Finite Group Databasebeta

← #37576141

Subgroups — #37576141

Order 24448.

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

normal subgroups only

This group has 952 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×#15690288C211yesmaximal, characteristic, complemented
2C2×#20259319C211yesmaximal, characteristic, complemented
3C2×#20259311C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic, complemented
5#27132573C214yesmaximal, complemented
6#27132564C211yesmaximal, characteristic, complemented
7#27132541C214yesmaximal, complemented
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
9#391×C191C211yesmaximal, characteristic, complemented
10C2×#15690288—24no—
11C2×#15690288—24no—
12C2×#15690288—24no—
13C22×D764C2211yescharacteristic
14C22×D764C2212yes—
15C22×D764C2214yes—
16C22×D764C2211yescharacteristic
17C22×D764C2212yes—
18C22×C4×D191C2212yes—
19C22×C4×D191C2211yescharacteristic
20#20259319—24no—
21#20259311—24no—
22#20259297—24no—
23C2×(C2×D191)⋊C4C2212yes—
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2214yes—
26C2×(C2×D191)⋊C4C2214yes—
27C2×C4×Dic191C2211yescharacteristic
28C191⋊#95C2212yes—
29C22×Q8×C191C2211yescharacteristic
30C2×C4⋊C4×C191C2211yescharacteristic
31C2×C42×C191C2211yescharacteristic
32#95×C191C2214yes—
33C23×D191C2312yes—
34C23×D191C2311yescharacteristic
35C23×D191C2312yes—
36#15690288—232no—
37C2×D764—24no—
38C2×D764—24no—
39C2×D764—24no—
40C2×D764—28no—
41C2×D764—22no—
42C2×D764—24no—
43C2×D764—22no—
44C2×D764—22no—
45C2×D764—22no—
46C2×D764—22no—
47C2×D764—28no—
48C2×D764—44no—
49C2×D764—44no—
50C2×D764D414yescomplemented
51C22×Dic191C2312yes—
52C22×Dic191C2311yescharacteristic
53C2×C4×D191—24no—
54C2×C4×D191—24no—
55C2×C4×D191—22no—
56C2×C4×D191—24no—
57C2×C4×D191—22no—
58C2×C4×D191D414yescomplemented
59(C2×D191)⋊C4—24no—
60(C2×D191)⋊C4—28no—
61(C2×D191)⋊C4—24no—
62(C2×D191)⋊C4—44no—
63C4×Dic191—22no—
64C2×Q8×C191—22no—
65C2×Q8×C191—22no—
66C2×Q8×C191D414yescomplemented
67C22×C4×C191C2312yes—
68C22×C4×C191C2312yes—
69C22×C4×C191C2312yes—
70C22×C4×C191C2311yescharacteristic
71C4⋊C4×C191—22no—
72C42×C191—22no—
73C22×D191—22no—
74C22×D191—21no—
75C22×D191—22no—
76C22×D191—21no—
77C22×D191—22no—
78C22×D191—24no—
79C22×D191—24no—
80C22×D191—22no—
81C22×D191—22no—
82C22×D191—22no—
83C22×D191—22no—
84C22×D191—24no—
85C22×D191—24no—
86C22×D191C2×D414yes—
87C22×D191C2×D412yes—
88D764—216no—
89D764—28no—
90D764—416no—
91D764—48no—
92D764—44no—
93C2×Dic191—21no—
94C2×Dic191—21no—
95C2×Dic191—22no—
96C2×Dic191—22no—
97C2×Dic191—22no—
98C2×Dic191C2×D412yes—
99C4×D191—216no—
100C4×D191—28no—
101C23×C191C2411yescharacteristic
102Q8×C191—28no—
103C2×C4×C191—22no—
104C2×C4×C191—22no—
105C2×C4×C191—22no—
106C2×C4×C191—22no—
107C2×C4×C191—22no—
108C2×C4×C191—24no—
109C2×C4×C191C2×D412yes—
110C2×C4×C191C2×D412yes—
111C2×C4×C191C2×D414yes—
112C2×C4×C191C2×D412yes—
113C2×C4×C191#4114yescomplemented
114C22×C191C22×D411yescharacteristic
115C22×C191C22×D411yescharacteristic
116C22×C191C22×D411yescharacteristic
117C22×C191C2×#4112yes—
118C22×C191H(5,𝔽2)12yes—
119C2×D191—24no—
120C2×D191—22no—
121C2×D191—22no—
122C2×D191—22no—
123C2×D191—28no—
124C2×D191—42no—
125C2×D191—42no—
126C2×D191—42no—
127C2×D191—42no—
128C2×D191—44no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—44no—
133C2×D191—44no—
134C4×C191—24no—
135C4×C191—24no—
136C4×C191—42no—
137C4×C191#12014yes—
138Dic191—22no—
139Dic191—42no—
140C2×C191C2×#12011yescharacteristic
141C2×C191#53211yescharacteristic
142C2×C191#53511yescharacteristic
143C2×C191#54612yes—
144C2×C191#55012yes—
145D191—42no—
146D191—44no—
147D191—82no—
