Finite Group Databasebeta

← C2×#27132894 #37582082

Subgroups — C2×#27132894 #37582082

Order 24448.

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

normal subgroups only

This group has 564 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×#15690274C211yesmaximal, characteristic, complemented
2C23×C191⋊C8C211yesmaximal, characteristic, complemented
3C2×#20259296C211yesmaximal, characteristic
4#27132894C218yesmaximal, direct product factor
5C2×#20259253C211yesmaximal, characteristic, complemented
6C2×#20259246C211yesmaximal, characteristic, complemented
7C2×#20259242C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C191C211yesmaximal, characteristic, complemented
9C22×D764C2211yescharacteristic
10C22×C22⋊D191—21no—
11C2×#15690274—24no—
12C2×#15690274C2214yescomplemented
13C22×C191⋊C8—24no—
14C22×C191⋊C8C2211yescharacteristic
15C22×C191⋊C8C2211yescharacteristic
16C22×C191⋊C8C2214yescomplemented
17#20259296—22no—
18#20259296C2214yes—
19C2×C4⋊Dic191C2211yescharacteristic
20C2×(C2×D191)⋊C4—21no—
21#20259253C2214yescomplemented
22#20259246C2214yescomplemented
23#20259242C2214yescomplemented
24C22×Q8×C191C2211yescharacteristic
25C23×C4×C191C2211yescharacteristic
26C22⋊Q8×C191—22no—
27C22⋊Q8×C191C2214yescomplemented
28C2×C4⋊C4×C191—21no—
29C2×C4⋊C4×C191C2211yescharacteristic
30C2×C22⋊C4×C191—21no—
31C23×D191—21no—
32C2×D764—22no—
33C2×D764—21no—
34C2×D764—22no—
35C2×D764C2312yes—
36C2×C22⋊D191—24no—
37C2×C22⋊D191—44no—
38C22×Dic191—21no—
39#15690274—28no—
40C2×C191⋊C8—28no—
41C2×C191⋊C8—22no—
42C2×C191⋊C8D414yescomplemented
43C2×C191⋊C8C2312yes—
44C2×C191⋊C8C2312yes—
45C4⋊Dic191—21no—
46C4⋊Dic191C2312yes—
47(C2×D191)⋊C4—24no—
48C24×C191D411yescharacteristic
49C2×Q8×C191—21no—
50C2×Q8×C191—22no—
51C2×Q8×C191—22no—
52C2×Q8×C191C2312yes—
53C22×C4×C191—22no—
54C22×C4×C191—22no—
55C22×C4×C191—21no—
56C22×C4×C191—21no—
57C22×C4×C191D411yescharacteristic
58C22×C4×C191D412yes—
59C22×C4×C191C2312yes—
60C22×C4×C191C2311yescharacteristic
61C4⋊C4×C191—24no—
62C4⋊C4×C191—21no—
63C4⋊C4×C191C2312yes—
64C22⋊C4×C191—24no—
65C22×D191—22no—
66C22×D191—42no—
67C22×D191—42no—
68C22×D191—42no—
69D764—22no—
70D764—22no—
71D764—42no—
72C22⋊D191—48no—
73C2×Dic191—22no—
74C2×Dic191—42no—
75C191⋊C8—22no—
76C191⋊C8C2×D414yescomplemented
77C23×C191—22no—
78C23×C191—22no—
79C23×C191SD1614yescomplemented
80C23×C191C2×D412yes—
81C23×C191C2×D411yescharacteristic
82Q8×C191—22no—
83Q8×C191—22no—
84Q8×C191—42no—
85C2×C4×C191—22no—
86C2×C4×C191—24no—
87C2×C4×C191—22no—
88C2×C4×C191—22no—
89C2×C4×C191—22no—
90C2×C4×C191—22no—
91C2×C4×C191—42no—
92C2×C4×C191—42no—
93C2×C4×C191C2×D412yes—
94C2×C4×C191C2×D412yes—
95C2×C4×C191C2×D411yescharacteristic
96C2×C4×C191#4112yes—
97C2×C4×C191C2411yescharacteristic
98C22×C191—22no—
99C22×C191—22no—
100C22×C191—28no—
101C22×C191#12112yes—
102C22×C191C2×SD1612yes—
103C22×C191C2×SD1614yescomplemented
104C22×C191#13512yes—
105C22×C191C22×D411yescharacteristic
106C2×D191—42no—
107C2×D191—42no—
108C2×D191—42no—
109C2×D191—84no—
110C4×C191—22no—
111C4×C191—42no—
112C4×C191—42no—
113C4×C191#12112yes—
114C4×C191C22×D411yescharacteristic
115C4×C191C2×#4111yescharacteristic
116Dic191—42no—
117C2×C191—24no—
118C2×C191#46514yescomplemented
119C2×C191C2×#12111yescharacteristic
120C2×C191C22×SD1611yescharacteristic
121C2×C191C2×#13511yescharacteristic
122D191—82no—
123C191C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1911nomaximal, Sylow
125C22×SD16—1911no—
