Finite Group Databasebeta

← #37576142

Subgroups — #37576142

Order 24448.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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×#20259318C211yesmaximal, characteristic, complemented
3C2×#20259293C211yesmaximal, characteristic, complemented
4#27132572C214yesmaximal, complemented
5#27132570C211yesmaximal, characteristic, complemented
6#27132565C211yesmaximal, characteristic, complemented
7#27132539C214yesmaximal, complemented
8C4×(C2×D191)⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C191C211yesmaximal, characteristic, complemented
10C2×#15690288—24no—
11C2×#15690288—24no—
12C2×#15690288—24no—
13C22×D764C2212yes—
14C22×D764C2211yescharacteristic
15C22×C22⋊D191C2214yes—
16C22×C22⋊D191C2212yes—
17C22×C4×D191C2212yes—
18C22×C4×D191C2211yescharacteristic
19#20259318—24no—
20#20259293—24no—
21C2×C4⋊Dic191C2211yescharacteristic
22C2×C22⋊Dic191C2212yes—
23C2×(C2×D191)⋊C4C2212yes—
24C2×(C2×D191)⋊C4C2214yes—
25C2×C4×Dic191C2211yescharacteristic
26C191⋊#95C2212yes—
27C191⋊#95C2214yes—
28C22×Q8×C191C2211yescharacteristic
29C22×D4×C191C2211yescharacteristic
30#124×C191—24no—
31C2×C22⋊C4×C191C2214yes—
32C2×C42×C191C2211yescharacteristic
33C23×D191C2311yescharacteristic
34C23×D191C2312yes—
35#15690288—232no—
36C2×D764—24no—
37C2×D764—24no—
38C2×D764—22no—
39C2×D764—24no—
40C2×D764—22no—
41C2×D764D414yescomplemented
42C2×C22⋊D191—24no—
43C2×C22⋊D191—28no—
44C2×C22⋊D191—28no—
45C2×C22⋊D191—44no—
46C2×C22⋊D191—44no—
47C22×Dic191C2312yes—
48C22×Dic191C2311yescharacteristic
49C22×Dic191C2312yes—
50C2×C4×D191—24no—
51C2×C4×D191—24no—
52C2×C4×D191—22no—
53C2×C4×D191—24no—
54C2×C4×D191—22no—
55C2×C4×D191D414yescomplemented
56C4⋊Dic191—22no—
57C22⋊Dic191—24no—
58(C2×D191)⋊C4—24no—
59(C2×D191)⋊C4—44no—
60C4×Dic191—22no—
61C24×C191C2312yes—
62C2×Q8×C191—22no—
63C2×Q8×C191—22no—
64C2×Q8×C191D414yescomplemented
65C2×D4×C191—22no—
66C2×D4×C191—22no—
67C2×D4×C191—22no—
68C22×C4×C191C2312yes—
69C22×C4×C191C2312yes—
70C22×C4×C191C2311yescharacteristic
71C22⋊C4×C191—28no—
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×D191C2×D414yes—
82C22×D191C2×D412yes—
83D764—216no—
84D764—28no—
85C22⋊D191—416no—
86C22⋊D191—48no—
87C2×Dic191—22no—
88C2×Dic191—21no—
89C2×Dic191—21no—
90C2×Dic191—22no—
91C2×Dic191—22no—
92C2×Dic191—24no—
93C2×Dic191C2×D412yes—
94C2×Dic191#4114yescomplemented
95C4×D191—216no—
96C4×D191—28no—
97C23×C191—22no—
98C23×C191—22no—
99C23×C191—22no—
100C23×C191—24no—
101C23×C191—24no—
102C23×C191C2411yescharacteristic
103Q8×C191—28no—
104D4×C191—44no—
105C2×C4×C191—22no—
106C2×C4×C191—22no—
107C2×C4×C191—22no—
108C2×C4×C191—22no—
109C2×C4×C191—22no—
110C2×C4×C191C2×D412yes—
111C2×C4×C191C2×D412yes—
112C2×C4×C191C2×D414yes—
113C2×C4×C191C2×D412yes—
114C22×C191—42no—
115C22×C191—42no—
116C22×C191—42no—
117C22×C191—44no—
118C22×C191—44no—
119C22×C191C22×D411yescharacteristic
120C22×C191C22×D411yescharacteristic
121C22×C191C22×D411yescharacteristic
122C22×C191C2×#4112yes—
123C22×C191H(5,𝔽2)12yes—
124C2×D191—24no—
125C2×D191—22no—
126C2×D191—22no—
127C2×D191—22no—
128C2×D191—28no—
129C2×D191—42no—
130C2×D191—42no—
131C2×D191—42no—
132C2×D191—42no—
133C2×D191—44no—
134C4×C191—24no—
135C4×C191—42no—
136C4×C191#12014yes—
137Dic191—22no—
138Dic191—24no—
139Dic191—42no—
140C2×C191—82no—
141C2×C191C2×#12011yescharacteristic
142C2×C191#53211yescharacteristic
143C2×C191#53511yescharacteristic
144C2×C191#54612yes—
