Finite Group Databasebeta

← #37575901

Subgroups — #37575901

Order 24448.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#20259303C211yesmaximal, characteristic, complemented
2C2×#20259278C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic191C211yesmaximal, characteristic, complemented
4#27132586C212yesmaximal, complemented
5#27132577C212yesmaximal, complemented
6#27132566C212yesmaximal, complemented
7#27132551C212yesmaximal, complemented
8#27132543C211yesmaximal, characteristic, complemented
9#27132538C211yesmaximal, characteristic, complemented
10#27132519C211yesmaximal, characteristic, complemented
11C2×#120×C191C211yesmaximal, characteristic
12C23×Dic191C2211yescharacteristic
13C23×Dic191C2211yescharacteristic
14#20259303—24no—
15#20259278—24no—
16C2×C4⋊Dic191C2212yescomplemented
17C2×Dic191⋊C4C2211yescharacteristic, complemented
18C2×Dic191⋊C4C2212yescomplemented
19C2×Dic191⋊C4C2211yescharacteristic, complemented
20C2×C22⋊Dic191—24no—
21C2×C22⋊Dic191C2211yescharacteristic
22C2×C22⋊Dic191C2211yescharacteristic
23C2×C22⋊Dic191C2212yes—
24C2×C22⋊Dic191C2212yes—
25C2×C22⋊Dic191C2212yes—
26C2×C22⋊Dic191C2212yes—
27C2×C22⋊Dic191C2211yescharacteristic, complemented
28C2×C22⋊Dic191C2211yescharacteristic, complemented
29C2×C22⋊Dic191C2212yescomplemented
30C2×C4×Dic191C2212yescomplemented
31C191⋊#95C2212yescomplemented
32C191⋊#95C2212yescomplemented
33C25×C191C2211yescharacteristic
34C22×D4×C191C2211yescharacteristic
35C22×D4×C191C2212yes—
36#120×C191—24no—
37C2×C22⋊C4×C191C2212yes—
38C2×C22⋊C4×C191C2211yescharacteristic
39C22×Dic191—22no—
40C22×Dic191—22no—
41C22×Dic191—22no—
42C22×Dic191—42no—
43C22×Dic191D414yescomplemented
44C22×Dic191C2312yes—
45C22×Dic191C2311yescharacteristic
46C22×Dic191C2312yes—
47C22×Dic191C2311yescharacteristic
48C22×Dic191C2311yescharacteristic
49C22×Dic191C2311yescharacteristic
50C4⋊Dic191—42no—
51Dic191⋊C4—22no—
52Dic191⋊C4—22no—
53Dic191⋊C4—24no—
54C22⋊Dic191—24no—
55C22⋊Dic191—24no—
56C22⋊Dic191—22no—
57C22⋊Dic191—22no—
58C22⋊Dic191—22no—
59C22⋊Dic191—22no—
60C22⋊Dic191—24no—
61C22⋊Dic191—42no—
62C22⋊Dic191—42no—
63C4×Dic191—24no—
64C24×C191—22no—
65C24×C191—24no—
66C24×C191—22no—
67C24×C191—22no—
68C24×C191—42no—
69C24×C191C2311yescharacteristic
70C24×C191C2312yes—
71C24×C191C2311yescharacteristic
72C2×D4×C191—22no—
73C2×D4×C191—22no—
74C2×D4×C191—24no—
75C2×D4×C191—22no—
76C2×D4×C191—24no—
77C2×D4×C191—42no—
78C22×C4×C191C2311yescharacteristic
79C22×C4×C191C2312yes—
80C22⋊C4×C191—24no—
81C22⋊C4×C191D414yes—
82C2×Dic191—28no—
83C2×Dic191—21no—
84C2×Dic191—21no—
85C2×Dic191—21no—
86C2×Dic191—21no—
87C2×Dic191—22no—
88C2×Dic191—21no—
89C2×Dic191—21no—
90C2×Dic191—22no—
91C2×Dic191—22no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—22no—
