Finite Group Databasebeta

← #34116149

Subgroups — #34116149

Order 20096.

380 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 88 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×D157⋊C8C211yesmaximal, characteristic
2C2×#19097834C212yesmaximal, complemented
3#23758215C211yesmaximal, characteristic
4#23758146C212yesmaximal, complemented
5C22⋊Q8×D157C211yesmaximal, characteristic
6C2×D157⋊C8—24no—
7C2×D157⋊C8—21no—
8C2×D157⋊C8C2212yes—
9C22×#11929441—21no—
10C2×C4⋊#10597845C2212yes—
11C2×C22⋊#10597845—21no—
12#19097834—24no—
13(C2×#10597845)⋊C4—21no—
14C2×Q8×D157C2211yescharacteristic
15C22×C4×D157C2211yescharacteristic
16#19097751C411yescharacteristic, complemented
17#19097750C411yescharacteristic, complemented
18#19097749—21no—
19#19097748—21no—
20#19097747C411yescharacteristic, complemented
21C4⋊C4×D157—21no—
22C4⋊C4×D157C2211yescharacteristic
23C22⋊C4×D157—21no—
24C22⋊Q8×C157C411yescharacteristic, complemented
25C22×#10597845—22no—
26D157⋊C8—24no—
27D157⋊C8—22no—
28D157⋊C8D414yes—
29C2×#11929441—24no—
30C2×#11929441—22no—
31C4⋊#10597845—22no—
32C4⋊#10597845—22no—
33C22⋊#10597845—42no—
34C23×D157D411yescharacteristic
35C2×Dic314—21no—
36C2×Dic314C2×C411yescharacteristic
37Q8×D157—22no—
38Q8×D157—22no—
39C22×Dic157D411yescharacteristic
40C2×C4×D157—22no—
41C2×C4×D157—22no—
42C2×C4×D157—21no—
43C2×C4×D157—21no—
44C2×C4×D157C2×C411yescharacteristic
45C2×C4×D157D411yescharacteristic
46C2×C4×D157D411yescharacteristic
47C2×C4×D157C2311yescharacteristic
48C4⋊Dic157—21no—
49C4⋊Dic157C2×C411yescharacteristic
50#14578933—21no—
51#14578933—21no—
52#14578933—21no—
53C22⋊Dic157—21no—
54#14578931—21no—
55#14578931—21no—
56C2×Q8×C157C2×C411yescharacteristic
57C22×C4×C157C2×C411yescharacteristic
58C4⋊C4×C157—21no—
59C4⋊C4×C157C2×C411yescharacteristic
60C22⋊C4×C157—21no—
61C2×#10597845—42no—
62C2×#10597845—42no—
63C2×#10597845—42no—
64#11929441—24no—
65C22×D157—22no—
66C22×D157—22no—
67C22×D157C22⋊C411yescharacteristic
68C22×D157SD1612yes—
69C22×D157Q1612yes—
70C22×D157C2×D411yescharacteristic
71Dic314—22no—
72Dic314—41no—
73Dic314—41no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157C22⋊C411yescharacteristic
79C2×Dic157C2×D411yescharacteristic
80C4×D157—22no—
81C4×D157—22no—
82C4×D157—21no—
83C4×D157—21no—
84C4×D157—42no—
85C4×D157—42no—
86C4×D157C2×D412yes—
87C4×D157#4112yes—
88C23×C157C22⋊C411yescharacteristic
89Q8×C157—22no—
90C2×C4×C157—22no—
91C2×C4×C157—21no—
92C2×C4×C157—21no—
93C2×C4×C157C22⋊C411yescharacteristic
94C2×C4×C157C22×C411yescharacteristic
95C22×C157—21no—
96C22×C157—21no—
97C22×C157#10312yescomplemented
98C22×C157C2×C22⋊C411yescharacteristic
99C2×D157—21no—
100C2×D157—21no—
101C2×D157—24no—
102C2×D157—24no—
103C2×D157#10312yescomplemented
104C2×D157#12112yes—
105C2×D157C2×SD1611yescharacteristic
106C2×D157C2×Q1611yescharacteristic
107C2×D157#13512yes—
108#10597845—82no—
109C4×C157—21no—
110C4×C157—41no—
111C4×C157—41no—
112C4×C157C4×D412yes—
113Dic157—21no—
114Dic157—41no—
115Dic157—41no—
116Dic157#12011yescharacteristic
117Dic157#12311yescharacteristic
118C2×C157—22no—
119C2×C157#38611yescharacteristic
120C2×C157C2×#10311yescharacteristic
121C2×C157#41611yescharacteristic
122D157—22no—
