Finite Group Databasebeta

← #37581927

Subgroups — #37581927

Order 24448.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#27133499C211yesmaximal, characteristic
2#27133239C212yesmaximal, complemented
3#27133234C211yesmaximal, characteristic, complemented
4#27132854C218yesmaximal, complemented
5#27132636C212yesmaximal, complemented
6#410×C191C211yesmaximal, characteristic, complemented
7#20259370—21no—
8#20259369—22no—
9#20259368—22no—
10#20259367—21no—
11#20259367C2214yes—
12C2×#15690287—22no—
13C2×#15690287C2212yes—
14#41×D191—24no—
15C22×Dic382C2211yescharacteristic, complemented
16#20259322—22no—
17#20259320—21no—
18#20259310—22no—
19#20259309—21no—
20#20259306C2218yescomplemented
21#20259303C2214yescomplemented
22#20259302—24no—
23#20259301—22no—
24#20259281C2211yescharacteristic, complemented
25#20259236C2218yescomplemented
26C2×#41×C191C2211yescharacteristic
27#117×C191C2212yescomplemented
28#99×C191C2214yescomplemented
29#15690288—24no—
30#15690287—24no—
31#15690287—24no—
32#15690287—22no—
33#15690287—22no—
34#15690287—24no—
35#15690287—22no—
36#15690287—22no—
37#15690286—24no—
38#15690286—28no—
39C2×Dic382—22no—
40C2×Dic382—21no—
41C2×Dic382—22no—
42C2×Dic382—21no—
43C2×Dic382—22no—
44C2×Dic382D412yes—
45Q8×D191—24no—
46C2×D764—22no—
47C2×D764D412yes—
48C2×C22⋊D191—24no—
49C2×C22⋊D191C2314yes—
50D4×D191—24no—
51D4×D191—28no—
52C22×Dic191C2312yes—
53C2×C4×D191—24no—
54C2×C4×D191D414yes—
55C4⋊Dic191—28no—
56C4⋊Dic191—21no—
57Dic191⋊C4—24no—
58C22⋊Dic191—24no—
59C22⋊Dic191C2312yes—
60(C2×D191)⋊C4—28no—
61C4×Dic191—21no—
62#41×C191—22no—
63#41×C191—22no—
64C2×Q8×C191D411yescharacteristic
65C2×D4×C191D411yescharacteristic
66C2×D4×C191C2312yes—
67C22×C4×C191D412yes—
68C22×C4×C191C2311yescharacteristic
69C4⋊C4×C191—22no—
70C22⋊C4×C191C2314yes—
71C42×C191—22no—
72C22×D191—24no—
73C22×D191C2×D414yes—
74Dic382—24no—
75Dic382—22no—
76Dic382—22no—
77Dic382—41no—
78Dic382—41no—
79Dic382—42no—
80D764—24no—
81D764—22no—
82D764—22no—
83C22⋊D191—28no—
84C22⋊D191—24no—
85C22⋊D191—24no—
86C2×Dic191—24no—
87C2×Dic191—22no—
88C2×Dic191—24no—
89C2×Dic191—22no—
90C2×Dic191C2×D414yes—
91C4×D191—28no—
92C4×D191—24no—
93C4×D191—24no—
94C23×C191C2×D412yes—
95C23×C191C2411yescharacteristic
96Q8×C191—22no—
97D4×C191—24no—
98D4×C191—22no—
99C2×C4×C191—22no—
100C2×C4×C191—22no—
101C2×C4×C191—24no—
102C2×C4×C191C2×D412yes—
103C2×C4×C191C2×D412yes—
104C2×C4×C191C2×D412yes—
105C2×C4×C191C2×D411yescharacteristic
106C2×C4×C191C2×D411yescharacteristic
107C22×C191—22no—
108C22×C191—41no—
109C22×C191—42no—
110C22×C191C22×D412yes—
111C22×C191C22×D411yescharacteristic
112C2×D191—24no—
113C2×D191—44no—
114C2×D191—84no—
115C4×C191—22no—
116C4×C191—44no—
117C4×C191#12011yescharacteristic
118C4×C191#12011yescharacteristic
119C4×C191#12012yes—
120Dic191—24no—
121Dic191—42no—
122Dic191—42no—
123C2×C191—21no—
124C2×C191—22no—
125C2×C191—42no—
126C2×C191C2×#12011yescharacteristic
127D191—44no—
128C191#302811yescharacteristic, complemented, Sylow
129#3028—1911nomaximal, Sylow
130#585—1911no—
131#536—1911no—
132#536—1912no—
