Finite Group Databasebeta

← #37583291

Subgroups — #37583291

Order 24448.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 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×D191C212yesmaximal
2#27133000C212yesmaximal, complemented
3#27132997C212yesmaximal, complemented
4#27132904C211yesmaximal, characteristic, complemented
5#27132902C212yesmaximal, complemented
6#27132820C212yesmaximal, complemented
7#27132817C212yesmaximal, complemented
8C2×(C2×D191)⋊C8C211yesmaximal, characteristic
9#467×C191C211yesmaximal, characteristic, complemented
10C2×Q8×D191C2212yes—
11C22×C4×D191C2211yescharacteristic
12C2×Dic764C2211yescharacteristic, complemented
13C2×#15690277C2212yescomplemented
14C22×C191⋊C8C2211yescharacteristic
15#20259302C2212yes—
16#20259301C2212yes—
17#20259300—22no—
18#20259299—22no—
19#20259298C2212yes—
20C4⋊C4×D191—22no—
21C4⋊C4×D191C2212yes—
22C22⋊C4×D191—22no—
23#20259256C2211yescharacteristic, complemented
24#20259255C2212yescomplemented
25#20259247C2212yescomplemented
26#20259246C2212yescomplemented
27#20259245C2212yescomplemented
28(C2×D191)⋊C8C2212yes—
29(C2×D191)⋊C8C2212yes—
30C2×Q16×C191C2211yescharacteristic, complemented
31C22×C8×C191C2211yescharacteristic
32C22⋊Q8×C191C2212yes—
33#107×C191C2211yescharacteristic, complemented
34#103×C191C2212yescomplemented
35C23×D191D411yescharacteristic
36C2×Dic382—22no—
37C2×Dic382C2312yes—
38Q8×D191—42no—
39Q8×D191—42no—
40C22×Dic191D411yescharacteristic
41C2×C4×D191—22no—
42C2×C4×D191—22no—
43C2×C4×D191—41no—
44C2×C4×D191—41no—
45C2×C4×D191D412yes—
46C2×C4×D191C2312yes—
47Dic764—41no—
48#15690277—42no—
49C2×C191⋊C8—41no—
50C2×C191⋊C8C2312yes—
51C4⋊Dic191—22no—
52C4⋊Dic191C2312yes—
53Dic191⋊C4—22no—
54Dic191⋊C4—22no—
55Dic191⋊C4—22no—
56C22⋊Dic191—22no—
57(C2×D191)⋊C4—22no—
58(C2×D191)⋊C4—22no—
59C2×Q8×C191C2312yes—
60C22×C4×C191C2311yescharacteristic
61Q16×C191—41no—
62C2×C8×C191—41no—
63C2×C8×C191C2311yescharacteristic
64C2×C8×C191C2311yescharacteristic
65C4⋊C4×C191—22no—
66C4⋊C4×C191C2312yes—
67C22⋊C4×C191—22no—
68C22×D191—41no—
69C22×D191—41no—
70C22×D191Q1614yescomplemented
71C22×D191C2×D412yes—
72Dic382—22no—
73Dic382—22no—
74Dic382—44no—
75C2×Dic191—22no—
76C2×Dic191—22no—
77C2×Dic191—41no—
78C2×Dic191C2×D412yes—
79C4×D191—22no—
80C4×D191—22no—
81C4×D191—42no—
82C4×D191—44no—
83C4×D191—44no—
84C191⋊C8—22no—
85C23×C191C2×D411yescharacteristic
86Q8×C191—22no—
87Q8×C191—22no—
88C2×C4×C191—21no—
89C2×C4×C191—21no—
90C2×C4×C191—22no—
91C2×C4×C191—22no—
92C2×C4×C191C2×D411yescharacteristic
93C2×C4×C191C2411yescharacteristic
94C8×C191—21no—
95C8×C191—21no—
96C22×C191—21no—
97C22×C191—21no—
98C22×C191C2×Q1612yes—
99C22×C191C22×D411yescharacteristic
100C2×D191—22no—
101C2×D191—22no—
102C2×D191—44no—
103C2×D191C2×Q1614yes—
104C4×C191—21no—
105C4×C191—42no—
106C4×C191—42no—
107C4×C191H(5,𝔽2)11yescharacteristic
108C4×C191H(5,𝔽2)11yescharacteristic
109Dic191—22no—
110Dic191—42no—
111Dic191—42no—
112C2×C191—22no—
113C2×C191#53411yescharacteristic
114C2×C191C22×Q1611yescharacteristic
115C2×C191#57511yescharacteristic
116D191—24no—
117C191#318911yescharacteristic, complemented, Sylow
118#3189—1911nomaximal, Sylow
119C2×C22⋊Q8—1912no—