95C2×Dic191—22no—
96C2×Dic191—22no—
97C2×Dic191—44no—
98C2×Dic191C2×D412yes—
99C2×Dic191C2×D412yes—
100C23×C191—22no—
101C23×C191—22no—
102C23×C191—28no—
103C23×C191—22no—
104C23×C191—22no—
105C23×C191—22no—
106C23×C191—22no—
107C23×C191—22no—
108C23×C191—22no—
109C23×C191—21no—
110C23×C191—21no—
111C23×C191—21no—
112C23×C191—22no—
113C23×C191—22no—
114C23×C191—48no—
115C23×C191—48no—
116C23×C191—44no—
117C23×C191—44no—
118C23×C191C2×D412yes—
119C23×C191C2×D412yes—
120C23×C191#4114yes—
121C23×C191#4112yes—
122C23×C191C2411yescharacteristic
123D4×C191—48no—
124D4×C191—44no—
125C2×C4×C191—22no—
126C2×C4×C191—22no—
127C2×C4×C191—21no—
128C2×C4×C191—21no—
129C2×C4×C191—21no—
130C2×C4×C191C2×D414yes—
131C22×C191—22no—
132C22×C191—22no—
133C22×C191—24no—
134C22×C191—24no—
135C22×C191—24no—
136C22×C191—24no—
137C22×C191—24no—
138C22×C191—416no—
139C22×C191—42no—
140C22×C191—42no—
141C22×C191—42no—
142C22×C191—42no—
143C22×C191—41no—
144C22×C191—41no—
145C22×C191—41no—
146C22×C191—42no—
147C22×C191—42no—
148C22×C191#12114yes—
149C22×C191C22×D411yescharacteristic
150C22×C191C22×D411yescharacteristic
151C22×C191C2×#4111yescharacteristic
152C22×C191C2×#4112yes—
153C22×C191H(5,𝔽2)12yes—
154C4×C191—24no—
155C4×C191—41no—
156Dic191—22no—
157Dic191—22no—
158Dic191—41no—
159Dic191—41no—
160Dic191—42no—
161Dic191—42no—
162C2×C191—24no—
163C2×C191—44no—
164C2×C191—81no—
165C2×C191C2×#12111yescharacteristic
166C2×C191#52511yescharacteristic
167C2×C191#53511yescharacteristic
168C2×C191#54612yes—
169C2×C191#55112yes—
170C191#241111yescharacteristic, complemented, Sylow
171#2411—1911nomaximal, Sylow
172C2×#123—1911no—
173C2×#120D19111yescharacteristic
174C2×#117—1911no—
175C22×C22⋊C4—1911no—
176#396—1912no—
177#394—1912no—
178#390—1912no—
179#388—1912no—
180#386—1911no—
181#385—1911no—
182#379—1911no—
183C25C2×D19111yescharacteristic
184C22×D4C2×D19111yescharacteristic
185C22×D4C2×D19112yes—
186C23×C4—1911no—
187C23×C4—1911no—
188#123—3824no—
189#120—24no—
190#117—3824no—
191C2×C4⋊C4—1912no—
192C2×C4⋊C4—1911no—
193C2×C4⋊C4—1911no—
194C2×C4⋊C4—1912no—
195C2×C22⋊C4—3824no—
196C2×C22⋊C4—1912no—
197C2×C22⋊C4—1912no—
198C2×C22⋊C4—1911no—
199C2×C22⋊C4—1911no—
200C2×C22⋊C4—1911no—
201C2×C22⋊C4—1911no—
202C2×C22⋊C4—1912no—
203C2×C22⋊C4—1912no—
204C2×C22⋊C4—1912no—
205C2×C22⋊C4C2×D19112yes—
206C2×C22⋊C4C2×D19111yescharacteristic
207C2×C42—1912no—
208#95—1912no—
209#95—1912no—
210C24—22no—
211C24—24no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24C22×D19111yescharacteristic
216C24C22×D19112yes—
217C24C22×D19111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C22×C4—3822no—
225C22×C4—3822no—
226C22×C4—3822no—
227C22×C4—1911no—
