Finite Group Databasebeta

← C2×#26041439 #37542928

Subgroups — C2×#26041439 #37542928

Order 23168.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15668449C211yesmaximal, characteristic
2C22×#15668439C211yesmaximal, characteristic, complemented
3C22×#15668436C211yesmaximal, characteristic, complemented
4#26041439C218yesmaximal, direct product factor
5C2×#19153739C211yesmaximal, characteristic, complemented
6C2×#19153736C211yesmaximal, characteristic, complemented
7C2×C22⋊#13089678C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C181C211yesmaximal, characteristic, complemented
9C2×#15668449—24no—
10C2×#15668449—24no—
11C2×#15668449—22no—
12C2×#15668449C2214yes—
13C22×Dic362C2211yescharacteristic
14C22×D724C2211yescharacteristic
15C22×C22⋊D181—21no—
16C22×C4×D181—21no—
17C2×#15668439—24no—
18C2×#15668439C2214yescomplemented
19C2×#15668436—24no—
20C2×#15668436C2214yescomplemented
21C22×#13089678C2211yescharacteristic
22C22×#13089678C2211yescharacteristic
23#19153739C2214yescomplemented
24#19153736C2214yescomplemented
25C22⋊#13089678C2214yescomplemented
26C22×Q8×C181C2211yescharacteristic
27C23×C4×C181C2211yescharacteristic
28C22⋊Q8×C181—22no—
29C22⋊Q8×C181C2214yescomplemented
30C2×C4⋊C4×C181—21no—
31C2×C4⋊C4×C181C2211yescharacteristic
32C2×C22⋊C4×C181—21no—
33C23×D181—21no—
34#15668449—216no—
35#15668449—28no—
36#15668449—28no—
37C2×Dic362—22no—
38C2×Dic362—21no—
39C2×Dic362D414yes—
40C2×Dic362C2312yes—
41C2×D724—22no—
42C2×D724—21no—
43C2×D724D414yes—
44C2×D724C2312yes—
45C2×C22⋊D181—28no—
46C2×C22⋊D181—24no—
47C22×Dic181—21no—
48C2×C4×D181—28no—
49C2×C4×D181—24no—
50#15668439—28no—
51#15668436—28no—
52C2×#13089678—22no—
53C2×#13089678—22no—
54C2×#13089678C2312yes—
55C2×#13089678C2312yes—
56C24×C181D411yescharacteristic
57C2×Q8×C181—21no—
58C2×Q8×C181—22no—
59C2×Q8×C181—22no—
60C2×Q8×C181C2312yes—
61C22×C4×C181—22no—
62C22×C4×C181—22no—
63C22×C4×C181—21no—
64C22×C4×C181—21no—
65C22×C4×C181D411yescharacteristic
66C22×C4×C181D412yes—
67C22×C4×C181C2312yes—
68C22×C4×C181C2311yescharacteristic
69C4⋊C4×C181—24no—
70C4⋊C4×C181—21no—
71C4⋊C4×C181C2312yes—
72C22⋊C4×C181—24no—
73C22×D181—24no—
74C22×D181—22no—
75C22×D181—42no—
76C22×D181—42no—
77Dic362—24no—
78Dic362—22no—
79Dic362C2×D414yes—
80D724—24no—
81D724—22no—
82D724C2×D414yes—
83C22⋊D181—216no—
84C2×Dic181—24no—
85C2×Dic181—22no—
86C4×D181—216no—
87#13089678—22no—
88#13089678—22no—
89C23×C181—22no—
90C23×C181—22no—
91C23×C181—22no—
92C23×C181C2×D412yes—
93C23×C181C2×D411yescharacteristic
94Q8×C181—22no—
95Q8×C181—22no—
96Q8×C181—42no—
97C2×C4×C181—22no—
98C2×C4×C181—24no—
99C2×C4×C181—22no—
100C2×C4×C181—22no—
101C2×C4×C181—22no—
102C2×C4×C181—22no—
103C2×C4×C181—42no—
104C2×C4×C181—42no—
105C2×C4×C181C2×D412yes—
106C2×C4×C181C2×D412yes—
107C2×C4×C181C2×D412yes—
108C2×C4×C181C2×D411yescharacteristic
109C2×C4×C181C2411yescharacteristic
110C22×C181—22no—
111C22×C181—22no—
112C22×C181—22no—
113C22×C181—44no—
114C22×C181#12012yes—
115C22×C181#13512yes—
116C22×C181#13712yes—
117C22×C181C22×D411yescharacteristic
118C2×D181—24no—
119C2×D181—42no—
120C2×D181—42no—
121C2×D181—48no—
122C4×C181—22no—
123C4×C181—42no—
124C4×C181—42no—
125C4×C181#12012yes—
126C4×C181C22×D411yescharacteristic
127C4×C181C22×D411yescharacteristic
128Dic181—24no—
129C2×C181—42no—
130C2×C181#45214yescomplemented
131C2×C181C2×#12011yescharacteristic
