Finite Group Databasebeta

← #37575840

Subgroups — #37575840

Order 24448.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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×#20259292C212yesmaximal, complemented
3C2×C22⋊C4×D191C211yesmaximal, characteristic, complemented
4#27132574C211yesmaximal, characteristic, complemented
5#27132573C211yesmaximal, characteristic, complemented
6#27132572C211yesmaximal, characteristic, complemented
7#27132541C212yesmaximal, complemented
8#27132541C212yesmaximal, complemented
9#27132539C212yesmaximal, complemented
10#27132525C211yesmaximal, characteristic, complemented
11#393×C191C211yesmaximal, characteristic, complemented
12C24×D191C2211yescharacteristic
13C2×#15690288—28no—
14C2×#15690288—24no—
15C22×D764C2212yes—
16C22×D764C2212yes—
17C22×D764C2211yescharacteristic, complemented
18C22×C22⋊D191C2212yes—
19C22×C4×D191C2211yescharacteristic
20C22×C4×D191C2211yescharacteristic
21C22×C4×D191C2212yescomplemented
22#20259292—28no—
23C2×C22⋊Dic191C2211yescharacteristic
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2211yescharacteristic
26C2×(C2×D191)⋊C4C2212yes—
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2212yescomplemented
29C2×(C2×D191)⋊C4C2212yescomplemented
30C22⋊C4×D191—24no—
31C191⋊#95C2212yescomplemented
32C191⋊#95C2211yescharacteristic, complemented
33C191⋊#95C2212yescomplemented
34C22×Q8×C191C2211yescharacteristic, complemented
35C2×C22⋊C4×C191C2212yes—
36C2×C22⋊C4×C191C2211yescharacteristic
37#95×C191C2211yescharacteristic, complemented
38#95×C191C2212yescomplemented
39C23×D191—22no—
40C23×D191—22no—
41C23×D191—24no—
42C23×D191—22no—
43C23×D191—42no—
44C23×D191C2312yes—
45C23×D191C2311yescharacteristic
46C23×D191C2311yescharacteristic
47#15690288—232no—
48C2×D764—24no—
49C2×D764—24no—
50C2×D764—24no—
51C2×D764—24no—
52C2×D764—24no—
53C2×D764—22no—
54C2×D764—42no—
55C2×D764D418yescomplemented
56C2×C22⋊D191—24no—
57C2×C22⋊D191—24no—
58C2×C22⋊D191—42no—
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2312yes—
61C22×Dic191C2311yescharacteristic
62C2×C4×D191—24no—
63C2×C4×D191—24no—
64C2×C4×D191—24no—
65C2×C4×D191—22no—
66C2×C4×D191—24no—
67C2×C4×D191—42no—
68C2×C4×D191D414yescomplemented
69C22⋊Dic191—22no—
70(C2×D191)⋊C4—24no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73(C2×D191)⋊C4—24no—
74(C2×D191)⋊C4—42no—
75(C2×D191)⋊C4—42no—
76C24×C191C2311yescharacteristic
77C2×Q8×C191—24no—
78C2×Q8×C191—22no—
79C22×C4×C191C2311yescharacteristic
80C22×C4×C191C2312yes—
81C22×C4×C191C2311yescharacteristic
82C22×C4×C191C2312yes—
83C22⋊C4×C191—22no—
84C22⋊C4×C191—24no—
85C22×D191—21no—
86C22×D191—21no—
87C22×D191—22no—
88C22×D191—28no—
89C22×D191—21no—
90C22×D191—22no—
91C22×D191—22no—
92C22×D191—22no—
93C22×D191—22no—
94C22×D191—22no—
95C22×D191—21no—
96C22×D191—22no—
97C22×D191—22no—
98C22×D191—22no—
99C22×D191—48no—
100C22×D191—48no—
101C22×D191—44no—
102C22×D191—44no—
103C22×D191C2×D414yes—
104C22×D191C2×D414yes—
105C22×D191C2×D412yes—
106C22×D191#4112yescomplemented
