Finite Group Databasebeta

← #37576131

Subgroups — #37576131

Order 24448.

914 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 161 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×#20259307C211yesmaximal, characteristic, complemented
3C2×#20259297C212yesmaximal, complemented
4#27132580C211yesmaximal, characteristic, complemented
5#27132574C211yesmaximal, characteristic, complemented
6#27132573C211yesmaximal, characteristic, complemented
7#27132541C212yesmaximal, complemented
8#27132541C212yesmaximal, complemented
9#27132534C212yesmaximal, complemented
10#95×D191C211yesmaximal, characteristic, complemented
11#395×C191C211yesmaximal, characteristic, complemented
12C2×#15690288—28no—
13C2×#15690288—24no—
14C22×D764C2211yescharacteristic
15C22×D764C2211yescharacteristic
16C22×D764C2212yes—
17C22×D764C2212yes—
18C22×C4×D191C2212yes—
19C22×C4×D191C2212yes—
20C22×C4×D191C2211yescharacteristic
21#20259307—24no—
22#20259297—28no—
23C2×Dic191⋊C4C2211yescharacteristic
24C2×(C2×D191)⋊C4C2212yes—
25C2×(C2×D191)⋊C4C2212yes—
26C2×(C2×D191)⋊C4C2211yescharacteristic
27C2×(C2×D191)⋊C4C2211yescharacteristic
28C2×(C2×D191)⋊C4C2212yes—
29C2×(C2×D191)⋊C4C2211yescharacteristic
30C2×(C2×D191)⋊C4C2212yes—
31C191⋊#95C2212yes—
32C191⋊#95C2211yescharacteristic
33C191⋊#95C2212yes—
34C22×Q8×C191C2211yescharacteristic
35C2×C4⋊C4×C191C2212yes—
36C2×C4⋊C4×C191C2211yescharacteristic
37#95×C191C2211yescharacteristic
38#95×C191C2212yes—
39C23×D191C2311yescharacteristic
40C23×D191C2312yes—
41C23×D191C2311yescharacteristic
42#15690288—232no—
43C2×D764—24no—
44C2×D764—24no—
45C2×D764—24no—
46C2×D764—24no—
47C2×D764—24no—
48C2×D764—22no—
49C2×D764—22no—
50C2×D764—24no—
51C2×D764—42no—
52C2×D764D418yescomplemented
53C22×Dic191C2312yes—
54C22×Dic191C2311yescharacteristic
55C22×Dic191C2311yescharacteristic
56C2×C4×D191—24no—
57C2×C4×D191—24no—
58C2×C4×D191—24no—
59C2×C4×D191—24no—
60C2×C4×D191—24no—
61C2×C4×D191—24no—
62C2×C4×D191—42no—
63C2×C4×D191D414yescomplemented
64Dic191⋊C4—22no—
65(C2×D191)⋊C4—24no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—22no—
68(C2×D191)⋊C4—24no—
69(C2×D191)⋊C4—22no—
70(C2×D191)⋊C4—24no—
71(C2×D191)⋊C4—42no—
72C2×Q8×C191—24no—
73C2×Q8×C191—22no—
74C22×C4×C191C2311yescharacteristic
75C22×C4×C191C2311yescharacteristic
76C22×C4×C191C2312yes—
77C22×C4×C191C2311yescharacteristic
78C22×C4×C191C2312yes—
79C4⋊C4×C191—24no—
80C4⋊C4×C191—22no—
81C22×D191—22no—
82C22×D191—21no—
83C22×D191—21no—
84C22×D191—22no—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—22no—
88C22×D191—22no—
89C22×D191—22no—
90C22×D191—21no—
91C22×D191—22no—
92C22×D191—22no—
93C22×D191—21no—
94C22×D191—21no—
95C22×D191C2×D414yes—
96C22×D191C2×D412yes—
97C22×D191C2×D414yes—
98D764—28no—
99D764—216no—
100D764—48no—
101D764—44no—
102C2×Dic191—22no—
103C2×Dic191—21no—
104C2×Dic191—22no—
105C2×Dic191—22no—
106C2×Dic191—22no—
107C2×Dic191—22no—
108C2×Dic191C2×D412yes—
109C4×D191—216no—
110C4×D191—28no—
111C4×D191—48no—
112C23×C191C2411yescharacteristic
113Q8×C191—28no—
114C2×C4×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—22no—
117C2×C4×C191—22no—
118C2×C4×C191—22no—
119C2×C4×C191—21no—
120C2×C4×C191—22no—
121C2×C4×C191—21no—
122C2×C4×C191—21no—
123C2×C4×C191—21no—
124C2×C4×C191C2×D414yes—
125C2×C4×C191C2×D412yes—
126C2×C4×C191#4114yescomplemented
127C22×C191C22×D411yescharacteristic
128C22×C191C22×D412yes—
129C22×C191C2×#4112yes—
130C22×C191H(5,𝔽2)11yescharacteristic
131C22×C1912−1+411yescharacteristic
132C2×D191—22no—
133C2×D191—22no—
134C2×D191—24no—
135C2×D191—24no—
136C2×D191—24no—
137C2×D191—24no—
138C2×D191—24no—
139C2×D191—42no—
140C2×D191—42no—
141C2×D191—42no—
142C2×D191—42no—
143C2×D191—41no—
144C2×D191—41no—
145C2×D191—41no—
146C2×D191—42no—
147C2×D191—42no—
148C2×D191#12014yes—
149C4×C191—24no—
150C4×C191—22no—
151C4×C191—24no—
152C4×C191—41no—
153C4×C191—41no—
154Dic191—22no—
155Dic191—41no—
156Dic191—42no—
157C2×C191C2×#12011yescharacteristic
