Finite Group Databasebeta

← #37583340

Subgroups — #37583340

Order 24448.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#27132995C212yesmaximal, complemented
3#27132992C212yesmaximal, complemented
4#27132921C211yesmaximal, characteristic, complemented
5#27132919C212yesmaximal, complemented
6#27132820C212yesmaximal, complemented
7#27132817C212yesmaximal, complemented
8#27132626C211yesmaximal, characteristic
9#470×C191C211yesmaximal, characteristic, complemented
10C2×Q8×D191C2212yes—
11C22×C4×D191C2211yescharacteristic
12C2×Dic764C2211yescharacteristic, complemented
13C2×#15690277C2212yescomplemented
14C2×#15690271C2211yescharacteristic
15#20259302C2212yes—
16#20259301C2212yes—
17#20259300—22no—
18#20259299—22no—
19#20259298C2212yes—
20C4⋊C4×D191—22no—
21C4⋊C4×D191C2212yes—
22C22⋊C4×D191—22no—
23#20259254C2211yescharacteristic, complemented
24#20259253C2212yescomplemented
25#20259247C2212yescomplemented
26#20259246C2212yescomplemented
27#20259245C2212yescomplemented
28(C2×D191)⋊C8C2212yes—
29(C2×D191)⋊C8C2212yes—
30C2×Q16×C191C2211yescharacteristic, complemented
31C2×M16×C191C2211yescharacteristic
32C22⋊Q8×C191C2212yes—
33#106×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—
49#15690271—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—
62M16×C191—41no—
63C2×C8×C191C2311yescharacteristic
64C2×C8×C191C2311yescharacteristic
65C4⋊C4×C191—22no—
66C4⋊C4×C191C2312yes—
67C22⋊C4×C191—22no—
68C22×D191—24no—
69C22×D191—41no—
70C22×D191C2×D412yes—
71Dic382—22no—
72Dic382—22no—
73Dic382—44no—
74C2×Dic191—22no—
75C2×Dic191—22no—
76C2×Dic191—41no—
77C2×Dic191C2×D412yes—
78C4×D191—22no—
79C4×D191—22no—
80C4×D191—42no—
81C4×D191—44no—
82C4×D191—44no—
83C191⋊C8—22no—
84C23×C191C2×D411yescharacteristic
85Q8×C191—22no—
86Q8×C191—22no—
87C2×C4×C191—21no—
88C2×C4×C191—21no—
89C2×C4×C191—22no—
90C2×C4×C191—22no—
91C2×C4×C191C2×D411yescharacteristic
92C2×C4×C191C2411yescharacteristic
93C8×C191—21no—
94C8×C191—21no—
95C22×C191—21no—
96C22×C191—22no—
97C22×C191C22×D411yescharacteristic
98C2×D191—22no—
99C2×D191—22no—
100C2×D191—44no—
101C2×D191#13714yes—
102C4×C191—21no—
103C4×C191—42no—
104C4×C191—42no—
105C4×C191H(5,𝔽2)11yescharacteristic
106C4×C191H(5,𝔽2)11yescharacteristic
107Dic191—22no—
108Dic191—42no—
109Dic191—42no—
110C2×C191—41no—
111C2×C191#53411yescharacteristic
112C2×C191C2×#13712yes—
113D191—24no—
114C191#319511yescharacteristic, complemented, Sylow
115#3195—1911nomaximal, Sylow
116C2×C22⋊Q8—1912no—
117#483—1912no—
118#483—1912no—
119#470—1911no—
120#470—1912no—
121#470D19111yescharacteristic, complemented
122#451—1912no—
123#451—1912no—
124#407—1911no—
125C22×Q8—1912no—
126C23×C4—1911no—
127C2×Q16—1912no—
128C2×Q16—1911no—
129C2×Q16C2×D19111yescharacteristic, complemented
130C2×M16—1911no—
131C2×M16C2×D19111yescharacteristic
132C22⋊Q8—3822no—
133C22⋊Q8—3822no—
134C22⋊Q8—1912no—
135C22⋊Q8—1912no—
136C22⋊Q8—1912no—
137C22⋊Q8C2×D19112yes—
138C2×C4⋊C4—3822no—
139C2×C4⋊C4—1912no—
140C2×C22⋊C4—3822no—
141#106—1911no—
142#106—1912no—
143#106C2×D19111yescharacteristic, complemented
144#103—1912no—
145#103—1912no—
146#103—1912no—
147#103C2×D19112yescomplemented
148C22⋊C8—1912no—
149C22⋊C8—1912no—
150C24—1911no—
151C2×Q8—3822no—
152C2×Q8—1912no—
153C2×Q8—7642no—
154C2×Q8—7642no—
155C2×Q8C22×D19112yes—
156C22×C4—3822no—
157C22×C4—3822no—
158C22×C4—1912no—
159C22×C4—1911no—
160C22×C4—1912no—
161C22×C4—7641no—
162C22×C4—7641no—
163C22×C4C22×D19111yescharacteristic
164Q16—41no—
165Q16—7641no—
166Q16—7642no—
167M16—41no—
168M16—7641no—
169C2×C8—1912no—
170C2×C8C22×D19111yescharacteristic
171C2×C8C22×D19111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—3822no—
174C4⋊C4—3822no—
175C4⋊C4—3822no—
176C4⋊C4—3822no—
177C4⋊C4—1912no—
178C4⋊C4C22×D19112yes—
179C22⋊C4—22no—
180C22⋊C4—3822no—
181C22⋊C4—3822no—
182C22⋊C4—3822no—
183C23—3824no—
184C23—1912no—
185C23—7641no—
186C23D4×D19111yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—3822no—
190Q8—3822no—
191Q8—7644no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3822no—
197C2×C4—3822no—
198C2×C4—3822no—
199C2×C4—3822no—
200C2×C4—1912no—
201C2×C4—7642no—
202C2×C4—7644no—
203C2×C4—7644no—
204C2×C4—7641no—
205C2×C4D4×D19111yescharacteristic
206C2×C4C23×D19111yescharacteristic
207C8—21no—
208C8—21no—
209C8—3822no—
210C22—22no—
211C22—21no—
212C22—3822no—
213C22—3822no—
214C22—1914no—
215C22—7644no—
216C22C2×D4×D19111yescharacteristic
217C4—21no—
218C4—42no—
219C4—42no—
220C4—3822no—
221C4—7642no—
222C4—7642no—
223C4#2025936711yescharacteristic
224C4#2025936811yescharacteristic
225C2—41no—
226C2—3824no—
227C2#2713322111yescharacteristic
228C2#137×D19112yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#470×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#470D191
hypocenterC191#3195
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #27132995C2semidirect
3. #27132992C2semidirect
4. #27132921C2semidirect
5. #27132919C2semidirect
6. #27132820C2semidirect
7. #27132817C2semidirect
9. #470×C191C2semidirect
12. C2×Dic764C22semidirect
13. C2×#15690277C22semidirect
23. #20259254C22semidirect
24. #20259253C22semidirect
25. #20259247C22semidirect
26. #20259246C22semidirect
27. #20259245C22semidirect
30. C2×Q16×C191C22semidirect
33. #106×C191C22semidirect
34. #103×C191C22semidirect
114. C191#3195semidirect
121. #470D191semidirect
129. C2×Q16C2×D191semidirect
143. #106C2×D191semidirect
147. #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
#271329952
#271329922
#271329212
#271329192
#271328202
#271328172
#271326262
#470×C1912
#3195191