Finite Group Databasebeta

← C2×#27132800 #37581706

Subgroups — C2×#27132800 #37581706

Order 24448.

792 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 173 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×D191C211yesmaximal, characteristic, complemented
2C22×#15690275C211yesmaximal, characteristic, complemented
3C2×#20259293C211yesmaximal, characteristic
4#27132800C218yesmaximal, direct product factor
5C2×#20259247C211yesmaximal, characteristic, complemented
6C2×#20259246C211yesmaximal, characteristic, complemented
7C2×(C2×D191)⋊C8C211yesmaximal, characteristic, complemented
8C22×SD16×C191C211yesmaximal, characteristic, complemented
9C22×Dic382—21no—
10C22×Dic382C2211yescharacteristic
11C2×Q8×D191—24no—
12C2×Q8×D191—22no—
13C2×Q8×D191—24no—
14C2×Q8×D191C2214yescomplemented
15C22×C22⋊D191—21no—
16C22×C4×D191—21no—
17C22×C4×D191C2211yescharacteristic
18C2×#15690275—24no—
19C2×#15690275C2214yescomplemented
20C22×C191⋊C8C2211yescharacteristic
21#20259293—22no—
22#20259293C2214yes—
23C2×C4⋊Dic191C2211yescharacteristic
24C2×C22⋊Dic191—21no—
25#20259247C2214yescomplemented
26#20259246C2214yescomplemented
27(C2×D191)⋊C8C2214yescomplemented
28C22×Q8×C191C2211yescharacteristic
29C22×D4×C191C2211yescharacteristic
30C2×SD16×C191—24no—
31C2×SD16×C191C2214yescomplemented
32C22×C8×C191C2211yescharacteristic
33C23×D191D411yescharacteristic
34C2×Dic382—28no—
35C2×Dic382—24no—
36C2×Dic382—21no—
37C2×Dic382—22no—
38C2×Dic382D414yescomplemented
39C2×Dic382C2312yes—
40Q8×D191—216no—
41Q8×D191—28no—
42Q8×D191—28no—
43C2×C22⋊D191—24no—
44C2×C22⋊D191—44no—
45C22×Dic191—21no—
46C22×Dic191—21no—
47C22×Dic191D411yescharacteristic
48C2×C4×D191—28no—
49C2×C4×D191—24no—
50C2×C4×D191—22no—
51C2×C4×D191—22no—
52C2×C4×D191D412yes—
53C2×C4×D191C2312yes—
54#15690275—28no—
55C2×C191⋊C8—22no—
56C2×C191⋊C8C2312yes—
57C4⋊Dic191—21no—
58C4⋊Dic191C2312yes—
59C22⋊Dic191—24no—
60C24×C191—21no—
61C2×Q8×C191—22no—
62C2×Q8×C191—21no—
63C2×Q8×C191D414yescomplemented
64C2×Q8×C191C2312yes—
65C2×D4×C191—22no—
66C2×D4×C191—22no—
67C2×D4×C191—21no—
68C2×D4×C191C2312yes—
69C22×C4×C191—21no—
70C22×C4×C191C2311yescharacteristic
71SD16×C191—28no—
72C2×C8×C191—22no—
73C2×C8×C191C2312yes—
74C22×D191—22no—
75C22×D191—22no—
76C22×D191SD1614yescomplemented
77C22×D191C2×D412yes—
78Dic382—216no—
79Dic382—24no—
80Dic382—22no—
81Dic382C2×D414yescomplemented
82C22⋊D191—48no—
83C2×Dic191—24no—
84C2×Dic191—22no—
85C2×Dic191—22no—
86C2×Dic191—22no—
87C2×Dic191—42no—
88C2×Dic191C2×D412yes—
89C4×D191—216no—
90C4×D191—24no—
91C4×D191—22no—
92C4×D191—22no—
93C191⋊C8—22no—
94C23×C191—22no—
95C23×C191—42no—
96C23×C191—42no—
97C23×C191—42no—
98C23×C191C2×D411yescharacteristic
99Q8×C191—24no—
100Q8×C191—22no—
101Q8×C191C2×D414yescomplemented
102D4×C191—22no—
103D4×C191—22no—
104D4×C191—42no—
105C2×C4×C191—24no—
106C2×C4×C191—22no—
107C2×C4×C191C2×D411yescharacteristic
108C2×C4×C191C2×D412yes—
109C2×C4×C191C2×D412yes—
110C2×C4×C191C2411yescharacteristic
111C8×C191—22no—
112C22×C191—42no—
113C22×C191—42no—
114C22×C191—42no—
115C22×C191—84no—
116C22×C191#12012yes—
117C22×C191C2×SD1612yes—
118C22×C191#13712yes—
119C22×C191C22×D411yescharacteristic
120C2×D191—22no—
121C2×D191—22no—
122C2×D191—28no—
123C2×D191C2×SD1614yescomplemented
124C4×C191—24no—
125C4×C191#12012yes—
126C4×C191C22×D411yescharacteristic
127C4×C191C22×D411yescharacteristic
128Dic191—24no—
129Dic191—22no—
130Dic191—42no—
131C2×C191—82no—
132C2×C191#44814yescomplemented
133C2×C191C2×#12011yescharacteristic
134C2×C191C22×SD1611yescharacteristic
135C2×C191C2×#13711yescharacteristic
136D191—24no—
137C191C2×#44811yescharacteristic, complemented, Sylow
138C2×#448—1911nomaximal, Sylow
139C23×Q8—1911no—
140C22×SD16—1911no—
