Finite Group Databasebeta

← #37583362

Subgroups — #37583362

Order 24448.

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

normal subgroups only

This group has 418 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#27133174C212yesmaximal, complemented
2#27133005C212yesmaximal, complemented
3#27132988C212yesmaximal, complemented
4#27132921C211yesmaximal, characteristic, complemented
5#27132916C211yesmaximal, characteristic, complemented
6#27132909C212yesmaximal, complemented
7#27132827C212yesmaximal, complemented
8#27132810C212yesmaximal, complemented
9#407×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2212yes—
11C2×Dic764C2211yescharacteristic, complemented
12C2×#15690276C2212yescomplemented
13C2×D1528C2211yescharacteristic, complemented
14#20259305—22no—
15#20259301C2212yes—
16#20259301C2212yes—
17#20259296C2212yes—
18#20259296C2212yes—
19#20259291—22no—
20C4×C22⋊D191—22no—
21C4×C22⋊D191—22no—
22#20259281C2212yes—
23#20259256C2212yescomplemented
24#20259254C2211yescharacteristic, complemented
25#20259254C2211yescharacteristic, complemented
26#20259247C2212yescomplemented
27#20259247C2212yescomplemented
28#20259244C2212yescomplemented
29#20259244C2212yescomplemented
30C23×C4×C191C2211yescharacteristic
31C2×M16×C191C2211yescharacteristic
32C2×M16×C191C2211yescharacteristic
33C22⋊C8×C191C2212yes—
34C22⋊C8×C191C2212yes—
35#15690287—42no—
36#15690287—42no—
37C2×Dic382C2312yes—
38C2×D764C2312yes—
39C2×C22⋊D191—22no—
40C2×C22⋊D191—22no—
41C2×C22⋊D191—22no—
42C2×C4×D191—22no—
43Dic764—41no—
44#15690276—42no—
45D1528—41no—
46C4⋊Dic191C2312yes—
47C4⋊Dic191C2312yes—
48Dic191⋊C4—22no—
49Dic191⋊C4—22no—
50C22⋊Dic191—22no—
51C22⋊Dic191—22no—
52C22⋊Dic191—22no—
53(C2×D191)⋊C4—22no—
54(C2×D191)⋊C4—22no—
55C4×Dic191—22no—
56C24×C191D411yescharacteristic
57C22×C4×C191—41no—
58C22×C4×C191—41no—
59C22×C4×C191D411yescharacteristic
60C22×C4×C191D412yes—
61C22×C4×C191C2312yes—
62C22×C4×C191C2311yescharacteristic
63M16×C191—41no—
64M16×C191—41no—
65C2×C8×C191C2311yescharacteristic
66C2×C8×C191C2312yes—
67C2×C8×C191C2311yescharacteristic
68C22×D191—22no—
69Dic382—22no—
70Dic382—22no—
71D764—22no—
72D764—22no—
73C22⋊D191—44no—
74C22⋊D191—44no—
75C22⋊D191—44no—
76C2×Dic191—22no—
77C2×Dic191—22no—
78C2×Dic191—22no—
79C4×D191—44no—
80C23×C191—41no—
81C23×C191—42no—
82C23×C191C2×D411yescharacteristic
83C23×C191C2×D412yes—
84C2×C4×C191—21no—
85C2×C4×C191—21no—
86C2×C4×C191—22no—
87C2×C4×C191—22no—
88C2×C4×C191—42no—
89C2×C4×C191—41no—
90C2×C4×C191C2×D412yes—
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2411yescharacteristic
93C8×C191—21no—
94C8×C191—22no—
95C8×C191—21no—
96C22×C191—21no—
97C22×C191—22no—
98C22×C191—22no—
99C22×C191—22no—
100C22×C191—22no—
101C22×C191—44no—
102C22×C191C22×D411yescharacteristic
103C2×D191—42no—
104C2×D191—42no—
105C2×D191—42no—
106C4×C191—21no—
107C4×C191—22no—
108C4×C191H(5,𝔽2)11yescharacteristic
109C4×C191H(5,𝔽2)11yescharacteristic
110Dic191—42no—
111Dic191—42no—
112Dic191—42no—
113C2×C191—41no—
114C2×C191—42no—
115C2×C191#53411yescharacteristic
116C2×C191#57512yes—
117D191—82no—
118C191#319711yescharacteristic, complemented, Sylow
