Finite Group Databasebeta

← C2×#27132912 #37582111

Subgroups — C2×#27132912 #37582111

Order 24448.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#15690270C211yesmaximal, characteristic, complemented
3C2×#20259298C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic
5#27132912C218yesmaximal, direct product factor
6C2×#20259255C211yesmaximal, characteristic, complemented
7C2×#20259244C211yesmaximal, characteristic, complemented
8C2×#103×C191C211yesmaximal, characteristic, complemented
9C22×D764—21no—
10C22×D764C2211yescharacteristic
11C22×C4×D191C2211yescharacteristic
12C2×#15690274—24no—
13C2×#15690274C2214yescomplemented
14C2×#15690270—24no—
15C2×#15690270C2214yescomplemented
16C22×C191⋊C8C2211yescharacteristic
17#20259298—22no—
18#20259298C2214yescomplemented
19#20259297—22no—
20#20259297C2214yes—
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4—21no—
23C2×(C2×D191)⋊C4—21no—
24C2×(C2×D191)⋊C4—21no—
25#20259255C2214yescomplemented
26#20259244C2214yescomplemented
27C22×Q8×C191C2211yescharacteristic
28C22×C8×C191C2211yescharacteristic
29C2×C4⋊C4×C191C2211yescharacteristic
30#103×C191C2214yescomplemented
31C23×D191—21no—
32C23×D191D411yescharacteristic
33C2×D764—24no—
34C2×D764—22no—
35C2×D764—21no—
36C2×D764—22no—
37C2×D764—44no—
38C2×D764C2312yes—
39C22×Dic191—21no—
40C22×Dic191D411yescharacteristic
41C2×C4×D191—22no—
42C2×C4×D191—22no—
43C2×C4×D191D412yes—
44C2×C4×D191C2312yes—
45#15690274—28no—
46#15690270—28no—
47C2×C191⋊C8D414yescomplemented
48C2×C191⋊C8C2312yes—
49C4⋊Dic191—21no—
50C4⋊Dic191C2312yes—
51Dic191⋊C4—24no—
52(C2×D191)⋊C4—24no—
53(C2×D191)⋊C4—24no—
54C2×Q8×C191—21no—
55C2×Q8×C191—22no—
56C2×Q8×C191—22no—
57C2×Q8×C191C2312yes—
58C22×C4×C191—21no—
59C22×C4×C191—21no—
60C22×C4×C191C2311yescharacteristic
61C2×C8×C191—22no—
62C2×C8×C191C2312yes—
63C4⋊C4×C191—21no—
64C4⋊C4×C191C2312yes—
65C22×D191—22no—
66C22×D191—22no—
67C22×D191—22no—
68C22×D191—22no—
69C22×D191—42no—
70C22×D191—42no—
71C22×D191—42no—
72C22×D191C2×D412yes—
73D764—22no—
74D764—22no—
75D764—48no—
76D764—42no—
77C2×Dic191—22no—
78C2×Dic191—22no—
79C2×Dic191—42no—
80C2×Dic191C2×D412yes—
81C4×D191—24no—
82C4×D191—22no—
83C4×D191—22no—
84C191⋊C8C2×D414yescomplemented
85C23×C191C2×D411yescharacteristic
86Q8×C191—22no—
87Q8×C191—22no—
88Q8×C191—42no—
89C2×C4×C191—22no—
90C2×C4×C191—22no—
91C2×C4×C191—42no—
92C2×C4×C191—42no—
93C2×C4×C191C2×D411yescharacteristic
94C2×C4×C191C2×D412yes—
95C2×C4×C191#4112yes—
96C2×C4×C191C2411yescharacteristic
97C8×C191—22no—
98C22×C191#12112yes—
99C22×C191AGL(1,ℤ/8)12yes—
100C22×C191#13712yes—
101C22×C191C22×D411yescharacteristic
102C2×D191—22no—
103C2×D191—22no—
104C2×D191—22no—
105C2×D191—44no—
106C2×D191—42no—
107C2×D191—42no—
108C2×D191—42no—
109C2×D191—84no—
110C4×C191—42no—
111C4×C191—42no—
112C4×C191#12112yes—
113C4×C191C22×D411yescharacteristic
114C4×C191C2×#4111yescharacteristic
115Dic191—22no—
116Dic191—42no—
117C2×C191#46814yescomplemented
118C2×C191C2×#12111yescharacteristic
119C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C191C2×#13711yescharacteristic
121D191—42no—
122D191—82no—
123C191C2×#46811yescharacteristic, complemented, Sylow
124C2×#468—1911nomaximal, Sylow