141C22×SD16D19111yescharacteristic, complemented
142C2×#121—1911no—
143#448—1918no—
144C2×#103—1911no—
145C2×#103—1911no—
146C2×C22⋊C8—1911no—
147C22×Q8—3821no—
148C22×Q8—3824no—
149C22×Q8—3824no—
150C22×Q8—3822no—
151C22×Q8—1914no—
152C22×Q8—1911no—
153C22×Q8C2×D19111yescharacteristic
154C22×D4—3821no—
155C22×D4C2×D19111yescharacteristic
156C23×C4—3821no—
157C23×C4—1911no—
158C2×SD16—24no—
159C2×SD16—3824no—
160C2×SD16—1914no—
161C2×SD16C2×D19114yescomplemented
162C22×C8—1911no—
163C22×C8C2×D19111yescharacteristic
164#121—3822no—
165#121—1914no—
166C2×C4⋊C4—1911no—
167C2×C22⋊C4—3821no—
168#103—1914no—
169#103—1914no—
170C22⋊C8—1914no—
171C24—21no—
172C24—1911no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—3828no—
176C2×Q8—3824no—
177C2×Q8—38216no—
178C2×Q8—3828no—
179C2×Q8—3828no—
180C2×Q8—3821no—
181C2×Q8—3822no—
182C2×Q8—1914no—
183C2×Q8—1912no—
184C2×Q8C22⋊D19114yescomplemented
185C2×Q8C22×D19112yes—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—3824no—
190C2×D4—7644no—
191C2×D4C22×D19112yes—
192C22×C4—21no—
193C22×C4—3828no—
194C22×C4—3824no—
195C22×C4—3821no—
196C22×C4—3822no—
197C22×C4—3822no—
198C22×C4—3821no—
199C22×C4—1911no—
200C22×C4—1912no—
201C22×C4—1912no—
202C22×C4C22×D19111yescharacteristic
203SD16—28no—
204SD16—3828no—
205C2×C8—22no—
206C2×C8—3822no—
207C2×C8—1912no—
208C2×C8C22×D19112yes—
209C4⋊C4—3821no—
210C4⋊C4—1912no—
211C22⋊C4—3824no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—3822no—
217C23—3822no—
218C23—1914no—
219C23—1912no—
220C23D4×D19111yescharacteristic
221Q8—22no—
222Q8—24no—
223Q8—38216no—
224Q8—3822no—
225Q8—3824no—
226Q8—1914no—
227Q8C2×C22⋊D19114yescomplemented
228D4—22no—
229D4—22no—
230D4—42no—
231D4—7648no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—38216no—
235C2×C4—3822no—
236C2×C4—3824no—
237C2×C4—3822no—
238C2×C4—3822no—
239C2×C4—3822no—
240C2×C4—3824no—
241C2×C4—3822no—
242C2×C4—1912no—
243C2×C4—7642no—
244C2×C4D4×D19112yes—
245C2×C4D4×D19111yescharacteristic
246C2×C4C2×C22⋊D19112yes—
247C2×C4C23×D19111yescharacteristic
248C8—22no—
249C8—3822no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—84no—
254C22—3822no—
255C22—3828no—
256C22—3822no—
257C22—1914no—
258C22#2025929012yes—
259C22SD16×D19112yes—
260C22#2025935112yes—
261C22C2×D4×D19111yescharacteristic
262C4—24no—
263C4—3824no—
264C4—3822no—
265C4—7642no—
266C4#2025929012yes—
267C4C2×D4×D19111yescharacteristic
268C4C22×C22⋊D19111yescharacteristic
269C2—82no—
270C2—3824no—
271C2#2713280014yesdirect product factor
272C2C2×#2025929011yescharacteristic
273C2C2×SD16×D19111yescharacteristic
274C2C2×#2025935111yescharacteristic
subgroupG/H
centerC23D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×SD16×C191C2
socleC23×C191C2×D4
prime radicalC2×#20259293C2
hypercenterC22×SD16D191
hypocenterC191C2×#448
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D191C2semidirect
2. C22×#15690275C2semidirect
4. #27132800C2direct (N × H)
5. C2×#20259247C2semidirect
6. C2×#20259246C2semidirect
7. C2×(C2×D191)⋊C8C2semidirect
8. C22×SD16×C191C2semidirect
14. C2×Q8×D191C22semidirect
19. C2×#15690275C22semidirect
25. #20259247C22semidirect
26. #20259246C22semidirect
27. (C2×D191)⋊C8C22semidirect
31. C2×SD16×C191C22semidirect
38. C2×Dic382D4semidirect
63. C2×Q8×C191D4semidirect
76. C22×D191SD16semidirect
81. Dic382C2×D4semidirect
101. Q8×C191C2×D4semidirect
123. C2×D191C2×SD16semidirect
132. C2×C191#448semidirect
137. C191C2×#448semidirect
141. C22×SD16D191semidirect
161. C2×SD16C2×D191semidirect
184. C2×Q8C22⋊D191semidirect
227. Q8C2×C22⋊D191semidirect
271. C2#27132800direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1912
C22×#156902752
C2×#202592932
#271328002
C2×#202592472
C2×#202592462
C2×(C2×D191)⋊C82
C22×SD16×C1912
C2×#448191