119#3197—1911nomaximal, Sylow
120#525—1912no—
121#485—1912no—
122#482—1912no—
123#470—1911no—
124#469—1911no—
125#468—1912no—
126#452—1912no—
127#449—1912no—
128#407D19111yescharacteristic, complemented
129C2×#41—1912no—
130C23×C4C2×D19111yescharacteristic
131C2×Q16—1911no—
132C2×SD16—1912no—
133C2×D8—1911no—
134C2×M16C2×D19111yescharacteristic
135C2×M16C2×D19111yescharacteristic
136#123—3822no—
137C22⋊Q8—1912no—
138C22⋊Q8—1912no—
139#121—1912no—
140#121—1912no—
141#120—3822no—
142C4×D4—3822no—
143C4×D4—3822no—
144#117—1912no—
145#107—1912no—
146#106—1911no—
147#106—1911no—
148#103—1912no—
149#103—1912no—
150#102—1912no—
151#102—1912no—
152C22⋊C8C2×D19112yes—
153C22⋊C8C2×D19112yes—
154C24D76411yescharacteristic
155#41—7642no—
156#41—7642no—
157C2×Q8—1912no—
158C2×D4—3822no—
159C2×D4—3822no—
160C2×D4—3822no—
161C2×D4—1912no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—3822no—
165C22×C4D76411yescharacteristic
166C22×C4D76412yes—
167C22×C4C22×D19112yes—
168C22×C4C22×D19111yescharacteristic
169Q16—7641no—
170SD16—7642no—
171D8—7641no—
172M16—41no—
173M16—41no—
174C2×C8C22×D19111yescharacteristic
175C2×C8C22×D19112yes—
176C2×C8C22×D19111yescharacteristic
177C4⋊C4—3822no—
178C4⋊C4—3822no—
179C4⋊C4—1912no—
180C4⋊C4—1912no—
181C22⋊C4—3822no—
182C22⋊C4—3822no—
183C22⋊C4—3822no—
184C22⋊C4—3822no—
185C22⋊C4—3822no—
186C42—3822no—
187C23—41no—
188C23—42no—
189C23—3822no—
190C23C2×D76411yescharacteristic
191C23C2×D76412yes—
192Q8—3822no—
193Q8—3822no—
194D4—3822no—
195D4—3822no—
196D4—7644no—
197D4—7644no—
198D4—7644no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—3822no—
206C2×C4—3822no—
207C2×C4—3822no—
208C2×C4—7644no—
209C2×C4C2×D76412yes—
210C2×C4C2×D76411yescharacteristic
211C2×C4C23×D19111yescharacteristic
212C8—21no—
213C8—22no—
214C8—21no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—21no—
220C22—44no—
221C22—7642no—
222C22—7642no—
223C22—7642no—
224C22C22×D76411yescharacteristic
225C4—22no—
226C4—21no—
227C4—7642no—
228C4—7642no—
229C4—7642no—
230C4#2025936711yescharacteristic
231C4#2025936711yescharacteristic
232C2—41no—
233C2—42no—
234C2—15282no—
235C2#2713322511yescharacteristic
236C2#2713346312yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#407×C191C2
socleC22×C191C22×D4
hypercenter#407D191
hypocenterC191#3197
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #27133174C2semidirect
2. #27133005C2semidirect
3. #27132988C2semidirect
4. #27132921C2semidirect
5. #27132916C2semidirect
6. #27132909C2semidirect
7. #27132827C2semidirect
8. #27132810C2semidirect
9. #407×C191C2semidirect
11. C2×Dic764C22semidirect
12. C2×#15690276C22semidirect
13. C2×D1528C22semidirect
23. #20259256C22semidirect
24. #20259254C22semidirect
25. #20259254C22semidirect
26. #20259247C22semidirect
27. #20259247C22semidirect
28. #20259244C22semidirect
29. #20259244C22semidirect
118. C191#3197semidirect
128. #407D191semidirect

Maximal subgroups

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

subgroupindex
#271331742
#271330052
#271329882
#271329212
#271329162
#271329092
#271328272
#271328102
#407×C1912
#3197191