Finite Group Databasebeta

← #37571452

Subgroups — #37571452

Order 24448.

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

normal subgroups only

This group has 272 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#27133262C211yesmaximal, characteristic, complemented
2#27132813C211yesmaximal, characteristic, complemented
3#27132799C211yesmaximal, characteristic
4#27132581C211yesmaximal, characteristic, complemented
5#27132396C211yesmaximal, characteristic, complemented
6#27132388C211yesmaximal, characteristic, complemented
7#328×C191C211yesmaximal, characteristic, complemented
8C22×D764C2211yescharacteristic
9C2×#15690274—21no—
10C2×D1528—21no—
11#20259317—21no—
12#20259313—21no—
13#20259307—21no—
14#20259306—21no—
15#20259298—21no—
16#20259296C2211yescharacteristic
17(C22⋊D191)⋊C4—21no—
18C2×Dic191⋊C4C2211yescharacteristic
19C2×(C2×D191)⋊C4—21no—
20C2×(C2×D191)⋊C4—21no—
21#20259244—21no—
22#20259242—21no—
23C22⋊(C191⋊C8)C2211yescharacteristic
24C22⋊Q8×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic
26#95×C191C2211yescharacteristic
27C23×D191—21no—
28C2×D764—21no—
29C2×D764—21no—
30C2×D764—42no—
31C2×D764D411yescharacteristic
32C2×C22⋊D191—21no—
33C22×Dic191D411yescharacteristic
34C22×Dic191D411yescharacteristic
35C2×C4×D191—21no—
36#15690274—42no—
37D1528—42no—
38C2×C191⋊C8—21no—
39C4⋊Dic191—21no—
40C4⋊Dic191—21no—
41C4⋊Dic191D411yescharacteristic
42Dic191⋊C4—21no—
43Dic191⋊C4—21no—
44Dic191⋊C4—21no—
45(C2×D191)⋊C4—21no—
46(C2×D191)⋊C4—21no—
47(C2×D191)⋊C4—21no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—42no—
50(C2×D191)⋊C4—42no—
51C4×Dic191—21no—
52C2×Q8×C191D411yescharacteristic
53C22×C4×C191—21no—
54C22×C4×C191C2311yescharacteristic
55C2×C8×C191—21no—
56C4⋊C4×C191—21no—
57C4⋊C4×C191D411yescharacteristic
58C22⋊C4×C191—21no—
59C22×D191—22no—
60C22×D191—21no—
61C22×D191—41no—
62C22×D191—41no—
63C22×D191—41no—
64C22×D191—41no—
65C22×D191—41no—
66C22×D191—41no—
67D764—22no—
68D764—22no—
69D764—41no—
70D764—41no—
71D764—41no—
72C22⋊D191—41no—
73C22⋊D191—41no—
74C2×Dic191—21no—
75C2×Dic191—21no—
76C2×Dic191—21no—
77C2×Dic191—21no—
78C2×Dic191—21no—
79C2×Dic191—41no—
80C4×D191—81no—
81C191⋊C8—41no—
82C23×C191C2×D411yescharacteristic
83Q8×C191—41no—
84C2×C4×C191—21no—
85C2×C4×C191—21no—
86C2×C4×C191—41no—
87C2×C4×C191—41no—
88C2×C4×C191—41no—
89C2×C4×C191—41no—
90C2×C4×C191C2×D411yescharacteristic
91C2×C4×C191C2×D411yescharacteristic
92C8×C191—41no—
93C22×C191—21no—
94C22×C191—21no—
95C22×C191#12011yescharacteristic
96C22×C191#13511yescharacteristic
97C22×C191AGL(1,ℤ/8)11yescharacteristic
98C2×D191—42no—
99C2×D191—42no—
100C2×D191—41no—
101C2×D191—41no—
102C2×D191—41no—
103C2×D191—42no—
104C2×D191—81no—
105C2×D191—81no—
106C2×D191—81no—
107C2×D191—81no—
108C4×C191—21no—
109C4×C191—21no—
110C4×C191—41no—
111C4×C191—41no—
112C4×C191—81no—
113Dic191—41no—
114Dic191—41no—
115Dic191—41no—
116C2×C191—21no—
117C2×C191—21no—