148C191#243111yescharacteristic, complemented, Sylow
149#2431—1911nomaximal, Sylow
150C22×#41—1911no—
151C2×#127—1911no—
152C2×#124—1911no—
153C2×#121—1911no—
154#392—1914no—
155#391D19111yescharacteristic, complemented
156#390—1911no—
157#386—1914no—
158C4×C22⋊C4—1911no—
159C2×#41—3824no—
160C2×#41—3824no—
161C2×#41—3824no—
162C22×Q8C2×D19111yescharacteristic
163C22×D4—1911no—
164C22×D4—1912no—
165C22×D4—1911no—
166C22×D4—1914no—
167C22×D4—1912no—
168C23×C4—1912no—
169C23×C4—1911no—
170#127—3824no—
171#124—3824no—
172#121—3824no—
173C2×C4⋊C4C2×D19111yescharacteristic
174C2×C22⋊C4—1912no—
175C2×C22⋊C4—1914no—
176C2×C22⋊C4—1912no—
177C2×C22⋊C4—1914no—
178C2×C42—1911no—
179C2×C42C2×D19111yescharacteristic
180#95—1912no—
181#95C2×D19114yes—
182C24—1912no—
183C24—1911no—
184C24—1912no—
185#41—38232no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8D76414yescomplemented
189C2×D4—3824no—
190C2×D4—3824no—
191C2×D4—3824no—
192C2×D4—3828no—
193C2×D4—3822no—
194C2×D4—3822no—
195C2×D4—3822no—
196C2×D4—3824no—
197C2×D4—3822no—
198C2×D4—3822no—
199C2×D4—3828no—
200C2×D4—1914no—
201C2×D4—7644no—
202C2×D4—7644no—
203C22×C4—3824no—
204C22×C4—3824no—
205C22×C4—3822no—
206C22×C4—3824no—
207C22×C4—3822no—
208C22×C4—1912no—
209C22×C4—1911no—
210C22×C4—1914no—
211C22×C4C22×D19112yes—
212C22×C4C22×D19112yes—
213C22×C4C22×D19112yes—
214C22×C4C22×D19111yescharacteristic
215C4⋊C4—22no—
216C22⋊C4—3824no—
217C22⋊C4—3828no—
218C22⋊C4—3824no—
219C22⋊C4—7644no—
220C42—22no—
221C42—3822no—
222C23—3822no—
223C23—3821no—
224C23—3822no—
225C23—3821no—
226C23—3822no—
227C23—3824no—
228C23—3824no—
229C23—3822no—
230C23—3822no—
231C23—3822no—
232C23—3822no—
233C23—3824no—
234C23—3824no—
235C23—1914no—
236C23—1912no—
237C23C23×D19111yescharacteristic
238Q8—28no—
239D4—38216no—
240D4—3828no—
241D4—76416no—
242D4—7648no—
243D4—7644no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—38216no—
251C2×C4—3821no—
252C2×C4—3821no—
253C2×C4—3828no—
254C2×C4—3822no—
255C2×C4—3822no—
256C2×C4—3822no—
257C2×C4—1912no—
258C2×C4D4×D19112yes—
259C2×C4D4×D19112yes—
260C2×C4C2×D76414yes—
261C2×C4C2×D76412yes—
262C2×C4#1569028814yescomplemented
263C22—3822no—
264C22—3822no—
265C22—3828no—
266C22—3824no—
267C22—3822no—
268C22—7642no—
269C22—7642no—
270C22—7644no—
271C22—7644no—
272C22—7642no—
273C22—7642no—
274C22—7642no—
275C22—7642no—
276C22—7642no—
277C22—7644no—
278C22C2×D4×D19111yescharacteristic
279C22C2×D4×D19111yescharacteristic
280C22C22×D76411yescharacteristic
281C22C2×#1569028812yes—
282C22#2025936812yes—
283C4—24no—
284C4—24no—
285C4—42no—
286C4—3822no—
287C4—7642no—
288C4#2025929214yes—
289C2—15282no—
290C2—7642no—
291C2—7644no—
292C2C2×#2025929211yescharacteristic
293C2#2713321411yescharacteristic
294C2#2713323111yescharacteristic
295C2#2713330412yes—
296C2#2713333512yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#391×C191C2
hypercenter#391D191
hypocenterC191#2431
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259319C2semidirect
3. C2×#20259311C2semidirect
4. C2×#20259297C2semidirect
5. #27132573C2semidirect
6. #27132564C2semidirect
7. #27132541C2semidirect
8. C4×(C2×D191)⋊C4C2semidirect
9. #391×C191C2semidirect
50. C2×D764D4semidirect
58. C2×C4×D191D4semidirect
66. C2×Q8×C191D4semidirect
113. C2×C4×C191#41semidirect
148. C191#2431semidirect
155. #391D191semidirect
188. C2×Q8D764semidirect
262. C2×C4#15690288semidirect

Maximal subgroups

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

subgroupindex
C22×#156902882
C2×#202593192
C2×#202593112
C2×#202592972
#271325732
#271325642
#271325412
C4×(C2×D191)⋊C42
#391×C1912
#2431191