Finite Group Databasebeta

← #37576177

Subgroups — #37576177

Order 24448.

852 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D191C211yesmaximal, characteristic, complemented
2C2×#20259311C211yesmaximal, characteristic, complemented
3C2×#20259310C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic
5#27132576C214yesmaximal, complemented
6#27132568C211yesmaximal, characteristic, complemented
7#27132541C214yesmaximal, complemented
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
9#391×C191C211yesmaximal, characteristic, complemented
10C22×Dic382C2212yes—
11C22×Dic382C2211yescharacteristic
12C2×Q8×D191—24no—
13C2×Q8×D191—24no—
14C2×Q8×D191—24no—
15C22×D764C2212yes—
16C22×D764C2211yescharacteristic
17C22×C4×D191C2212yes—
18C22×C4×D191C2211yescharacteristic
19#20259311—24no—
20#20259310—24no—
21#20259297—24no—
22C2×(C2×D191)⋊C4C2212yes—
23C2×(C2×D191)⋊C4C2214yes—
24C2×(C2×D191)⋊C4C2214yes—
25C2×(C2×D191)⋊C4C2212yes—
26C2×C4×Dic191C2211yescharacteristic
27C191⋊#95C2212yes—
28C191⋊#95C2214yes—
29C22×Q8×C191C2211yescharacteristic
30C2×C4⋊C4×C191C2211yescharacteristic
31C2×C42×C191C2211yescharacteristic
32#95×C191C2214yes—
33C23×D191C2312yes—
34C23×D191C2311yescharacteristic
35C2×Dic382—24no—
36C2×Dic382—24no—
37C2×Dic382—22no—
38C2×Dic382—24no—
39C2×Dic382—22no—
40C2×Dic382D414yescomplemented
41Q8×D191—232no—
42C2×D764—24no—
43C2×D764—28no—
44C2×D764—22no—
45C2×D764—22no—
46C2×D764—22no—
47C22×Dic191C2312yes—
48C22×Dic191C2312yes—
49C22×Dic191C2311yescharacteristic
50C2×C4×D191—24no—
51C2×C4×D191—24no—
52C2×C4×D191—22no—
53C2×C4×D191—24no—
54C2×C4×D191—22no—
55C2×C4×D191D414yes—
56(C2×D191)⋊C4—24no—
57(C2×D191)⋊C4—24no—
58(C2×D191)⋊C4—28no—
59(C2×D191)⋊C4—28no—
60C4×Dic191—22no—
61C2×Q8×C191—22no—
62C2×Q8×C191—22no—
63C2×Q8×C191D414yescomplemented
64C22×C4×C191C2312yes—
65C22×C4×C191C2312yes—
66C22×C4×C191C2312yes—
67C22×C4×C191C2311yescharacteristic
68C4⋊C4×C191—22no—
69C42×C191—22no—
70C22×D191—21no—
71C22×D191—21no—
72C22×D191—22no—
73C22×D191—22no—
74C22×D191—22no—
75C22×D191—22no—
76C22×D191—24no—
77C22×D191—24no—
78C22×D191C2×D412yes—
79C22×D191#4114yes—
80Dic382—216no—
81Dic382—28no—
82D764—48no—
83D764—44no—
84C2×Dic191—22no—
85C2×Dic191—21no—
86C2×Dic191—22no—
87C2×Dic191—21no—
88C2×Dic191—22no—
89C2×Dic191—22no—
90C2×Dic191—22no—
91C2×Dic191C2×D414yes—
92C2×Dic191C2×D412yes—
93C4×D191—216no—
94C4×D191—28no—
95C23×C191C2411yescharacteristic
96Q8×C191—28no—
97C2×C4×C191—22no—
98C2×C4×C191—22no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—22no—
102C2×C4×C191—24no—
103C2×C4×C191—22no—
104C2×C4×C191C2×D412yes—
105C2×C4×C191C2×D412yes—
106C2×C4×C191C2×D414yes—
107C2×C4×C191C2×D412yes—
108C22×C191C22×D411yescharacteristic
109C22×C191C22×D411yescharacteristic
110C22×C191C22×D411yescharacteristic
111C22×C191C2×#4112yes—
112C22×C1912−1+412yes—
113C2×D191—22no—
114C2×D191—22no—
115C2×D191—24no—
116C2×D191—24no—
117C2×D191—42no—
118C2×D191—42no—
119C2×D191—42no—
120C2×D191—44no—
121C2×D191—44no—
122C2×D191#12414yescomplemented
123C4×C191—24no—
124C4×C191—42no—
125C4×C191—42no—
126C4×C191#12014yes—
127Dic191—24no—
128Dic191—22no—
129Dic191—42no—
130C2×C191C2×#12011yescharacteristic
131C2×C191C2×#12411yescharacteristic
132C2×C191#53611yescharacteristic
133C2×C191#54812yes—
134C2×C191#54812yes—
135D191—24no—
136D191—82no—