125C22×SD16—1911no—
126C22×M16—1911no—
127C2×C22⋊Q8—1911no—
128C2×#121—1911no—
129#468—1918no—
130C2×#107—1911no—
131C2×#103D19111yescharacteristic, complemented
132C2×#102—1911no—
133C22×Q8C2×D19111yescharacteristic
134C22×D4—3821no—
135C22×D4—1911no—
136C23×C4—1911no—
137C2×SD16—3824no—
138C2×SD16—1914no—
139C2×M16—3824no—
140C2×M16—1914no—
141C22×C8—1911no—
142C22×C8C2×D19111yescharacteristic
143C22⋊Q8—3822no—
144C22⋊Q8—1914no—
145#121—3822no—
146#121—1914no—
147C2×C4⋊C4—3821no—
148C2×C4⋊C4—1911no—
149C2×C4⋊C4C2×D19111yescharacteristic
150C2×C22⋊C4—3821no—
151C2×C22⋊C4—3821no—
152#107—1914no—
153#103C2×D19114yescomplemented
154#102—1914no—
155C24—3821no—
156C24—1911no—
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—21no—
168C22×C4—21no—
169C22×C4—3822no—
170C22×C4—3822no—
171C22×C4—3821no—
172C22×C4—1911no—
173C22×C4—1912no—
174C22×C4—1912no—
175C22×C4C22×D19111yescharacteristic
176SD16—3828no—
177M16—3828no—
178C2×C8—22no—
179C2×C8—1914no—
180C2×C8—1912no—
181C2×C8C22×D19112yes—
182C4⋊C4—21no—
183C4⋊C4—3824no—
184C4⋊C4—3821no—
185C4⋊C4—1912no—
186C4⋊C4C22×D19112yes—
187C22⋊C4—3824no—
188C22⋊C4—3824no—
189C23—3822no—
190C23—3822no—
191C23—3822no—
192C23—3822no—
193C23—1912no—
194C23—7642no—
195C23—7642no—
196C23—7642no—
197C23D4×D19111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—3822no—
202D4—3822no—
203D4—7648no—
204D4—7642no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—3822no—
210C2×C4—3822no—
211C2×C4—3822no—
212C2×C4—3824no—
213C2×C4—3822no—
214C2×C4—1912no—
215C2×C4—7642no—
216C2×C4D4×D19112yes—
217C2×C4D4×D19111yescharacteristic
218C2×C4#1569028712yes—
219C2×C4C23×D19111yescharacteristic
220C8—22no—
221C8—1914no—
222C22—3822no—
223C22—3822no—
224C22—3822no—
225C22—15284no—
226C22—7642no—
227C22—7642no—
228C22—7642no—
229C22—7644no—
230C22#2025929512yes—
231C22#2025934912yes—
232C22#2025935412yes—
233C22C2×D4×D19111yescharacteristic
234C4—42no—
235C4—42no—
236C4—3822no—
237C4—7642no—
238C4#2025929512yes—
239C4C2×D4×D19111yescharacteristic
240C4C2×#1569028711yescharacteristic
241C2—15282no—
242C2—7642no—
243C2#2713291214yesdirect product factor
244C2C2×#2025929511yescharacteristic
245C2C2×#2025934911yescharacteristic
246C2C2×#2025935411yescharacteristic
subgroupG/H
centerC23D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#103×C191C2
socleC23×C191C2×D4
prime radicalC2×#20259297C2
hypercenterC2×#103D191
hypocenterC191C2×#468
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#15690274C2semidirect
2. C22×#15690270C2semidirect
3. C2×#20259298C2semidirect
5. #27132912C2direct (N × H)
6. C2×#20259255C2semidirect
7. C2×#20259244C2semidirect
8. C2×#103×C191C2semidirect
13. C2×#15690274C22semidirect
15. C2×#15690270C22semidirect
18. #20259298C22semidirect
25. #20259255C22semidirect
26. #20259244C22semidirect
30. #103×C191C22semidirect
47. C2×C191⋊C8D4semidirect
84. C191⋊C8C2×D4semidirect
117. C2×C191#468semidirect
123. C191C2×#468semidirect
131. C2×#103D191semidirect
153. #103C2×D191semidirect
243. C2#27132912direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156902742
C22×#156902702
C2×#202592982
C2×#202592972
#271329122
C2×#202592552
C2×#202592442
C2×#103×C1912
C2×#468191