Finite Group Databasebeta

← #37576173

Subgroups — #37576173

Order 24448.

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

normal subgroups only

This group has 852 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×Q8×D191C211yesmaximal, characteristic, complemented
2C2×#20259308C212yesmaximal, complemented
3C2×#20259295C211yesmaximal, characteristic
4#27132576C211yesmaximal, characteristic, complemented
5#27132575C211yesmaximal, characteristic, complemented
6#27132574C211yesmaximal, characteristic, complemented
7#27132567C212yesmaximal, complemented
8#27132541C212yesmaximal, complemented
9#27132539C212yesmaximal, complemented
10#27132510C211yesmaximal, characteristic, complemented
11#393×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C22×Dic382C2212yes—
14C2×Q8×D191—28no—
15C2×Q8×D191—24no—
16C22×D764C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C22×C22⋊D191C2211yescharacteristic
19C22×C4×D191C2212yes—
20C22×C4×D191C2211yescharacteristic
21#20259308—28no—
22#20259295—24no—
23C2×Dic191⋊C4C2211yescharacteristic
24C2×C22⋊Dic191C2212yes—
25C2×(C2×D191)⋊C4C2212yes—
26C2×(C2×D191)⋊C4C2212yes—
27C2×(C2×D191)⋊C4C2212yes—
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×C4×Dic191C2212yes—
30C191⋊#95C2211yescharacteristic
31C191⋊#95C2212yes—
32C191⋊#95C2212yes—
33C191⋊#95C2212yes—
34C22×Q8×C191C2211yescharacteristic
35C2×C22⋊C4×C191C2212yes—
36C2×C22⋊C4×C191C2211yescharacteristic
37#95×C191C2211yescharacteristic
38#95×C191C2212yes—
39C23×D191C2311yescharacteristic
40C23×D191C2311yescharacteristic
41C2×Dic382—24no—
42C2×Dic382—24no—
43C2×Dic382—24no—
44C2×Dic382—22no—
45C2×Dic382D418yescomplemented
46Q8×D191—232no—
47C2×D764—22no—
48C2×D764—24no—
49C2×C22⋊D191—22no—
50C2×C22⋊D191—24no—
51C2×C22⋊D191—22no—
52C2×C22⋊D191—24no—
53C22×Dic191C2311yescharacteristic
54C22×Dic191C2312yes—
55C22×Dic191C2312yes—
56C22×Dic191C2311yescharacteristic
57C2×C4×D191—24no—
58C2×C4×D191—24no—
59C2×C4×D191—24no—
60C2×C4×D191—24no—
61C2×C4×D191D414yes—
62Dic191⋊C4—22no—
63C22⋊Dic191—24no—
64(C2×D191)⋊C4—24no—
65(C2×D191)⋊C4—24no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—24no—
68C4×Dic191—24no—
69C24×C191C2311yescharacteristic
70C2×Q8×C191—24no—
71C2×Q8×C191—22no—
72C22×C4×C191C2311yescharacteristic
73C22×C4×C191C2312yes—
74C22×C4×C191C2312yes—
75C22×C4×C191C2311yescharacteristic
76C22⋊C4×C191—22no—
77C22⋊C4×C191—24no—
78C22×D191—22no—
79C22×D191—21no—
80C22×D191—21no—
81C22×D191—22no—
82C22×D191—21no—
83C22×D191—22no—
84C22×D191—21no—
85C22×D191—21no—
86C22×D191C2×D412yes—
87C22×D191#4114yes—
88Dic382—28no—
89Dic382—216no—
90D764—44no—
91C22⋊D191—44no—
92C22⋊D191—44no—
93C2×Dic191—22no—
94C2×Dic191—22no—
95C2×Dic191—22no—
96C2×Dic191—21no—
97C2×Dic191—22no—
98C2×Dic191—22no—
99C2×Dic191—22no—
100C2×Dic191C2×D414yes—
101C2×Dic191C2×D414yes—
102C2×Dic191C2×D412yes—
103C4×D191—216no—
104C4×D191—28no—
105C23×C191—22no—
106C23×C191—21no—
107C23×C191—22no—
108C23×C191—21no—
109C23×C191—21no—
110C23×C191C2411yescharacteristic
111Q8×C191—28no—
112C2×C4×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191—22no—
116C2×C4×C191—21no—
117C2×C4×C191—22no—
118C2×C4×C191—22no—
119C2×C4×C191—22no—
120C2×C4×C191C2×D414yes—
121C2×C4×C191C2×D412yes—
122C22×C191—41no—
123C22×C191—41no—
124C22×C191—41no—
125C22×C191—42no—
126C22×C191—42no—
127C22×C191C22×D411yescharacteristic
128C22×C191C22×D412yes—
129C22×C191C2×#4112yes—
130C22×C1912−1+411yescharacteristic
131C22×C1912−1+411yescharacteristic
132C2×D191—22no—
133C2×D191—22no—
134C2×D191—24no—
135C2×D191—24no—
136C2×D191—41no—
137C2×D191—41no—
138C2×D191—41no—
139C2×D191—42no—
140C2×D191—42no—
141C2×D191#12414yescomplemented
142C4×C191—24no—
143C4×C191—22no—
144C4×C191—41no—
145C4×C191—42no—
146Dic191—24no—
147Dic191—41no—
148Dic191—42no—