126C23×C8—1911no—
127C2×C22⋊Q8D19111yescharacteristic, complemented
128C2×#121—1911no—
129#465—1918no—
130C2×#106—1911no—
131C2×#103—1911no—
132C2×#102—1911no—
133C22×Q8C2×D19111yescharacteristic
134C22×D4—3821no—
135C22×D4—1911no—
136C23×C4C2×D19111yescharacteristic
137C2×SD16—3824no—
138C2×SD16—1914no—
139C22×C8—3824no—
140C22×C8—1911no—
141C22×C8—1914no—
142C22×C8—1911no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D19114yescomplemented
145#121—3822no—
146#121—1914no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—1911no—
149C2×C4⋊C4C2×D19111yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—3821no—
152#106—1914no—
153#103—1914no—
154#102—1914no—
155C24—3821no—
156C24C22⋊D19111yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D19112yes—
161C2×D4—3824no—
162C2×D4—3822no—
163C2×D4—3821no—
164C2×D4—3822no—
165C2×D4—1912no—
166C2×D4—7644no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—3821no—
172C22×C4C22⋊D19112yes—
173C22×C4C22⋊D19111yescharacteristic
174C22×C4C22×D19112yes—
175C22×C4C22×D19111yescharacteristic
176SD16—3828no—
177C2×C8—3822no—
178C2×C8—3828no—
179C2×C8—1912no—
180C2×C8—1914no—
181C2×C8—1912no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—3821no—
185C4⋊C4—1912no—
186C4⋊C4C22×D19112yes—
187C22⋊C4—24no—
188C22⋊C4—3824no—
189C23—22no—
190C23—22no—
191C23—3822no—
192C23—7642no—
193C23—7642no—
194C23—7642no—
195C23#1569027414yescomplemented
196C23C2×C22⋊D19112yes—
197C23C2×C22⋊D19111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—3822no—
202D4—3822no—
203D4—7648no—
204D4—7642no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—3822no—
214C2×C4—7642no—
215C2×C4D4×D19112yes—
216C2×C4C2×C22⋊D19112yes—
217C2×C4C2×C22⋊D19111yescharacteristic
218C2×C4#1569028612yes—
219C2×C4C23×D19111yescharacteristic
220C8—3822no—
221C8—1914no—
222C22—22no—
223C22—28no—
224C22—22no—
225C22—15284no—
226C22—7642no—
227C22—7642no—
228C22—7642no—
229C22#2025929412yes—
230C22C2×#1569027412yes—
231C22C2×#1569027414yescomplemented
232C22#2025934212yes—
233C22C22×C22⋊D19111yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—7642no—
238C4#2025929412yes—
239C4C2×D4×D19111yescharacteristic
240C4C2×#1569028611yescharacteristic
241C2—24no—
242C2—15282no—
243C2#2713289414yesdirect product factor
244C2C2×#2025929411yescharacteristic
245C2C22×#1569027411yescharacteristic
246C2C2×#2025934211yescharacteristic
subgroupG/H
centerC23C2×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×C22⋊Q8×C191C2
socleC23×C191C2×D4
prime radicalC2×#20259296C2
hypercenterC2×C22⋊Q8D191
hypocenterC191C2×#465
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#15690274C2semidirect
2. C23×C191⋊C8C2semidirect
4. #27132894C2direct (N × H)
5. C2×#20259253C2semidirect
6. C2×#20259246C2semidirect
7. C2×#20259242C2semidirect
8. C2×C22⋊Q8×C191C2semidirect
12. C2×#15690274C22semidirect
16. C22×C191⋊C8C22semidirect
21. #20259253C22semidirect
22. #20259246C22semidirect
23. #20259242C22semidirect
27. C22⋊Q8×C191C22semidirect
42. C2×C191⋊C8D4semidirect
76. C191⋊C8C2×D4semidirect
79. C23×C191SD16semidirect
103. C22×C191C2×SD16semidirect
118. C2×C191#465semidirect
123. C191C2×#465semidirect
127. C2×C22⋊Q8D191semidirect
144. C22⋊Q8C2×D191semidirect
195. C23#15690274semidirect
231. C22C2×#15690274semidirect
243. C2#27132894direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156902742
C23×C191⋊C82
C2×#202592962
#271328942
C2×#202592532
C2×#202592462
C2×#202592422
C2×C22⋊Q8×C1912
C2×#465191