145C2×C191#55012yes—
146D191—42no—
147D191—44no—
148C191#243111yescharacteristic, complemented, Sylow
149#2431—1911nomaximal, Sylow
150C22×#41—1911no—
151C2×#127—1911no—
152C2×#124D19111yescharacteristic, complemented
153C2×#121—1911no—
154#392—1914no—
155#391—1911no—
156#390—1911no—
157#386—1914no—
158C4×C22⋊C4—1911no—
159C2×#41—3824no—
160C2×#41—3824no—
161C2×#41—3824no—
162C22×Q8C2×D19111yescharacteristic
163C22×D4—1914no—
164C22×D4—1912no—
165C22×D4—1912no—
166C22×D4—1911no—
167C22×D4C2×D19111yescharacteristic
168C23×C4—1912no—
169C23×C4—1911no—
170#127—3824no—
171#124—24no—
172#121—3824no—
173C2×C4⋊C4—1911no—
174C2×C22⋊C4—1912no—
175C2×C22⋊C4—1912no—
176C2×C22⋊C4—1914no—
177C2×C22⋊C4C2×D19114yes—
178C2×C42—1911no—
179C2×C42C2×D19111yescharacteristic
180#95—1912no—
181#95—1914no—
182C24—1911no—
183C24—1912no—
184C24C22×D19112yes—
185#41—38232no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8C22⋊D19114yescomplemented
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—3824no—
193C2×D4—3824no—
194C2×D4—3824no—
195C2×D4—3828no—
196C2×D4—3828no—
197C2×D4—3822no—
198C2×D4—3824no—
199C2×D4—3822no—
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—1912no—
210C22×C4—1911no—
211C22×C4—1914no—
212C22×C4C22×D19112yes—
213C22×C4C22×D19112yes—
214C22×C4C22×D19111yescharacteristic
215C4⋊C4—3822no—
216C22⋊C4—28no—
217C22⋊C4—3824no—
218C22⋊C4—3824no—
219C22⋊C4—7644no—
220C42—22no—
221C42—3822no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—24no—
226C23—24no—
227C23—3822no—
228C23—3821no—
229C23—3822no—
230C23—3821no—
231C23—3822no—
232C23—3824no—
233C23—3824no—
234C23—3822no—
235C23—1914no—
236C23—1912no—
237C23C23×D19111yescharacteristic
238Q8—28no—
239D4—44no—
240D4—38216no—
241D4—3828no—
242D4—76416no—
243D4—7648no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—38216no—
250C2×C4—3822no—
251C2×C4—3821no—
252C2×C4—3821no—
253C2×C4—3822no—
254C2×C4—3828no—
255C2×C4—3822no—
256C2×C4—3824no—
257C2×C4—1912no—
258C2×C4—1914no—
259C2×C4D4×D19112yes—
260C2×C4D4×D19112yes—
261C2×C4C2×C22⋊D19114yes—
262C2×C4C2×C22⋊D19112yes—
263C22—42no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—3822no—
269C22—3822no—
270C22—3828no—
271C22—3824no—
272C22—3822no—
273C22—7642no—
274C22—7642no—
275C22—7642no—
276C22—7642no—
277C22—7644no—
278C22C2×D4×D19111yescharacteristic
279C22C2×D4×D19111yescharacteristic
280C22C22×C22⋊D19111yescharacteristic
281C22#41×D19112yes—
282C22#2025936812yes—
283C4—24no—
284C4—42no—
285C4—3824no—
286C4—3822no—
287C4—7642no—
288C4#2025929014yes—
289C2—82no—
290C2—7642no—
291C2—7644no—
292C2C2×#2025929011yescharacteristic
293C2#2713321211yescharacteristic
294C2#2713323111yescharacteristic
295C2#2713330612yes—
296C2#2713333212yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
FittingC2×#124×C191C2
hypercenterC2×#124D191
hypocenterC191#2431
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259318C2semidirect
3. C2×#20259293C2semidirect
4. #27132572C2semidirect
5. #27132570C2semidirect
6. #27132565C2semidirect
7. #27132539C2semidirect
8. C4×(C2×D191)⋊C4C2semidirect
9. C2×#124×C191C2semidirect
41. C2×D764D4semidirect
55. C2×C4×D191D4semidirect
64. C2×Q8×C191D4semidirect
94. C2×Dic191#41semidirect
148. C191#2431semidirect
152. C2×#124D191semidirect
188. C2×Q8C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C22×#156902882
C2×#202593182
C2×#202592932
#271325722
#271325702
#271325652
#271325392
C4×(C2×D191)⋊C42
C2×#124×C1912
#2431191