Finite Group Databasebeta

← #37571502

Subgroups — #37571502

Order 24448.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#27133251C211yesmaximal, characteristic
2#27132821C211yesmaximal, characteristic, complemented
3#27132804C211yesmaximal, characteristic, complemented
4#27132575C211yesmaximal, characteristic, complemented
5#27132401C211yesmaximal, characteristic, complemented
6#27132398C211yesmaximal, characteristic, complemented
7#323×C191C211yesmaximal, characteristic, complemented
8C22×Dic382C2211yescharacteristic
9C2×Dic764—21no—
10C2×#15690276—21no—
11#20259315—21no—
12#20259308—21no—
13#20259304—21no—
14#20259301C2211yescharacteristic
15#20259296C2211yescharacteristic
16#20259295—21no—
17#20259291—21no—
18C4×C22⋊D191—21no—
19C2×C22⋊Dic191—21no—
20#20259247—21no—
21#20259247—21no—
22C191⋊#95—21no—
23C191⋊#95C2211yescharacteristic
24C2×C22⋊C4×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26C22⋊C8×C191C2211yescharacteristic
27C2×Dic382—21no—
28C2×Dic382—21no—
29C2×Dic382—42no—
30C2×Dic382D411yescharacteristic
31C2×D764D411yescharacteristic
32C2×C22⋊D191—21no—
33C2×C22⋊D191—21no—
34C2×C22⋊D191—21no—
35C22×Dic191—21no—
36C22×Dic191—21no—
37C2×C4×D191—21no—
38Dic764—42no—
39#15690276—42no—
40C4⋊Dic191D411yescharacteristic
41C4⋊Dic191D411yescharacteristic
42Dic191⋊C4—21no—
43Dic191⋊C4—21no—
44C22⋊Dic191—21no—
45C22⋊Dic191—21no—
46C22⋊Dic191—21no—
47C22⋊Dic191—42no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—21no—
50C4×Dic191—21no—
51C24×C191D411yescharacteristic
52C22×C4×C191D411yescharacteristic
53C22×C4×C191C2311yescharacteristic
54C2×C8×C191—21no—
55C2×C8×C191—21no—
56C22⋊C4×C191—21no—
57C22⋊C4×C191—21no—
58C22×D191—21no—
59Dic382—22no—
60Dic382—22no—
61Dic382—41no—
62Dic382—41no—
63Dic382—41no—
64D764—41no—
65C22⋊D191—41no—
66C22⋊D191—41no—
67C22⋊D191—81no—
68C22⋊D191—81no—
69C2×Dic191—22no—
70C2×Dic191—21no—
71C2×Dic191—21no—
72C2×Dic191—21no—
73C2×Dic191—41no—
74C2×Dic191—41no—
75C2×Dic191—41no—
76C2×Dic191—41no—
77C2×Dic191—41no—
78C4×D191—81no—
79C23×C191—21no—
80C23×C191—21no—
81C23×C191—21no—
82C23×C191—41no—
83C23×C191—41no—
84C23×C191C2×D411yescharacteristic
85C2×C4×C191—21no—
86C2×C4×C191—41no—
87C2×C4×C191—41no—
88C2×C4×C191C2×D411yescharacteristic
89C2×C4×C191C2×D411yescharacteristic
90C8×C191—41no—
91C8×C191—41no—
92C22×C191—21no—
93C22×C191—21no—
94C22×C191—42no—
95C22×C191—41no—
96C22×C191—41no—
97C22×C191—41no—
98C22×C191—81no—
99C22×C191#12011yescharacteristic
100C22×C191#13511yescharacteristic
101C22×C191#13711yescharacteristic
102C2×D191—41no—
103C2×D191—41no—
104C2×D191—41no—
105C4×C191—21no—
106C4×C191—21no—
107C4×C191—41no—
108Dic191—41no—
109Dic191—41no—
110Dic191—42no—
111Dic191—41no—
112Dic191—81no—
113C2×C191—21no—
114C2×C191—21no—
115C2×C191—81no—
116C2×C191#45211yescharacteristic
117C2×C191#45511yescharacteristic
118C2×C191B(4,𝔽2)11yescharacteristic
119D191—81no—
120C191#160611yescharacteristic, complemented, Sylow
121#1606—1911nomaximal, Sylow
122#538—1911no—
123#451—1911no—
124#448—1911no—
125#393—1911no—
126#328—1911no—
127#327—1911no—
128#323D19111yescharacteristic, complemented
129C22×Q8—1911no—
130C2×Q16—3821no—
131C2×SD16—3821no—
132#126—3821no—
133#124—3821no—
134#123—3821no—
135C22⋊Q8—1911no—
136#121—3821no—
137#121—1911no—
138#120—3821no—
139C4×D4—3821no—
140C2×C22⋊C4—3821no—
141C2×C22⋊C4C2×D19111yescharacteristic
142#103—3821no—
143#103—3821no—
144C22⋊C8C2×D19111yescharacteristic
145C22⋊C8C2×D19111yescharacteristic
146#95—3821no—
147#95—1911no—
148C24D76411yescharacteristic
149C2×Q8—3821no—
150C2×Q8—3821no—
151C2×Q8—1911no—
152C2×Q8—7642no—
153C2×D4—3821no—
154C2×D4—3821no—
155C2×D4—3821no—
156C2×D4—1911no—
157C22×C4—3821no—
158C22×C4—3821no—
159C22×C4—3821no—
160C22×C4D76411yescharacteristic
161C22×C4C22×D19111yescharacteristic
162Q16—7642no—
163SD16—7642no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—3821no—
167C4⋊C4—3821no—
168C4⋊C4—1911no—
169C4⋊C4—1911no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3821no—
173C22⋊C4—3821no—
174C22⋊C4—3821no—
175C22⋊C4—3821no—
176C22⋊C4—3821no—
177C22⋊C4—7642no—
178C42—3821no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3821no—
185C23C2×D76411yescharacteristic
186Q8—3822no—
187Q8—3822no—
188Q8—7641no—
189Q8—7641no—
190Q8—7641no—
191D4—15281no—
192D4—15281no—
193D4—7641no—
194D4—7641no—
195D4—7641no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3822no—
200C2×C4—3821no—
201C2×C4—3821no—
202C2×C4—3821no—
203C2×C4—15281no—
204C2×C4—7641no—
205C2×C4—7641no—
206C2×C4—7641no—
207C2×C4—7641no—
208C2×C4—7641no—
209C2×C4D4×D19111yescharacteristic
210C2×C4D4×D19111yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—7641no—
221C22—7641no—
222C22—7641no—
223C22#2025929211yescharacteristic
224C22#2025934411yescharacteristic
225C22#2025935511yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—15281no—
230C4—7641no—
231C4—7641no—
232C4—7642no—
233C4—7641no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—15281no—
238C2#2713282711yescharacteristic
239C2#2713284311yescharacteristic
240C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#323×C191C2
socleC22×C191#120
prime radical#27133251C2
hypercenter#323D191
hypocenterC191#1606
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27132821C2semidirect
3. #27132804C2semidirect
4. #27132575C2semidirect
5. #27132401C2semidirect
6. #27132398C2semidirect
7. #323×C191C2semidirect
120. C191#1606semidirect
128. #323D191semidirect

Maximal subgroups

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

subgroupindex
#271332512
#271328212
#271328042
#271325752
#271324012
#271323982
#323×C1912
#1606191