Finite Group Databasebeta

← C2×#27132840 #37581849

Subgroups — C2×#27132840 #37581849

Order 24448.

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

normal subgroups only

This group has 760 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
1C2×#20259367C211yesmaximal, characteristic
2C2×#20259353C212yesmaximal, complemented
3#27132840C218yesmaximal, direct product factor
4C2×#20259250C212yesmaximal, complemented
5C2×#20259239C211yesmaximal, characteristic, complemented
6C2×#124×C191C211yesmaximal, characteristic, complemented
7#20259367—24no—
8#20259367—22no—
9#20259367C2214yes—
10C2×#15690287C2212yes—
11C2×#15690286—22no—
12C22×C22⋊D191—22no—
13C2×D4×D191—22no—
14#20259353—24no—
15#20259353C2218yescomplemented
16C2×#15690277—22no—
17C2×#15690274—22no—
18C2×#15690271C2212yescomplemented
19#20259250C2218yescomplemented
20#20259239C2214yescomplemented
21C22×Q8×C191C2211yescharacteristic, complemented
22C22×D4×C191C2211yescharacteristic
23#124×C191—22no—
24#124×C191C2214yescomplemented
25C2×C22⋊C4×C191—22no—
26C2×C42×C191C2211yescharacteristic, complemented
27C23×D191—22no—
28#15690287—22no—
29#15690287—22no—
30#15690287D414yes—
31#15690287C2314yes—
32#15690286—28no—
33#15690286—28no—
34C2×Dic382D412yes—
35C2×D764D412yes—
36C2×C22⋊D191—24no—
37C2×C22⋊D191—28no—
38C2×C22⋊D191—22no—
39C2×C22⋊D191—28no—
40D4×D191—28no—
41D4×D191—28no—
42C22×Dic191—22no—
43C2×C4×D191—22no—
44#15690277—28no—
45#15690274—28no—
46#15690271—22no—
47#15690271C2314yescomplemented
48C2×C191⋊C8—22no—
49C24×C191D412yes—
50C2×Q8×C191—21no—
51C2×Q8×C191—22no—
52C2×Q8×C191—22no—
53C2×Q8×C191C2312yescomplemented
54C2×D4×C191—22no—
55C2×D4×C191—22no—
56C2×D4×C191D412yes—
57C2×D4×C191C2312yes—
58C22×C4×C191—21no—
59C22×C4×C191—21no—
60C22×C4×C191C2311yescharacteristic
61C22⋊C4×C191—28no—
62C42×C191—21no—
63C42×C191C2312yescomplemented
64C22×D191—22no—
65C22×D191—28no—
66C22×D191—84no—
67Dic382—22no—
68Dic382C2×D414yes—
69D764—22no—
70D764C2×D414yes—
71C22⋊D191—216no—
72C22⋊D191—24no—
73C22⋊D191—24no—
74C2×Dic191—28no—
75C2×Dic191—22no—
76C4×D191—24no—
77C4×D191—24no—
78C191⋊C8—24no—
79C23×C191—22no—
80C23×C191—22no—
81C23×C191—44no—
82C23×C191C2×D411yescharacteristic
83C23×C191C2×D414yes—
84Q8×C191—24no—
85Q8×C191—42no—
86D4×C191—24no—
87D4×C191—24no—
88C2×C4×C191—22no—
89C2×C4×C191—22no—
90C2×C4×C191—42no—
91C2×C4×C191—41no—
92C2×C4×C191—41no—
93C2×C4×C191C2×D412yes—
94C2×C4×C191C2×D412yes—
95C2×C4×C191C2×D411yescharacteristic
96C2×C4×C191C2411yescharacteristic
97C22×C191—22no—
98C22×C191—24no—
99C22×C191—44no—
100C22×C191—44no—
101C22×C191—82no—
102C22×C191#12011yescharacteristic
103C22×C191#12011yescharacteristic
104C22×C191C22×D411yescharacteristic
105C2×D191—24no—
106C2×D191—44no—
107C2×D191—88no—
108C4×C191—42no—
109C4×C191—42no—
110C4×C191#12012yes—
111C4×C191C22×D412yes—
112Dic191—24no—
113C2×C191—21no—
114C2×C191—21no—
115C2×C191—44no—
116C2×C191#45512yescomplemented
117C2×C191C2×#12011yescharacteristic
118D191—44no—
119C191C2×#45511yescharacteristic, complemented, Sylow
120C2×#455—1911nomaximal, Sylow
121C2×H(5,𝔽2)—1911no—
122C2×#137—1912no—
123C2×#124D19111yescharacteristic, complemented
124#455—1918no—