118C2×C191#44911yescharacteristic
119C2×C191#45411yescharacteristic
120C2×C191B(4,𝔽2)11yescharacteristic
121D191—81no—
122D191—82no—
123C191#159711yescharacteristic, complemented, Sylow
124#1597—1911nomaximal, Sylow
125#539—1911no—
126#450—1911no—
127#447—1911no—
128#394—1911no—
129#328D19111yescharacteristic, complemented
130#327—1911no—
131#324—1911no—
132C22×D4—1911no—
133C2×SD16—3821no—
134C2×D8—3821no—
135#126—3821no—
136#125—3821no—
137#123—3821no—
138#123—3821no—
139C22⋊Q8—3821no—
140C22⋊Q8C2×D19111yescharacteristic
141#121—1911no—
142C4×D4—3821no—
143C2×C4⋊C4—1911no—
144C2×C22⋊C4—3821no—
145C2×C22⋊C4—3821no—
146#102—3821no—
147#102—3821no—
148C22⋊C8—1911no—
149C22⋊C8C2×D19111yescharacteristic
150#95C2×D19111yescharacteristic
151C24—3821no—
152C2×Q8D76411yescharacteristic
153C2×D4—3821no—
154C2×D4—3821no—
155C2×D4—3821no—
156C2×D4—1911no—
157C2×D4—7642no—
158C22×C4—21no—
159C22×C4—3821no—
160C22×C4—1911no—
161C22×C4—1911no—
162C22×C4C22×D19111yescharacteristic
163SD16—7642no—
164D8—7642no—
165C2×C8—21no—
166C2×C8—3821no—
167C4⋊C4—21no—
168C4⋊C4—3821no—
169C4⋊C4—3821no—
170C4⋊C4—3821no—
171C4⋊C4—3821no—
172C4⋊C4—3821no—
173C4⋊C4—1911no—
174C4⋊C4D76411yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—3821no—
177C22⋊C4—3821no—
178C22⋊C4—3821no—
179C22⋊C4—3821no—
180C22⋊C4—7642no—
181C22⋊C4—7642no—
182C42—3821no—
183C23—3822no—
184C23—3821no—
185C23—7641no—
186C23—7641no—
187C23—7641no—
188C23—7641no—
189C23—7641no—
190C23—7641no—
191C23D4×D19111yescharacteristic
192Q8—41no—
193D4—3822no—
194D4—3822no—
195D4—7641no—
196D4—7641no—
197D4—7641no—
198D4—7641no—
199D4—7641no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—3821no—
207C2×C4—3821no—
208C2×C4—3821no—
209C2×C4—3821no—
210C2×C4—3821no—
211C2×C4—15281no—
212C2×C4—7641no—
213C2×C4D4×D19111yescharacteristic
214C2×C4C2×D76411yescharacteristic
215C8—41no—
216C8—7641no—
217C22—21no—
218C22—21no—
219C22—15281no—
220C22—15281no—
221C22—15281no—
222C22—15281no—
223C22—7642no—
224C22—7642no—
225C22—7641no—
226C22—7641no—
227C22—7641no—
228C22—7642no—
229C22#2025929211yescharacteristic
230C22#2025934511yescharacteristic
231C22#2025934911yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—7641no—
238C4—7641no—
239C4—7641no—
240C2—21no—
241C2—21no—
242C2—15282no—
243C2—15281no—
244C2#2713281111yescharacteristic
245C2#2713283711yescharacteristic
246C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#328×C191C2
socleC22×C191#120
prime radical#27132799C2
hypercenter#328D191
hypocenterC191#1597
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #27133262C2semidirect
2. #27132813C2semidirect
4. #27132581C2semidirect
5. #27132396C2semidirect
6. #27132388C2semidirect
7. #328×C191C2semidirect
123. C191#1597semidirect
129. #328D191semidirect

Maximal subgroups

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

subgroupindex
#271332622
#271328132
#271327992
#271325812
#271323962
#271323882
#328×C1912
#1597191