120#484—1912no—
121#484—1912no—
122#467—1911no—
123#467—1912no—
124#467D19111yescharacteristic, complemented
125#451—1912no—
126#451—1912no—
127C2×C22⋊C8—1911no—
128C22×Q8—1912no—
129C23×C4—1911no—
130C2×Q16—1912no—
131C2×Q16—1911no—
132C2×Q16C2×D19111yescharacteristic, complemented
133C22×C8—1911no—
134C22×C8C2×D19111yescharacteristic
135C22⋊Q8—3822no—
136C22⋊Q8—3822no—
137C22⋊Q8—1912no—
138C22⋊Q8—1912no—
139C22⋊Q8—1912no—
140C22⋊Q8C2×D19112yes—
141C2×C4⋊C4—3822no—
142C2×C4⋊C4—1912no—
143C2×C22⋊C4—3822no—
144#107—1911no—
145#107—1912no—
146#107C2×D19111yescharacteristic, complemented
147#103—1912no—
148#103—1912no—
149#103—1912no—
150#103C2×D19112yescomplemented
151C22⋊C8—1912no—
152C22⋊C8—1912no—
153C24—1911no—
154C2×Q8—3822no—
155C2×Q8—1912no—
156C2×Q8—7642no—
157C2×Q8—7642no—
158C2×Q8C22×D19112yes—
159C22×C4—3822no—
160C22×C4—3822no—
161C22×C4—1912no—
162C22×C4—1911no—
163C22×C4—1912no—
164C22×C4—7641no—
165C22×C4—7641no—
166C22×C4C22×D19111yescharacteristic
167Q16—41no—
168Q16—7641no—
169Q16—7642no—
170C2×C8—41no—
171C2×C8—1912no—
172C2×C8—7641no—
173C2×C8C22×D19111yescharacteristic
174C2×C8C22×D19111yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—3822no—
177C4⋊C4—3822no—
178C4⋊C4—3822no—
179C4⋊C4—3822no—
180C4⋊C4—1912no—
181C4⋊C4C22×D19112yes—
182C22⋊C4—22no—
183C22⋊C4—3822no—
184C22⋊C4—3822no—
185C22⋊C4—3822no—
186C23—1912no—
187C23—1914no—
188C23—7641no—
189C23—7641no—
190C23D4×D19111yescharacteristic
191Q8—22no—
192Q8—22no—
193Q8—3822no—
194Q8—3822no—
195Q8—7644no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—3822no—
201C2×C4—3822no—
202C2×C4—3822no—
203C2×C4—3822no—
204C2×C4—1912no—
205C2×C4—7642no—
206C2×C4—7644no—
207C2×C4—7644no—
208C2×C4—7641no—
209C2×C4D4×D19111yescharacteristic
210C2×C4C23×D19111yescharacteristic
211C8—21no—
212C8—21no—
213C8—3822no—
214C22—21no—
215C22—21no—
216C22—3822no—
217C22—3822no—
218C22—1914no—
219C22—7644no—
220C22Q16×D19112yes—
221C22C2×D4×D19111yescharacteristic
222C4—21no—
223C4—42no—
224C4—42no—
225C4—3822no—
226C4—7642no—
227C4—7642no—
228C4#2025936711yescharacteristic
229C4#2025936811yescharacteristic
230C2—22no—
231C2—3824no—
232C2#2713322111yescharacteristic
233C2C2×Q16×D19111yescharacteristic
234C2#2713346011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#467×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#467D191
hypocenterC191#3189
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #27133000C2semidirect
3. #27132997C2semidirect
4. #27132904C2semidirect
5. #27132902C2semidirect
6. #27132820C2semidirect
7. #27132817C2semidirect
9. #467×C191C2semidirect
12. C2×Dic764C22semidirect
13. C2×#15690277C22semidirect
23. #20259256C22semidirect
24. #20259255C22semidirect
25. #20259247C22semidirect
26. #20259246C22semidirect
27. #20259245C22semidirect
30. C2×Q16×C191C22semidirect
33. #107×C191C22semidirect
34. #103×C191C22semidirect
70. C22×D191Q16semidirect
117. C191#3189semidirect
124. #467D191semidirect
132. C2×Q16C2×D191semidirect
146. #107C2×D191semidirect
150. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1912
#271330002
#271329972
#271329042
#271329022
#271328202
#271328172
C2×(C2×D191)⋊C82
#467×C1912
#3189191