137C191#243611yescharacteristic, complemented, Sylow
138#2436—1911nomaximal, Sylow
139C23×Q8—1911no—
140C2×#124—1911no—
141C2×#124—1911no—
142C2×#121—1911no—
143#393—1914no—
144#391—1911no—
145#391D19111yescharacteristic, complemented
146#386—1914no—
147C4×C22⋊C4—1911no—
148C22×Q8—3824no—
149C22×Q8—3824no—
150C22×Q8—3824no—
151C22×Q8—1912no—
152C22×Q8—1911no—
153C22×Q8C2×D19111yescharacteristic
154C22×D4—1911no—
155C22×D4—1912no—
156C23×C4—1912no—
157C23×C4—1911no—
158#124—3824no—
159#124—3824no—
160#121—3824no—
161C2×C4⋊C4C2×D19111yescharacteristic
162C2×C22⋊C4—1912no—
163C2×C22⋊C4—1914no—
164C2×C22⋊C4—1914no—
165C2×C22⋊C4—1912no—
166C2×C42—1911no—
167C2×C42C2×D19111yescharacteristic
168#95—1912no—
169#95—1914no—
170#95C2×D19114yes—
171C24—1912no—
172C24—1911no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—3824no—
176C2×Q8—3824no—
177C2×Q8—38232no—
178C2×Q8—3822no—
179C2×Q8—3824no—
180C2×Q8—3822no—
181C2×Q8—1914no—
182C2×Q8D76414yescomplemented
183C2×D4—3824no—
184C2×D4—3828no—
185C2×D4—3822no—
186C2×D4—3822no—
187C2×D4—3822no—
188C22×C4—3824no—
189C22×C4—3824no—
190C22×C4—3822no—
191C22×C4—3824no—
192C22×C4—3822no—
193C22×C4—1912no—
194C22×C4—1912no—
195C22×C4—1911no—
196C22×C4—1914no—
197C22×C4C22×D19112yes—
198C22×C4C22×D19112yes—
199C22×C4C22×D19112yes—
200C22×C4C22×D19111yescharacteristic
201C4⋊C4—22no—
202C22⋊C4—3824no—
203C22⋊C4—3824no—
204C22⋊C4—3828no—
205C22⋊C4—3828no—
206C42—22no—
207C42—3822no—
208C23—3821no—
209C23—3821no—
210C23—3822no—
211C23—3822no—
212C23—3822no—
213C23—3822no—
214C23—3824no—
215C23—3824no—
216C23—1912no—
217C23—1914no—
218C23C23×D19111yescharacteristic
219Q8—28no—
220Q8—38216no—
221Q8—3828no—
222D4—7648no—
223D4—7644no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—38216no—
232C2×C4—3822no—
233C2×C4—3821no—
234C2×C4—3822no—
235C2×C4—3821no—
236C2×C4—3828no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—3822no—
240C2×C4—1914no—
241C2×C4—1912no—
242C2×C4D4×D19112yes—
243C2×C4D4×D19112yes—
244C2×C4C2×D76414yes—
245C2×C4C2×D76412yes—
246C22—3822no—
247C22—3824no—
248C22—3824no—
249C22—3822no—
250C22—1914no—
251C22—7642no—
252C22—7644no—
253C22—7644no—
254C22—7642no—
255C22—7642no—
256C22C2×D4×D19111yescharacteristic
257C22C2×D4×D19111yescharacteristic
258C22C22×D76411yescharacteristic
259C22#41×D19112yes—
260C22#2025937012yes—
261C4—24no—
262C4—42no—
263C4—42no—
264C4—3824no—
265C4—3822no—
266C4—7642no—
267C4#2025929214yes—
268C2—3824no—
269C2—15282no—
270C2C2×#2025929211yescharacteristic
271C2#124×D19111yescharacteristic
272C2#2713323811yescharacteristic
273C2#2713332312yes—
274C2#2713332512yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#391×C191C2
prime radicalC2×#20259297C2
hypercenter#391D191
hypocenterC191#2436
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D191C2semidirect
2. C2×#20259311C2semidirect
3. C2×#20259310C2semidirect
5. #27132576C2semidirect
6. #27132568C2semidirect
7. #27132541C2semidirect
8. C4×(C2×D191)⋊C4C2semidirect
9. #391×C191C2semidirect
40. C2×Dic382D4semidirect
63. C2×Q8×C191D4semidirect
122. C2×D191#124semidirect
137. C191#2436semidirect
145. #391D191semidirect
182. C2×Q8D764semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C2×#202593112
C2×#202593102
C2×#202592972
#271325762
#271325682
#271325412
C4×(C2×D191)⋊C42
#391×C1912
#2436191