123D157#46512yes—
124D157#46712yes—
125C157#189211yescharacteristic, complemented, Sylow
126#1892—1571nomaximal, Sylow
127C23×C8—1571no—
128C2×C22⋊Q8—1571no—
129C2×#103—1572no—
130#380—1571no—
131#340—1572no—
132C22×Q8—1571no—
133C23×C4—1571no—
134C22×C8—3141no—
135C22×C8—3141no—
136C22×C8—3144no—
137C22×C8—1572no—
138C22⋊Q8—3141no—
139C22⋊Q8—3141no—
140C22⋊Q8—1571no—
141C22⋊Q8—1571no—
142C22⋊Q8—1571no—
143C22⋊Q8#1059784511yescharacteristic, complemented
144C2×C4⋊C4—3141no—
145C2×C4⋊C4—1572no—
146C2×C4⋊C4—1571no—
147C2×C22⋊C4—3141no—
148C2×C22⋊C4—3141no—
149#103—3144no—
150#95—3141no—
151C24—1571no—
152C2×Q8—3141no—
153C2×Q8—3142no—
154C2×Q8—3142no—
155C2×Q8—1571no—
156C2×Q8C2×#1059784511yescharacteristic
157C22×C4—3142no—
158C22×C4—3142no—
159C22×C4—3141no—
160C22×C4—3141no—
161C22×C4—3142no—
162C22×C4—1571no—
163C22×C4—1571no—
164C22×C4—1571no—
165C22×C4—1571no—
166C22×C4—1571no—
167C22×C4C2×#1059784511yescharacteristic
168C2×C8—3144no—
169C2×C8—3144no—
170C2×C8—3142no—
171C2×C8—3142no—
172C2×C8—1574no—
173C4⋊C4—21no—
174C4⋊C4—3141no—
175C4⋊C4—3141no—
176C4⋊C4—3141no—
177C4⋊C4—3141no—
178C4⋊C4—3142no—
179C4⋊C4—3142no—
180C4⋊C4—1571no—
181C4⋊C4C2×#1059784511yescharacteristic
182C22⋊C4—21no—
183C22⋊C4—3141no—
184C22⋊C4—3141no—
185C22⋊C4—3141no—
186C22⋊C4—6282no—
187C23—3142no—
188C23—3142no—
189C23—1572no—
190C23—1572no—
191C23—1571no—
192C23—1571no—
193C23C22⋊#1059784511yescharacteristic
194Q8—22no—
195Q8—3142no—
196Q8—6281no—
197Q8—6281no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—3142no—
202C2×C4—3141no—
203C2×C4—3141no—
204C2×C4—3141no—
205C2×C4—3141no—
206C2×C4—3142no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—6282no—
210C2×C4—6282no—
211C2×C4—6282no—
212C2×C4—6282no—
213C2×C4—6282no—
214C2×C4—1571no—
215C2×C4—1571no—
216C2×C4—1572no—
217C2×C4—1572no—
218C2×C4C22⋊#1059784511yescharacteristic
219C2×C4C22×#1059784511yescharacteristic
220C8—3144no—
221C22—21no—
222C22—21no—
223C22—3144no—
224C22—3144no—
225C22—3141no—
226C22—3141no—
227C22—1572no—
228C22—1572no—
229C22—1571no—
230C22—1571no—
231C22—1572no—
232C22#1909783412yescomplemented
233C22C2×C22⋊#1059784511yescharacteristic
234C4—21no—
235C4—41no—
236C4—41no—
237C4—3141no—
238C4—6281no—
239C4—6281no—
240C4—1571no—
241C4—1571no—
242C4—12562no—
243C4D4×#1059784512yes—
244C2—22no—
245C2—3142no—
246C2—1572no—
247C2—1572no—
248C2#2375822111yescharacteristic
249C2C2×#1909783411yescharacteristic
250C2#2375826311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597845
commutatorC2×C4×C157C22×C4
FrattiniC2×C4C22×#10597845
FittingC22⋊Q8×C157C4
socleC22×C157C2×C22⋊C4
prime radicalC23×D157D4
cosocleC2×C4×D157C23
hypercenterC22⋊Q8#10597845
hypocenterC157#1892
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#19097834C2semidirect
4. #23758146C2semidirect
16. #19097751C4semidirect
17. #19097750C4semidirect
20. #19097747C4semidirect
24. C22⋊Q8×C157C4semidirect
97. C22×C157#103semidirect
103. C2×D157#103semidirect
125. C157#1892semidirect
143. C22⋊Q8#10597845semidirect
232. C22#19097834semidirect

Maximal subgroups

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

subgroupindex
C22×D157⋊C82
C2×#190978342
#237582152
#237581462
C22⋊Q8×D1572
#1892157