149Dic191#12014yes—
150C2×C191—81no—
151C2×C191C2×#12011yescharacteristic
152C2×C191C2×#12411yescharacteristic
153C2×C191#53611yescharacteristic
154C2×C191#54812yes—
155C2×C191#54812yes—
156D191—24no—
157D191—81no—
158C191#243611yescharacteristic, complemented, Sylow
159#2436—1911nomaximal, Sylow
160C23×Q8—1911no—
161C2×#124—1912no—
162C2×#121—1911no—
163#393—1911no—
164#393—1911no—
165#393—1911no—
166#393D19111yescharacteristic, complemented
167#391—1912no—
168#386—1912no—
169#386—1912no—
170C4×C22⋊C4—1911no—
171C22×Q8—3828no—
172C22×Q8—3824no—
173C22×Q8—1911no—
174C22×Q8—1912no—
175C22×Q8C2×D19111yescharacteristic
176C22×D4—1911no—
177C22×D4—1911no—
178C22×D4—1911no—
179C23×C4—1912no—
180C23×C4—1911no—
181#124—3828no—
182#121—3824no—
183C2×C4⋊C4—1911no—
184C2×C22⋊C4—1912no—
185C2×C22⋊C4—1911no—
186C2×C22⋊C4—1912no—
187C2×C22⋊C4—1912no—
188C2×C22⋊C4—1912no—
189C2×C22⋊C4C2×D19111yescharacteristic
190C2×C22⋊C4C2×D19112yes—
191C2×C42—1912no—
192#95—1911no—
193#95—1912no—
194#95—1912no—
195#95—1912no—
196#95C2×D19111yescharacteristic
197#95C2×D19112yes—
198C24—1911no—
199C24—1911no—
200C24C22×D19111yescharacteristic
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—3824no—
204C2×Q8—3824no—
205C2×Q8—3824no—
206C2×Q8—38232no—
207C2×Q8—3822no—
208C2×Q8—1918no—
209C2×D4—3822no—
210C2×D4—3824no—
211C2×D4—3822no—
212C2×D4—3824no—
213C2×D4—3822no—
214C2×D4—3824no—
215C22×C4—3824no—
216C22×C4—3824no—
217C22×C4—3824no—
218C22×C4—3824no—
219C22×C4—1911no—
220C22×C4—1912no—
221C22×C4—1912no—
222C22×C4—1914no—
223C22×C4—1911no—
224C22×C4C22×D19111yescharacteristic
225C22×C4C22×D19112yes—
226C22×C4C22×D19112yes—
227C22×C4C22×D19111yescharacteristic
228C4⋊C4—3822no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—3824no—
232C22⋊C4—3824no—
233C22⋊C4—3824no—
234C22⋊C4—3822no—
235C22⋊C4—3824no—
236C42—3824no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—3822no—
243C23—3821no—
244C23—3821no—
245C23—3822no—
246C23—3821no—
247C23—3822no—
248C23—3821no—
249C23—3821no—
250C23—1912no—
251C23—1914no—
252C23C23×D19111yescharacteristic
253Q8—28no—
254Q8—3828no—
255Q8—38216no—
256D4—7644no—
257D4—7644no—
258D4—7644no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—38216no—
268C2×C4—3828no—
269C2×C4—3822no—
270C2×C4—3822no—
271C2×C4—3822no—
272C2×C4—3821no—
273C2×C4—3822no—
274C2×C4—3822no—
275C2×C4—3822no—
276C2×C4—1914no—
277C2×C4—1914no—
278C2×C4—1912no—
279C2×C4D4×D19114yes—
280C2×C4D4×D19112yes—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—3822no—
287C22—3824no—
288C22—3824no—
289C22—3822no—
290C22—1914no—
291C22—7641no—
292C22—7642no—
293C22—7642no—
294C22—7641no—
295C22—7641no—
296C22C2×D4×D19111yescharacteristic
297C22C2×D4×D19112yes—
298C22#41×D19112yes—
299C22#2025936911yescharacteristic
300C22#2025937011yescharacteristic
301C4—24no—
302C4—22no—
303C4—42no—
304C4—41no—
305C4—3824no—
306C4—1914no—
307C4—7642no—
308C4—7641no—
309C2—81no—
310C2—3824no—
311C2—15281no—
312C2#120×D19111yescharacteristic
313C2#124×D19111yescharacteristic
314C2#2713323611yescharacteristic
315C2#2713332112yes—
316C2#2713332312yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#393×C191C2
prime radicalC2×#20259295C2
hypercenter#393D191
hypocenterC191#2436
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D191C2semidirect
2. C2×#20259308C2semidirect
4. #27132576C2semidirect
5. #27132575C2semidirect
6. #27132574C2semidirect
7. #27132567C2semidirect
8. #27132541C2semidirect
9. #27132539C2semidirect
10. #27132510C2semidirect
11. #393×C191C2semidirect
45. C2×Dic382D4semidirect
141. C2×D191#124semidirect
158. C191#2436semidirect
166. #393D191semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C2×#202593082
C2×#202592952
#271325762
#271325752
#271325742
#271325672
#271325412
#271325392
#271325102
#393×C1912
#2436191