Finite Group Databasebeta

← #37571473

Subgroups — #37571473

Order 24448.

308 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 51 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#27133221C211yesmaximal, characteristic
2#27132804C211yesmaximal, characteristic, complemented
3#27132801C211yesmaximal, characteristic, complemented
4#27132576C211yesmaximal, characteristic, complemented
5#27132396C211yesmaximal, characteristic, complemented
6#27132386C211yesmaximal, characteristic, complemented
7#327×C191C211yesmaximal, characteristic, complemented
8C22×Dic382C2211yescharacteristic, complemented
9C2×D4×D191—21no—
10C2×#15690276—21no—
11C2×#15690275—21no—
12#20259307—21no—
13#20259304—21no—
14#20259296C2211yescharacteristic
15#20259294—21no—
16#20259292—21no—
17#20259290—21no—
18C2×(C2×D191)⋊C4—21no—
19C2×(C2×D191)⋊C4C2211yescharacteristic
20#20259247—21no—
21#20259245—21no—
22C22⋊(C191⋊C8)C2211yescharacteristic, complemented
23C191⋊#95—21no—
24#121×C191C2211yescharacteristic
25C22⋊C8×C191C2211yescharacteristic, complemented
26#95×C191C2211yescharacteristic, complemented
27C23×D191—21no—
28C23×D191D411yescharacteristic
29C2×Dic382—21no—
30C2×Dic382—21no—
31C2×Dic382—42no—
32C2×D764—21no—
33C2×D764D411yescharacteristic
34C2×C22⋊D191—21no—
35C2×C22⋊D191—21no—
36C2×C22⋊D191—21no—
37D4×D191—42no—
38D4×D191—42no—
39C22×Dic191—21no—
40C22×Dic191D411yescharacteristic
41C2×C4×D191—21no—
42#15690276—42no—
43#15690275—42no—
44C2×C191⋊C8—21no—
45C4⋊Dic191D411yescharacteristic
46Dic191⋊C4—21no—
47C22⋊Dic191—21no—
48(C2×D191)⋊C4—21no—
49(C2×D191)⋊C4—21no—
50(C2×D191)⋊C4—21no—
51(C2×D191)⋊C4—21no—
52(C2×D191)⋊C4—42no—
53C2×D4×C191—21no—
54C2×D4×C191D411yescharacteristic
55C22×C4×C191—21no—
56C22×C4×C191C2311yescharacteristic
57C2×C8×C191—21no—
58C4⋊C4×C191D411yescharacteristic
59C22⋊C4×C191—21no—
60C22×D191—21no—
61C22×D191—21no—
62C22×D191—21no—
63C22×D191—42no—
64C22×D191—41no—
65C22×D191—41no—
66C22×D191—81no—
67C22×D191—81no—
68C22×D191SD1612yescomplemented
69Dic382—22no—
70Dic382—22no—
71Dic382—41no—
72Dic382—41no—
73D764—41no—
74D764—42no—
75C22⋊D191—42no—
76C22⋊D191—41no—
77C22⋊D191—41no—
78C22⋊D191—81no—
79C2×Dic191—22no—
80C2×Dic191—21no—
81C2×Dic191—21no—
82C2×Dic191—41no—
83C2×Dic191—41no—
84C2×Dic191—41no—
85C4×D191—42no—
86C191⋊C8—41no—
87C23×C191—21no—
88C23×C191C2×D411yescharacteristic
89D4×C191—41no—
90D4×C191—41no—
91D4×C191—41no—
92C2×C4×C191—21no—
93C2×C4×C191—41no—
94C2×C4×C191—41no—
95C2×C4×C191—41no—
96C2×C4×C191—41no—
97C2×C4×C191C2×D411yescharacteristic
98C2×C4×C191C2×D411yescharacteristic
99C8×C191—41no—
100C22×C191—21no—
101C22×C191—21no—
102C22×C191—41no—
103C22×C191—41no—
104C22×C191—41no—
105C22×C191#12011yescharacteristic
106C22×C191C2×SD1611yescharacteristic
107C22×C191AGL(1,ℤ/8)11yescharacteristic
108C2×D191—22no—
109C2×D191—41no—
110C2×D191—41no—
111C2×D191—42no—
112C2×D191—41no—
113C2×D191—41no—
114C2×D191—41no—
115C2×D191—81no—
116C2×D191—84no—
117C4×C191—21no—
118C4×C191—21no—
119C4×C191—41no—
120C4×C191—81no—
121Dic191—41no—
122Dic191—41no—
123Dic191—42no—
124C2×C191—21no—
125C2×C191—21no—
126C2×C191—81no—
127C2×C191#45011yescharacteristic
128C2×C191#45511yescharacteristic
129C2×C191B(4,𝔽2)11yescharacteristic
