Finite Group Databasebeta

← #37582903

Subgroups — #37582903

Order 24448.

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

normal subgroups only

This group has 400 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#27133219C211yesmaximal, characteristic, complemented
2#124×D191C211yesmaximal, characteristic
3#27133024C212yesmaximal, complemented
4#27133021C212yesmaximal, complemented
5#27133020C211yesmaximal, characteristic, complemented
6#27133018C211yesmaximal, characteristic, complemented
7#27133017C211yesmaximal, characteristic, complemented
8#27132670C212yesmaximal, complemented
9#27132666C212yesmaximal
10C4×#15690270C211yesmaximal, characteristic, complemented
11#488×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic
13C2×D4×D191C2211yescharacteristic
14C2×#15690270C2212yes—
15#20259321C2212yescomplemented
16#20259314C2211yescharacteristic, complemented
17#20259311C2211yescharacteristic
18#20259310C2211yescharacteristic
19#20259309C2211yescharacteristic
20#20259308—22no—
21#20259300—22no—
22#20259287—22no—
23#20259280C2212yes—
24C22⋊C4×D191—22no—
25C42×D191C2211yescharacteristic
26#20259247C2212yescomplemented
27#20259246C2212yescomplemented
28#20259245C2212yescomplemented
29#20259244C2212yes—
30#20259243C2212yes—
31#20259242C2212yes—
32#20259232C2212yescomplemented
33C4×C191⋊C8C2211yescharacteristic, complemented
34#125×C191C2211yescharacteristic, complemented
35#124×C191C2211yescharacteristic
36#103×C191C2212yescomplemented
37#102×C191C2212yes—
38C4×C8×C191C2211yescharacteristic, complemented
39C23×D191—21no—
40C2×Dic382—21no—
41C2×Dic382—22no—
42C2×Dic382C2311yescharacteristic
43Q8×D191—24no—
44C2×D764C2311yescharacteristic
45C2×C22⋊D191—21no—
46D4×D191—24no—
47C2×C4×D191—21no—
48C2×C4×D191D412yes—
49C2×C4×D191C2311yescharacteristic
50#15690270—24no—
51C2×C191⋊C8C2312yes—
52C4⋊Dic191C2312yes—
53Dic191⋊C4—22no—
54Dic191⋊C4—22no—
55C22⋊Dic191—22no—
56(C2×D191)⋊C4—22no—
57(C2×D191)⋊C4—22no—
58(C2×D191)⋊C4—22no—
59C4×Dic191—22no—
60C4×Dic191D412yes—
61C4×Dic191C2311yescharacteristic
62C2×Q8×C191C2311yescharacteristic
63C2×D4×C191C2311yescharacteristic
64C2×C8×C191C2312yes—
65C4⋊C4×C191—22no—
66C4⋊C4×C191C2312yes—
67C22⋊C4×C191—22no—
68C42×C191C2311yescharacteristic
69C22×D191—21no—
70C22×D191—44no—
71C22×D191—42no—
72C22×D191C2×D411yescharacteristic
73Dic382—22no—
74Dic382—44no—
75Dic382—42no—
76D764—22no—
77C22⋊D191—42no—
78C2×Dic191—22no—
79C2×Dic191—21no—
80C2×Dic191C2×D412yes—
81C2×Dic191C2×D411yescharacteristic
82C4×D191—42no—
83C4×D191—42no—
84C4×D191#4114yes—
85C191⋊C8—22no—
86C23×C191—21no—
87Q8×C191—22no—
88D4×C191—22no—
89C2×C4×C191—21no—
90C2×C4×C191—22no—
91C2×C4×C191C2×D412yes—
92C2×C4×C191C2411yescharacteristic
93C8×C191—22no—
94C22×C191—41no—
95C22×C191—42no—
96C22×C191C22×D411yescharacteristic
97C2×D191—22no—
98C2×D191—41no—
99C2×D191—42no—
100C2×D191—84no—
101C2×D191#12412yes—
102C4×C191—22no—
103C4×C191—41no—
104C4×C191—42no—
105C4×C191C2×#4112yes—
106Dic191—22no—
107Dic191—41no—
108Dic191—42no—
109Dic191#12412yes—
110C2×C191—81no—
111C2×C191C2×#12411yescharacteristic
112C2×C191#57512yes—
113D191—22no—
114D191—81no—
115C191#314011yescharacteristic, complemented, Sylow