107D764—28no—
108D764—216no—
109D764—48no—
110C22⋊D191—48no—
111C2×Dic191—22no—
112C2×Dic191—22no—
113C2×Dic191—21no—
114C2×Dic191—22no—
115C2×Dic191—22no—
116C2×Dic191—22no—
117C2×Dic191C2×D412yes—
118C4×D191—216no—
119C4×D191—28no—
120C4×D191—44no—
121C23×C191—21no—
122C23×C191—22no—
123C23×C191—21no—
124C23×C191—22no—
125C23×C191#4112yescomplemented
126C23×C191C2411yescharacteristic
127Q8×C191—28no—
128C2×C4×C191—22no—
129C2×C4×C191—22no—
130C2×C4×C191—22no—
131C2×C4×C191—21no—
132C2×C4×C191—22no—
133C2×C4×C191—22no—
134C2×C4×C191—22no—
135C2×C4×C191—22no—
136C2×C4×C191C2×D414yes—
137C2×C4×C191C2×D412yes—
138C22×C191—22no—
139C22×C191—24no—
140C22×C191—41no—
141C22×C191—41no—
142C22×C191—42no—
143C22×C191C22×D411yescharacteristic
144C22×C191C22×D412yes—
145C22×C191C2×#4111yescharacteristic
146C22×C191C2×#4111yescharacteristic
147C22×C191H(5,𝔽2)12yes—
148C2×D191—22no—
149C2×D191—22no—
150C2×D191—24no—
151C2×D191—24no—
152C2×D191—24no—
153C2×D191—22no—
154C2×D191—24no—
155C2×D191—24no—
156C2×D191—24no—
157C2×D191—416no—
158C2×D191—41no—
159C2×D191—42no—
160C2×D191—42no—
161C2×D191—41no—
162C2×D191—42no—
163C2×D191—42no—
164C2×D191—42no—
165C2×D191#12014yes—
166C4×C191—24no—
167C4×C191—22no—
168C4×C191—41no—
169C4×C191—42no—
170Dic191—22no—
171Dic191—41no—
172Dic191—42no—
173C2×C191—42no—
174C2×C191C2×#12011yescharacteristic
175C2×C191#52511yescharacteristic
176C2×C191#53511yescharacteristic
177C2×C191#54612yes—
178C2×C191#54612yes—
179D191—24no—
180D191—44no—
181D191—42no—
182C191#240411yescharacteristic, complemented, Sylow
183#2404—1911nomaximal, Sylow
184C22×#41—1911no—
185C2×#120—1912no—
186C22×C22⋊C4—1911no—
187#393—1911no—
188#393D19111yescharacteristic, complemented
189#392—1911no—
190#392—1911no—
191#386—1912no—
192#386—1912no—
193#386—1912no—
194#381—1911no—
195C25—1911no—
196C2×#41—3828no—
197C2×#41—3824no—
198C22×Q8C2×D19111yescharacteristic, complemented
199C22×D4—1912no—
200C22×D4—1911no—
201C22×D4—1912no—
202C22×D4—1912no—
203C23×C4—1912no—
204C23×C4—1911no—
205C23×C4—1911no—
206#120—3828no—
207C2×C22⋊C4—3824no—
208C2×C22⋊C4—1911no—
209C2×C22⋊C4—1911no—
210C2×C22⋊C4—1911no—
211C2×C22⋊C4—1912no—
212C2×C22⋊C4—1912no—
213C2×C22⋊C4—1912no—
214C2×C22⋊C4—1912no—
215C2×C22⋊C4C2×D19111yescharacteristic
216C2×C22⋊C4C2×D19112yes—
217#95—1912no—
218#95—1911no—
219#95—1912no—
220#95C2×D19111yescharacteristic, complemented
221#95C2×D19112yescomplemented
222C24—3822no—
223C24—3822no—
224C24—3824no—
225C24—3822no—
226C24—1911no—
227C24—1912no—
228C24—1911no—
229C24—7642no—
230C24C22×D19111yescharacteristic
231#41—38232no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×D4—3824no—
235C2×D4—3824no—
236C2×D4—3824no—
237C2×D4—3824no—
238C2×D4—3824no—
239C2×D4—3824no—
240C2×D4—3824no—
241C2×D4—3822no—
242C2×D4—1918no—
243C2×D4—7642no—
244C2×D4—7642no—
245C22×C4—3824no—
246C22×C4—3824no—
247C22×C4—3824no—
248C22×C4—3822no—
249C22×C4—3824no—
250C22×C4—1912no—
251C22×C4—1911no—
252C22×C4—1911no—
253C22×C4—1914no—
254C22×C4—7642no—