158C2×C191#53211yescharacteristic
159C2×C191#53711yescharacteristic
160C2×C191#54612yes—
161C2×C191#54712yes—
162D191—24no—
163D191—44no—
164D191—81no—
165C191#243011yescharacteristic, complemented, Sylow
166#2430—1911nomaximal, Sylow
167C22×#41—1911no—
168C2×#123—1911no—
169C2×#121—1912no—
170#395D19111yescharacteristic, complemented
171#394—1911no—
172#393—1911no—
173#392—1911no—
174#386—1912no—
175#386—1912no—
176#385—1912no—
177C2×#95—1911no—
178C2×#41—3828no—
179C2×#41—3824no—
180C22×Q8C2×D19111yescharacteristic
181C22×D4—1911no—
182C22×D4—1911no—
183C22×D4—1912no—
184C22×D4—1912no—
185C23×C4—1912no—
186C23×C4—1911no—
187C23×C4—1912no—
188#123—3824no—
189#121—3828no—
190C2×C4⋊C4—1911no—
191C2×C4⋊C4C2×D19112yes—
192C2×C4⋊C4C2×D19111yescharacteristic
193C2×C22⋊C4—1912no—
194C2×C22⋊C4—1912no—
195C2×C22⋊C4—1911no—
196C2×C22⋊C4—1911no—
197C2×C22⋊C4—1911no—
198C2×C22⋊C4—1912no—
199C2×C22⋊C4—1912no—
200#95—1912no—
201#95—1911no—
202#95—1912no—
203#95C2×D19111yescharacteristic
204#95C2×D19112yes—
205C24—1911no—
206C24—1912no—
207C24—1911no—
208#41—38232no—
209C2×Q8—24no—
210C2×Q8—22no—
211C2×D4—3824no—
212C2×D4—3824no—
213C2×D4—3824no—
214C2×D4—3824no—
215C2×D4—3824no—
216C2×D4—3822no—
217C2×D4—3822no—
218C2×D4—3824no—
219C2×D4—1918no—
220C2×D4—7642no—
221C22×C4—3824no—
222C22×C4—3824no—
223C22×C4—3824no—
224C22×C4—3824no—
225C22×C4—3824no—
226C22×C4—3824no—
227C22×C4—1912no—
228C22×C4—1911no—
229C22×C4—1911no—
230C22×C4—1914no—
231C22×C4—7642no—
232C22×C4C22×D19111yescharacteristic
233C22×C4C22×D19111yescharacteristic
234C22×C4C22×D19112yes—
235C22×C4C22×D19111yescharacteristic
236C22×C4C22×D19112yes—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—3822no—
240C22⋊C4—3824no—
241C22⋊C4—3822no—
242C22⋊C4—3822no—
243C22⋊C4—3824no—
244C22⋊C4—3822no—
245C22⋊C4—3824no—
246C22⋊C4—7642no—
247C23—3822no—
248C23—3821no—
249C23—3821no—
250C23—3822no—
251C23—3822no—
252C23—3822no—
253C23—3822no—
254C23—3822no—
255C23—3822no—
256C23—3821no—
257C23—3822no—
258C23—3822no—
259C23—3821no—
260C23—3821no—
261C23—1914no—
262C23—1912no—
263C23—1914no—
264C23C23×D19111yescharacteristic
265Q8—28no—
266D4—3828no—
267D4—38216no—
268D4—7648no—
269D4—7644no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—38216no—
281C2×C4—3828no—
282C2×C4—3822no—
283C2×C4—3821no—
284C2×C4—3822no—
285C2×C4—3822no—
286C2×C4—3822no—
287C2×C4—3822no—
288C2×C4—1912no—
289C2×C4—7648no—
290C2×C4D4×D19114yes—
291C2×C4D4×D19112yes—
292C2×C4#1569028814yescomplemented
293C22—3822no—
294C22—3822no—
295C22—3824no—
296C22—3824no—
297C22—3824no—
298C22—3824no—
299C22—3824no—
300C22—1914no—
301C22—7641no—
302C22—7641no—
303C22—7642no—
304C22—7642no—
305C22—7641no—
306C22—7642no—
307C22—7642no—
308C22—7642no—
309C22—7642no—
310C22C2×D4×D19111yescharacteristic
311C22C2×D4×D19112yes—
312C22C2×#1569028812yes—
313C22#2025936811yescharacteristic
314C22#2025937011yescharacteristic
315C4—24no—
316C4—24no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—3822no—
321C4—7641no—
322C4—7642no—
323C2—3824no—
324C2—15281no—
325C2—7644no—
326C2#120×D19111yescharacteristic
327C2#2713321411yescharacteristic
328C2#2713324311yescharacteristic
329C2#2713330412yes—
330C2#2713331612yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#395×C191C2
hypercenter#395D191
hypocenterC191#2430
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C22×#15690288C2semidirect
2. C2×#20259307C2semidirect
3. C2×#20259297C2semidirect
4. #27132580C2semidirect
5. #27132574C2semidirect
6. #27132573C2semidirect
7. #27132541C2semidirect
8. #27132541C2semidirect
9. #27132534C2semidirect
10. #95×D191C2semidirect
11. #395×C191C2semidirect
52. C2×D764D4semidirect
63. C2×C4×D191D4semidirect
126. C2×C4×C191#41semidirect
165. C191#2430semidirect
170. #395D191semidirect
292. 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×#202593072
C2×#202592972
#271325802
#271325742
#271325732
#271325412
#271325342
#95×D1912
#395×C1912
#2430191