Finite Group Databasebeta

← #37582694

Subgroups — #37582694

Order 24448.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#27133199C211yesmaximal, characteristic
2C2×#20259287C211yesmaximal, characteristic, complemented
3#27132981C211yesmaximal, characteristic
4#27132969C212yesmaximal, complemented
5#27132888C212yesmaximal, complemented
6#27132818C211yesmaximal, characteristic, complemented
7#27132804C211yesmaximal, characteristic
8#27132767C212yesmaximal, complemented
9#27132762C212yesmaximal, complemented
10#27132717C211yesmaximal, characteristic, complemented
11#484×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C2×#15690277C2212yes—
14C2×#15690276C2212yes—
15#20259317—21no—
16#20259313C2211yescharacteristic
17#20259311C2211yescharacteristic
18#20259306—21no—
19#20259296C2211yescharacteristic
20Q8×Dic191C2211yescharacteristic
21#20259287—21no—
22#20259287—21no—
23#20259287C2212yes—
24#20259287C2212yes—
25#20259284C2211yescharacteristic
26(C22⋊D191)⋊C4—21no—
27#20259280—21no—
28C2×Dic191⋊C4—21no—
29C2×C4×Dic191—21no—
30C2×C4×Dic191C2211yescharacteristic
31#20259253C2212yes—
32Dic191⋊C8C2212yes—
33#20259247C2212yes—
34#20259245C2212yes—
35#20259242C2212yes—
36C22⋊(C191⋊C8)C2211yescharacteristic
37#20259232C2212yes—
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yes—
41#103×C191C2212yes—
42C22⋊C8×C191C2211yescharacteristic
43C2×Dic382—24no—
44C2×Dic382—21no—
45C2×Dic382C2312yes—
46C2×D764C2311yescharacteristic
47C2×C22⋊D191—21no—
48C22×Dic191—21no—
49C22×Dic191D411yescharacteristic
50C22×Dic191D411yescharacteristic
51C2×C4×D191—21no—
52#15690277—24no—
53#15690276—24no—
54C2×C191⋊C8C2312yes—
55C4⋊Dic191—21no—
56C4⋊Dic191—21no—
57C4⋊Dic191C2311yescharacteristic
58Dic191⋊C4—22no—
59Dic191⋊C4—22no—
60Dic191⋊C4—21no—
61(C2×D191)⋊C4—21no—
62(C2×D191)⋊C4—21no—
63(C2×D191)⋊C4—21no—
64(C2×D191)⋊C4—21no—
65C4×Dic191—22no—
66C4×Dic191—22no—
67C4×Dic191—21no—
68C4×Dic191D411yescharacteristic
69C4×Dic191D411yescharacteristic
70C4×Dic191C2311yescharacteristic
71C4×Dic191C2311yescharacteristic
72C2×Q8×C191C2311yescharacteristic
73C22×C4×C191—21no—
74C22×C4×C191C2311yescharacteristic
75C2×C8×C191C2312yes—
76C4⋊C4×C191—21no—
77C4⋊C4×C191—21no—
78C4⋊C4×C191C2312yes—
79C4⋊C4×C191C2311yescharacteristic
80C22⋊C4×C191—21no—
81C22×D191—21no—
82Dic382—24no—
83Dic382—22no—
84Dic382#4114yes—
85D764—22no—
86C22⋊D191—42no—
87C2×Dic191—24no—
88C2×Dic191—22no—
89C2×Dic191—22no—
90C2×Dic191—21no—
91C2×Dic191—41no—
92C2×Dic191C2×D411yescharacteristic
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C2×Dic191C2×D411yescharacteristic
96C4×D191—42no—
97C191⋊C8—22no—
98C23×C191C2×D411yescharacteristic
99Q8×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—21no—
102C2×C4×C191—21no—
103C2×C4×C191—22no—
104C2×C4×C191—42no—
105C2×C4×C191C2×D411yescharacteristic
106C2×C4×C191C2411yescharacteristic
107C8×C191—22no—
108C22×C191—21no—
109C22×C191—21no—
110C22×C191#13712yes—
111C22×C191C22×D411yescharacteristic
112C2×D191—41no—
113C2×D191—41no—
114C2×D191—41no—
115C4×C191—21no—
116C4×C191—41no—
117C4×C191—42no—
118C4×C191—41no—
119C4×C191C2×#4112yes—
120Dic191—24no—
121Dic191—21no—
122Dic191—21no—
123Dic191—21no—
124Dic191—21no—
125Dic191—41no—
126C2×C191—22no—