133#458—1918no—
134#410—1912no—
135#410D19111yescharacteristic, complemented
1362−1+4—3822no—
1372−1+4—3821no—
138H(5,𝔽2)—3822no—
139H(5,𝔽2)—3821no—
140H(5,𝔽2)—1914no—
141C2×#41—3824no—
142C2×#41—3822no—
143C2×#41—1912no—
144C2×#41C2×D19111yescharacteristic
145C22×Q8—1911no—
146C4⋊Q8—3822no—
147C4⋊Q8—3821no—
148#124—3822no—
149#124—3821no—
150#123—1914no—
151#123—1918no—
152C22⋊Q8—3824no—
153C22⋊Q8—3822no—
154#117—1911no—
155#117C2×D19112yescomplemented
156#99—1918no—
157#99C2×D19114yescomplemented
158#41—22no—
159#41—22no—
160#41—3824no—
161#41—3824no—
162#41—3824no—
163#41—3824no—
164#41—3828no—
165#41—3822no—
166#41—3822no—
167#41—3822no—
168#41—3824no—
169#41—3822no—
170C2×Q8—3822no—
171C2×Q8—3824no—
172C2×Q8—3822no—
173C2×Q8—3822no—
174C2×Q8—3821no—
175C2×Q8—3821no—
176C2×Q8—1912no—
177C2×Q8D76411yescharacteristic
178C2×D4—3822no—
179C2×D4—3824no—
180C2×D4—3828no—
181C2×D4—3824no—
182C2×D4—1914no—
183C2×D4—1912no—
184C2×D4D76411yescharacteristic
185C2×D4C22×D19112yes—
186C22×C4—3824no—
187C22×C4—1912no—
188C22×C4—1914no—
189C22×C4D76412yes—
190C22×C4C22×D19111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3828no—
193C4⋊C4—3824no—
194C4⋊C4—3821no—
195C22⋊C4—3828no—
196C22⋊C4—3824no—
197C22⋊C4—1912no—
198C22⋊C4C22×D19114yes—
199C42—22no—
200C42—3821no—
201C23—3824no—
202C23—1914no—
203C23C2×D76412yes—
204C23C23×D19111yescharacteristic
205Q8—22no—
206Q8—3824no—
207Q8—3822no—
208Q8—3822no—
209Q8—7641no—
210Q8—7641no—
211Q8—7642no—
212D4—24no—
213D4—22no—
214D4—3824no—
215D4—3828no—
216D4—3824no—
217D4—3824no—
218D4—3822no—
219D4—3822no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—3828no—
224C2×C4—3824no—
225C2×C4—3824no—
226C2×C4—3824no—
227C2×C4—3824no—
228C2×C4—3822no—
229C2×C4—3822no—
230C2×C4—1914no—
231C2×C4D4×D19112yes—
232C2×C4D4×D19112yes—
233C2×C4C2×D76412yes—
234C2×C4C2×D76411yescharacteristic
235C2×C4C2×D76411yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—3824no—
240C22—15284no—
241C22—7644no—
242C22C2×D4×D19112yes—
243C22C22×D76411yescharacteristic
244C4—22no—
245C4—44no—
246C4—3824no—
247C4—7642no—
248C4—7642no—
249C4#2025929211yescharacteristic
250C4#2025929211yescharacteristic
251C4#2025929212yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—7644no—
256C2C2×#2025929211yescharacteristic
subgroupG/H
centerC2C2×#20259292
commutator, cosocleC23×C191C24
FrattiniC23C23×D191
Fitting#410×C191C2
socleC2×C191C2×#120
prime radical#27133499C2
hypercenter#410D191
hypocenterC191#3028
LCSG ▷ C23×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27133239C2semidirect
3. #27133234C2semidirect
4. #27132854C2semidirect
5. #27132636C2semidirect
6. #410×C191C2semidirect
15. C22×Dic382C22semidirect
20. #20259306C22semidirect
21. #20259303C22semidirect
24. #20259281C22semidirect
25. #20259236C22semidirect
27. #117×C191C22semidirect
28. #99×C191C22semidirect
128. C191#3028semidirect
135. #410D191semidirect
155. #117C2×D191semidirect
157. #99C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334992
#271332392
#271332342
#271328542
#271326362
#410×C1912
#3028191