255C22×C4C22×D19112yes—
256C22×C4C22×D19111yescharacteristic
257C22×C4C22×D19112yes—
258C22×C4C22×D19111yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—3824no—
262C22⋊C4—3822no—
263C22⋊C4—3822no—
264C22⋊C4—3822no—
265C22⋊C4—3824no—
266C22⋊C4—7642no—
267C22⋊C4—7642no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—3828no—
273C23—3821no—
274C23—3821no—
275C23—3822no—
276C23—3821no—
277C23—3822no—
278C23—3822no—
279C23—3822no—
280C23—3822no—
281C23—3822no—
282C23—3821no—
283C23—3822no—
284C23—3822no—
285C23—3822no—
286C23—1914no—
287C23—1912no—
288C23—1914no—
289C23—1912no—
290C23—7648no—
291C23—7648no—
292C23—7644no—
293C23—7644no—
294C23#1569028812yescomplemented
295C23C23×D19111yescharacteristic
296Q8—28no—
297D4—3828no—
298D4—38216no—
299D4—7648no—
300D4—7648no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—38216no—
310C2×C4—3828no—
311C2×C4—3822no—
312C2×C4—3821no—
313C2×C4—3822no—
314C2×C4—3822no—
315C2×C4—3822no—
316C2×C4—3822no—
317C2×C4—1912no—
318C2×C4—7644no—
319C2×C4D4×D19114yes—
320C2×C4D4×D19112yes—
321C22—24no—
322C22—22no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—3822no—
327C22—3824no—
328C22—3824no—
329C22—3822no—
330C22—3824no—
331C22—3824no—
332C22—3824no—
333C22—3824no—
334C22—3822no—
335C22—1914no—
336C22—76416no—
337C22—7641no—
338C22—7642no—
339C22—7642no—
340C22—7642no—
341C22—7642no—
342C22—7641no—
343C22—7642no—
344C22C2×D4×D19111yescharacteristic
345C22C2×D4×D19112yes—
346C22#41×D19111yescharacteristic
347C22C2×#1569028811yescharacteristic
348C22#2025936812yes—
349C4—24no—
350C4—22no—
351C4—41no—
352C4—42no—
353C4—3822no—
354C4—7641no—
355C4—7642no—
356C2—42no—
357C2—3824no—
358C2—7642no—
359C2—7644no—
360C2#120×D19111yescharacteristic
361C2#2713317611yescharacteristic
362C2#2713323111yescharacteristic
363C2#2713330412yes—
364C2#2713330612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#393×C191C2
hypercenter#393D191
hypocenterC191#2404
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259292C2semidirect
3. C2×C22⋊C4×D191C2semidirect
4. #27132574C2semidirect
5. #27132573C2semidirect
6. #27132572C2semidirect
7. #27132541C2semidirect
8. #27132541C2semidirect
9. #27132539C2semidirect
10. #27132525C2semidirect
11. #393×C191C2semidirect
17. C22×D764C22semidirect
21. C22×C4×D191C22semidirect
28. C2×(C2×D191)⋊C4C22semidirect
29. C2×(C2×D191)⋊C4C22semidirect
31. C191⋊#95C22semidirect
32. C191⋊#95C22semidirect
33. C191⋊#95C22semidirect
34. C22×Q8×C191C22semidirect
37. #95×C191C22semidirect
38. #95×C191C22semidirect
55. C2×D764D4semidirect
68. C2×C4×D191D4semidirect
106. C22×D191#41semidirect
125. C23×C191#41semidirect
182. C191#2404semidirect
188. #393D191semidirect
198. C22×Q8C2×D191semidirect
220. #95C2×D191semidirect
221. #95C2×D191semidirect
294. C23#15690288semidirect

Maximal subgroups

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

subgroupindex
C22×#156902882
C2×#202592922
C2×C22⋊C4×D1912
#271325742
#271325732
#271325722
#271325412
#271325392
#271325252
#393×C1912
#2404191