228C22×C4—1912no—
229C22×C4—1912no—
230C22×C4—1911no—
231C22×C4—1914no—
232C22×C4—1911no—
233C22×C4—1911no—
234C22×C4—7642no—
235C22×C4C22×D19111yescharacteristic
236C22×C4C22×D19112yes—
237C4⋊C4—3822no—
238C4⋊C4—3822no—
239C4⋊C4—3824no—
240C4⋊C4—7642no—
241C22⋊C4—24no—
242C22⋊C4—3822no—
243C22⋊C4—3822no—
244C22⋊C4—3824no—
245C22⋊C4—3824no—
246C22⋊C4—3824no—
247C22⋊C4—3822no—
248C22⋊C4—3822no—
249C22⋊C4—7642no—
250C22⋊C4—7642no—
251C22⋊C4C22⋊D19114yes—
252C42—3824no—
253C23—22no—
254C23—22no—
255C23—28no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—21no—
266C23—22no—
267C23—48no—
268C23—48no—
269C23—44no—
270C23—44no—
271C23D4×D19112yes—
272C23C2×C22⋊D19112yes—
273C23#1569028614yes—
274C23#1569028612yes—
275C23C23×D19111yescharacteristic
276D4—48no—
277D4—44no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—3821no—
284C2×C4—3821no—
285C2×C4—3821no—
286C2×C4—3821no—
287C2×C4—3828no—
288C2×C4—3821no—
289C2×C4—3821no—
290C2×C4—3822no—
291C2×C4—3822no—
292C2×C4—3822no—
293C2×C4—3822no—
294C2×C4—3822no—
295C2×C4—3822no—
296C2×C4—3822no—
297C2×C4—3822no—
298C2×C4—1912no—
299C2×C4—1912no—
300C2×C4—7644no—
301C2×C4C2×C22⋊D19114yes—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22#2025929314yes—
320C22C2×D4×D19111yescharacteristic
321C22C22×C22⋊D19111yescharacteristic
322C22C2×#1569028612yes—
323C22C2×#1569028611yescharacteristic
324C22#2025936712yes—
325C4—24no—
326C4—41no—
327C4—3822no—
328C4—3822no—
329C4—7642no—
330C4—7641no—
331C4—7642no—
332C4—7641no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2C2×#2025929311yescharacteristic
337C2#2713316911yescharacteristic
338C2#2713322811yescharacteristic
339C2#2713330712yes—
340C2#2713334412yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting, prime radicalC2×#120×C191C2
hypercenterC2×#120D191
hypocenterC191#2411
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#20259303C2semidirect
2. C2×#20259278C2semidirect
3. C22×C22⋊Dic191C2semidirect
4. #27132586C2semidirect
5. #27132577C2semidirect
6. #27132566C2semidirect
7. #27132551C2semidirect
8. #27132543C2semidirect
9. #27132538C2semidirect
10. #27132519C2semidirect
16. C2×C4⋊Dic191C22semidirect
17. C2×Dic191⋊C4C22semidirect
18. C2×Dic191⋊C4C22semidirect
19. C2×Dic191⋊C4C22semidirect
27. C2×C22⋊Dic191C22semidirect
28. C2×C22⋊Dic191C22semidirect
29. C2×C22⋊Dic191C22semidirect
30. C2×C4×Dic191C22semidirect
31. C191⋊#95C22semidirect
32. C191⋊#95C22semidirect
43. C22×Dic191D4semidirect
170. C191#2411semidirect

Maximal subgroups

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

subgroupindex
C2×#202593032
C2×#202592782
C22×C22⋊Dic1912
#271325862
#271325772
#271325662
#271325512
#271325432
#271325382
#271325192
C2×#120×C1912
#2411191