116#3140—1911nomaximal, Sylow
117#533—1911no—
118C2×#124—1911no—
119#489—1912no—
120#489—1912no—
121#488—1911no—
122#488—1911no—
123#488—1911no—
124#488D19111yescharacteristic, complemented
125#419—1912no—
126#418—1912no—
127C4×M16—1911no—
128C22×Q8—1911no—
129C22×D4—1911no—
130C2×M16—1912no—
131C4⋊Q8—1912no—
132#125—1911no—
133#125C2×D19111yescharacteristic, complemented
134#124—3822no—
135#124—1911no—
136#124—1911no—
137#124—1911no—
138#124C2×D19111yescharacteristic
139C22⋊Q8—3822no—
140C4×Q8—3822no—
141#117—1912no—
142C2×C22⋊C4—3822no—
143C2×C42—1911no—
144#103—1912no—
145#103—1912no—
146#103—1912no—
147#103C2×D19112yescomplemented
148#102—1912no—
149#102—1912no—
150#102—1912no—
151#102C2×D19112yes—
152#97—1912no—
153C4×C8—1911no—
154C4×C8C2×D19111yescharacteristic, complemented
155C24—3821no—
156C2×Q8—3822no—
157C2×Q8—3821no—
158C2×Q8—3824no—
159C2×Q8—1911no—
160C2×Q8C22×D19111yescharacteristic
161C2×D4—3824no—
162C2×D4—3821no—
163C2×D4—1911no—
164C2×D4C22×D19111yescharacteristic
165C22×C4—3821no—
166C22×C4—1911no—
167C22×C4—1912no—
168M16—3824no—
169C2×C8—1912no—
170C2×C8C22×D19112yes—
171C4⋊C4—22no—
172C4⋊C4—3822no—
173C4⋊C4—3822no—
174C4⋊C4—1912no—
175C4⋊C4C22×D19112yes—
176C22⋊C4—22no—
177C22⋊C4—3822no—
178C22⋊C4—3822no—
179C22⋊C4—3822no—
180C22⋊C4—3822no—
181C42—3822no—
182C42—1912no—
183C42—1911no—
184C42C22×D19111yescharacteristic
185C23—21no—
186C23—3821no—
187C23—1911no—
188C23—7644no—
189C23—7642no—
190Q8—22no—
191Q8—3822no—
192Q8—7644no—
193Q8—7642no—
194D4—22no—
195D4—3822no—
196D4—7642no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3822no—
200C2×C4—3821no—
201C2×C4—1912no—
202C2×C4—1911no—
203C2×C4—1914no—
204C2×C4—7642no—
205C2×C4—7642no—
206C2×C4D4×D19112yes—
207C2×C4C23×D19111yescharacteristic
208C8—22no—
209C8—3822no—
210C22—42no—
211C22—41no—
212C22—3822no—
213C22—1912no—
214C22—15284no—
215C22—7642no—
216C22—7641no—
217C22C2×D4×D19111yescharacteristic
218C4—22no—
219C4—42no—
220C4—41no—
221C4—3822no—
222C4—1912no—
223C4—7642no—
224C4—7641no—
225C4#41×D19112yes—
226C2—81no—
227C2—3822no—
228C2—15281no—
229C2#124×D19111yescharacteristic
230C2#2713346012yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#488×C191C2
socleC22×C191C22×D4
prime radicalC2×D4×D191C22
hypercenter#488D191
hypocenterC191#3140
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #27133219C2semidirect
3. #27133024C2semidirect
4. #27133021C2semidirect
5. #27133020C2semidirect
6. #27133018C2semidirect
7. #27133017C2semidirect
8. #27132670C2semidirect
10. C4×#15690270C2semidirect
11. #488×C191C2semidirect
15. #20259321C22semidirect
16. #20259314C22semidirect
26. #20259247C22semidirect
27. #20259246C22semidirect
28. #20259245C22semidirect
32. #20259232C22semidirect
33. C4×C191⋊C8C22semidirect
34. #125×C191C22semidirect
36. #103×C191C22semidirect
38. C4×C8×C191C22semidirect
115. C191#3140semidirect
124. #488D191semidirect
133. #125C2×D191semidirect
147. #103C2×D191semidirect
154. C4×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332192
#124×D1912
#271330242
#271330212
#271330202
#271330182
#271330172
#271326702
#271326662
C4×#156902702
#488×C1912
#3140191