125C2×#104—1912no—
126C2×#100—1911no—
127H(5,𝔽2)—3824no—
128H(5,𝔽2)—3822no—
129H(5,𝔽2)—1914no—
130C2×#41—3822no—
131C2×#41—1912no—
132C22×Q8C2×D19111yescharacteristic, complemented
133C22×D4—3822no—
134C22×D4—3822no—
135C22×D4C2×D19111yescharacteristic
136#137—3824no—
137#137—1918no—
138C2×Q16—3822no—
139C2×SD16—3822no—
140C2×M16—1912no—
141#124—22no—
142#124C2×D19114yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D19111yescharacteristic, complemented
145#104—1918no—
146#100—1914no—
147C24—3822no—
148C24C22⋊D19112yes—
149#41—3828no—
150#41—3828no—
151#41—3822no—
152#41—3822no—
153#41—1914no—
154#41—1914no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—1912no—
159C2×Q8C22×D19112yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—3828no—
163C2×D4—3828no—
164C2×D4—3824no—
165C2×D4—3828no—
166C2×D4—3822no—
167C2×D4—3828no—
168C2×D4—1912no—
169C2×D4C22⋊D19112yes—
170C2×D4C22×D19112yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—3822no—
174C22×C4—3822no—
175C22×C4C22×D19111yescharacteristic
176Q16—3828no—
177SD16—3828no—
178M16—3822no—
179M16—1914no—
180C2×C8—3822no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D19112yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—3828no—
188C23—3822no—
189C23—15284no—
190C23C2×C22⋊D19111yescharacteristic
191C23C2×C22⋊D19114yes—
192Q8—24no—
193Q8—42no—
194Q8—3822no—
195Q8—1914no—
196D4—24no—
197D4—24no—
198D4—38216no—
199D4—3824no—
200D4—3824no—
201D4—3822no—
202D4—1914no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3828no—
209C2×C4—3822no—
210C2×C4—3824no—
211C2×C4—3824no—
212C2×C4D4×D19112yes—
213C2×C4D4×D19112yes—
214C2×C4C2×C22⋊D19111yescharacteristic
215C2×C4C23×D19111yescharacteristic
216C8—3824no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—3824no—
223C22—15288no—
224C22—7644no—
225C22#2025929011yescharacteristic
226C22#2025929011yescharacteristic
227C22C22×C22⋊D19111yescharacteristic
228C4—42no—
229C4—42no—
230C4—3824no—
231C4#2025929012yes—
232C4C2×D4×D19112yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—7644no—
237C2#2713284012yesdirect product factor
238C2C2×#2025929011yescharacteristic
subgroupG/H
centerC22#20259290
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#124×C191C2
socleC22×C191#120
prime radicalC2×#20259367C2
hypercenterC2×#124D191
hypocenterC191C2×#455
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#20259353C2semidirect
3. #27132840C2direct (N × H)
4. C2×#20259250C2semidirect
5. C2×#20259239C2semidirect
6. C2×#124×C191C2semidirect
15. #20259353C22semidirect
18. C2×#15690271C22semidirect
19. #20259250C22semidirect
20. #20259239C22semidirect
21. C22×Q8×C191C22semidirect
24. #124×C191C22semidirect
26. C2×C42×C191C22semidirect
47. #15690271C23semidirect
53. C2×Q8×C191C23semidirect
63. C42×C191C23semidirect
116. C2×C191#455semidirect
119. C191C2×#455semidirect
123. C2×#124D191semidirect
132. C22×Q8C2×D191semidirect
142. #124C2×D191semidirect
144. C2×C42C2×D191semidirect
159. C2×Q8C22×D191semidirect
183. C42C22×D191semidirect
237. C2#27132840direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202593672
C2×#202593532
#271328402
C2×#202592502
C2×#202592392
C2×#124×C1912
C2×#455191