127C2×C191#52511yescharacteristic
128C2×C191C2×#13711yescharacteristic
129C2×C191#57511yescharacteristic
130D191—81no—
131C191#311711yescharacteristic, complemented, Sylow
132#3117—1911nomaximal, Sylow
133#529—1911no—
134C2×C4×Q8—1911no—
135#484D19111yescharacteristic, complemented
136#481—1911no—
137#479—1912no—
138#464—1912no—
139#451—1911no—
140#448—1911no—
141#441—1912no—
142#440—1912no—
143#432—1911no—
144C22×Q8—1911no—
145C2×Q16—1912no—
146C2×SD16—1912no—
147#126—3821no—
148#125—1911no—
149#124—1911no—
150#123—3821no—
151C22⋊Q8C2×D19111yescharacteristic
152#121—1911no—
153C4×Q8—3821no—
154C4×Q8—3821no—
155C4×Q8—1911no—
156C4×Q8—1912no—
157C4×Q8—1912no—
158C4×D4—3821no—
159C4×D4—1911no—
160#117—3821no—
161C2×C4⋊C4—3821no—
162C2×C4⋊C4C2×D19111yescharacteristic
163C2×C42—3821no—
164C2×C42—1911no—
165#107C2×D19112yes—
166#106—1912no—
167C4⋊C8—1912no—
168#103—1912no—
169#103—1912no—
170#103C2×D19112yes—
171#102—1912no—
172C22⋊C8—1911no—
173C22⋊C8C2×D19111yescharacteristic
174#97—1912no—
175C2×Q8—3824no—
176C2×Q8—3821no—
177C2×Q8—1912no—
178C2×Q8C22×D19111yescharacteristic
179C2×D4—3821no—
180C2×D4—1911no—
181C22×C4—21no—
182C22×C4—3821no—
183C22×C4—3821no—
184C22×C4—1911no—
185C22×C4—1911no—
186C22×C4C22×D19111yescharacteristic
187Q16—3824no—
188SD16—3824no—
189C2×C8—1912no—
190C2×C8C22×D19112yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—3822no—
194C4⋊C4—3822no—
195C4⋊C4—3821no—
196C4⋊C4—3821no—
197C4⋊C4—3821no—
198C4⋊C4—1911no—
199C4⋊C4C22×D19112yes—
200C4⋊C4C22×D19111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—3821no—
203C22⋊C4—3821no—
204C22⋊C4—3821no—
205C22⋊C4—3821no—
206C42—3822no—
207C42—3822no—
208C42—3821no—
209C42—1911no—
210C42—1911no—
211C42—1911no—
212C42—1911no—
213C23—3821no—
214C23D4×D19111yescharacteristic
215Q8—22no—
216Q8—3824no—
217Q8—3822no—
218Q8—1914no—
219D4—3822no—
220D4—7642no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—3824no—
227C2×C4—3822no—
228C2×C4—3822no—
229C2×C4—3821no—
230C2×C4—1911no—
231C2×C4—1911no—
232C2×C4—1911no—
233C2×C4—1911no—
234C2×C4—7642no—
235C2×C4—7641no—
236C2×C4D4×D19111yescharacteristic
237C2×C4C23×D19111yescharacteristic
238C8—22no—
239C8—3822no—
240C22—21no—
241C22—21no—
242C22—7641no—
243C22—7641no—
244C22—7641no—
245C22#2025935412yes—
246C22C2×D4×D19111yescharacteristic
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—3821no—
252C4—3821no—
253C4—3821no—
254C4—3821no—
255C4—3824no—
256C4—7641no—
257C4#41×D19112yes—
258C2—22no—
259C2—15281no—
260C2#2713317111yescharacteristic
261C2C2×#2025935411yescharacteristic
262C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#484×C191C2
socleC22×C191C22×D4
prime radical#20259296C22
hypercenter#484D191
hypocenterC191#3117
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#20259287C2semidirect
4. #27132969C2semidirect
5. #27132888C2semidirect
6. #27132818C2semidirect
8. #27132767C2semidirect
9. #27132762C2semidirect
10. #27132717C2semidirect
11. #484×C191C2semidirect
131. C191#3117semidirect
135. #484D191semidirect

Maximal subgroups

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

subgroupindex
#271331992
C2×#202592872
#271329812
#271329692
#271328882
#271328182
#271328042
#271327672
#271327622
#271327172
#484×C1912
#3117191