×#101—1572no—
117#344—1572no—
118C2×#41—1571no—
119C22×Q8—1571no—
120#131—3144no—
121C2×M16—3141no—
122C2×M16—3141no—
123C2×M16—1572no—
124C2×M16—1572no—
125C4⋊Q8—3141no—
126#124—3141no—
127#123—1571no—
128#123—1571no—
129#123—1571no—
130#123#1059784511yescharacteristic
131C22⋊Q8—3141no—
132C22⋊Q8—3141no—
133#117—1571no—
134#101—3141no—
135#101—3144no—
136#100—3141no—
137#41—3142no—
138#41—3142no—
139C2×Q8—3142no—
140C2×Q8—3141no—
141C2×Q8—3141no—
142C2×Q8—3142no—
143C2×Q8—1571no—
144C2×D4—1571no—
145C2×D4—1571no—
146C2×D4C2×#1059784511yescharacteristic
147C22×C4—1571no—
148C22×C4—1571no—
149C22×C4—1571no—
150C22×C4—1571no—
151C22×C4C2×#1059784511yescharacteristic
152M16—3142no—
153M16—3142no—
154M16—3142no—
155M16—3142no—
156M16—3142no—
157M16—1574no—
158C2×C8—3142no—
159C2×C8—3142no—
160C2×C8—3142no—
161C2×C8—3142no—
162C4⋊C4—21no—
163C4⋊C4—3141no—
164C4⋊C4—3141no—
165C4⋊C4—3141no—
166C4⋊C4—3141no—
167C22⋊C4—21no—
168C22⋊C4—3141no—
169C22⋊C4—3141no—
170C22⋊C4—3141no—
171C22⋊C4—1571no—
172C22⋊C4C2×#1059784511yescharacteristic
173C42—3141no—
174C23—1571no—
175C23C22⋊#1059784511yescharacteristic
176C23C22×#1059784511yescharacteristic
177Q8—3142no—
178Q8—6281no—
179Q8—6282no—
180Q8—6281no—
181D4—22no—
182D4—3142no—
183D4—3142no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—3141no—
188C2×C4—3141no—
189C2×C4—3141no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—3141no—
193C2×C4—3142no—
194C2×C4—1571no—
195C2×C4—1571no—
196C2×C4—1571no—
197C2×C4—1572no—
198C2×C4—1572no—
199C2×C4C22⋊#1059784511yescharacteristic
200C8—3144no—
201C8—6282no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—3141no—
206C22—6281no—
207C22C2×C22⋊#1059784511yescharacteristic
208C22D4×#1059784512yes—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—3141no—
213C4—6281no—
214C4—6281no—
215C4—1571no—
216C4—1572no—
217C4—1571no—
218C2—21no—
219C2—22no—
220C2—41no—
221C2—6281no—
222C2#2375822111yescharacteristic
subgroupG/H
centerC2#23758221
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#123×C157C4
socleC2×C157#386
prime radicalC2×#14578953C22
cosocleC22×Dic157C23
hypercenter#123#10597845
hypocenterC157#1915
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. C2×#19097832C2semidirect
4. #23758154C2semidirect
111. C157#1915semidirect

Maximal subgroups

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

subgroupindex
#237583512
#237582602
C2×#190978322
#237581542
#237578452
#1915157