130D191—42no—
131D191—81no—
132C191#160111yescharacteristic, complemented, Sylow
133#1601—1911nomaximal, Sylow
134#534—1911no—
135#448—1911no—
136#448—1911no—
137#393—1911no—
138#327—1911no—
139#327D19111yescharacteristic, complemented
140#323—1911no—
141C22×Q8—1911no—
142C22×D4—3821no—
143C2×SD16—3821no—
144C2×SD16—3821no—
145#123—3821no—
146#123—3821no—
147#121—3821no—
148#121—1911no—
149#121C2×D19111yescharacteristic
150#120—3821no—
151#120—3821no—
152C2×C22⋊C4—3821no—
153C2×C22⋊C4—1911no—
154#103—3821no—
155#103—3821no—
156C22⋊C8—1911no—
157C22⋊C8C2×D19111yescharacteristic, complemented
158#95—3821no—
159#95C2×D19111yescharacteristic, complemented
160C24—3821no—
161C24—1911no—
162C2×Q8—3821no—
163C2×Q8—3821no—
164C2×Q8—7642no—
165C2×D4—21no—
166C2×D4—3821no—
167C2×D4—3821no—
168C2×D4—3821no—
169C2×D4—3821no—
170C2×D4—1911no—
171C2×D4—7642no—
172C2×D4—7642no—
173C2×D4D76411yescharacteristic
174C22×C4—21no—
175C22×C4—3821no—
176C22×C4—3821no—
177C22×C4—1911no—
178C22×C4C22×D19111yescharacteristic
179SD16—7642no—
180SD16—7642no—
181C2×C8—21no—
182C2×C8—3821no—
183C4⋊C4—3821no—
184C4⋊C4—1911no—
185C4⋊C4D76411yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3821no—
188C22⋊C4—3821no—
189C22⋊C4—3821no—
190C22⋊C4—3821no—
191C22⋊C4—3821no—
192C22⋊C4—7642no—
193C23—21no—
194C23—3821no—
195C23—3821no—
196C23—3821no—
197C23—1912no—
198C23—15281no—
199C23—15281no—
200C23—7641no—
201C23—7641no—
202C23—7642no—
203C23D4×D19111yescharacteristic
204Q8—3822no—
205Q8—3822no—
206Q8—7641no—
207Q8—7641no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—15281no—
212D4—7641no—
213D4—7642no—
214D4—7642no—
215D4—7641no—
216D4—7641no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—3822no—
223C2×C4—3821no—
224C2×C4—3821no—
225C2×C4—7642no—
226C2×C4—7641no—
227C2×C4—7641no—
228C2×C4—7641no—
229C2×C4D4×D19111yescharacteristic
230C2×C4C2×D76411yescharacteristic
231C8—41no—
232C8—7641no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—3822no—
239C22—15281no—
240C22—15284no—
241C22—7642no—
242C22—7641no—
243C22—7641no—
244C22—7641no—
245C22—7641no—
246C22—7641no—
247C22#2025929211yescharacteristic
248C22SD16×D19111yescharacteristic
249C22#2025934611yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—7641no—
255C4—7641no—
256C4—7642no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—15281no—
261C2—7642no—
262C2#2713281511yescharacteristic
263C2#2713284211yescharacteristic
264C2#2713285111yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC22×C4×C191C23
FrattiniC22×C4C22×D191
Fitting#327×C191C2
socleC22×C191#120
prime radical#27133221C2
hypercenter#327D191
hypocenterC191#1601
LCSG ▷ C22×C4×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27132804C2semidirect
3. #27132801C2semidirect
4. #27132576C2semidirect
5. #27132396C2semidirect
6. #27132386C2semidirect
7. #327×C191C2semidirect
8. C22×Dic382C22semidirect
22. C22⋊(C191⋊C8)C22semidirect
25. C22⋊C8×C191C22semidirect
26. #95×C191C22semidirect
68. C22×D191SD16semidirect
132. C191#1601semidirect
139. #327D191semidirect
157. C22⋊C8C2×D191semidirect
159. #95C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332212
#271328042
#271328012
#271325762
#271323962
#271323862
#327×C1912
#1601191