132C2×C181C2×#13511yescharacteristic
133C2×C181C2×#13711yescharacteristic
134D181—44no—
135C181C2×#45211yescharacteristic, complemented, Sylow
136C2×#452—1811nomaximal, Sylow
137C22×#41—1811no—
138C22×Q16—1811no—
139C22×SD16—1811no—
140C2×C22⋊Q8D18111yescharacteristic, complemented
141#452—1818no—
142C2×#103—1811no—
143C2×#102—1811no—
144C2×C22⋊C8—1811no—
145C2×#41—3624no—
146C2×#41—3624no—
147C2×#41—3622no—
148C2×#41—1814no—
149C22×Q8—1811no—
150C22×Q8C2×D18111yescharacteristic
151C22×D4—3621no—
152C22×D4—1811no—
153C23×C4—3621no—
154C23×C4C2×D18111yescharacteristic
155C2×Q16—3624no—
156C2×Q16—1814no—
157C2×SD16—3624no—
158C2×SD16—1814no—
159C22×C8—1811no—
160C22×C8—1811no—
161C22⋊Q8—22no—
162C22⋊Q8C2×D18114yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×D18111yescharacteristic
165C2×C22⋊C4—21no—
166#103—1814no—
167#102—1814no—
168C22⋊C8—1814no—
169C24—3621no—
170C24C22⋊D18111yescharacteristic
171#41—36216no—
172#41—3628no—
173#41—3628no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—3622no—
178C2×Q8—3621no—
179C2×Q8—1814no—
180C2×Q8—1812no—
181C2×Q8C22×D18112yes—
182C2×D4—3628no—
183C2×D4—3624no—
184C2×D4—3622no—
185C2×D4—3621no—
186C2×D4—1814no—
187C2×D4—1812no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—3628no—
193C22×C4—3621no—
194C22×C4—3624no—
195C22×C4C22⋊D18112yes—
196C22×C4C22⋊D18111yescharacteristic
197C22×C4C22×D18112yes—
198C22×C4C22×D18111yescharacteristic
199Q16—3628no—
200SD16—3628no—
201C2×C8—3622no—
202C2×C8—3622no—
203C2×C8—1812no—
204C2×C8—1812no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D18112yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—3624no—
213C23—3622no—
214C23—7242no—
215C23—7242no—
216C23C2×C22⋊D18112yes—
217C23C2×C22⋊D18111yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—3624no—
222Q8—3622no—
223Q8—1814no—
224D4—36216no—
225D4—3624no—
226D4—3622no—
227D4—1814no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—36216no—
237C2×C4—3624no—
238C2×C4—3622no—
239C2×C4D4×D18112yes—
240C2×C4D4×D18112yes—
241C2×C4C2×C22⋊D18112yes—
242C2×C4C2×C22⋊D18111yescharacteristic
243C2×C4C23×D18111yescharacteristic
244C8—3622no—
245C8—3622no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—3624no—
251C22—7248no—
252C22—7242no—
253C22—7242no—
254C22#1915378412yes—
255C22#1915383612yes—
256C22#1915384712yes—
257C22C22×C22⋊D18111yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—3624no—
262C4#1915378412yes—
263C4C2×D4×D18111yescharacteristic
264C4C2×D4×D18111yescharacteristic
265C2—42no—
266C2—7244no—
267C2#2604143914yesdirect product factor
268C2C2×#1915378411yescharacteristic
269C2C2×#1915383611yescharacteristic
270C2C2×#1915384711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×C22⋊Q8×C181C2
socleC23×C181C2×D4
prime radicalC22×#15668449C2
hypercenterC2×C22⋊Q8D181
hypocenterC181C2×#452
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#15668439C2semidirect
3. C22×#15668436C2semidirect
4. #26041439C2direct (N × H)
5. C2×#19153739C2semidirect
6. C2×#19153736C2semidirect
7. C2×C22⋊#13089678C2semidirect
8. C2×C22⋊Q8×C181C2semidirect
18. C2×#15668439C22semidirect
20. C2×#15668436C22semidirect
23. #19153739C22semidirect
24. #19153736C22semidirect
25. C22⋊#13089678C22semidirect
29. C22⋊Q8×C181C22semidirect
130. C2×C181#452semidirect
135. C181C2×#452semidirect
140. C2×C22⋊Q8D181semidirect
162. C22⋊Q8C2×D181semidirect
267. C2#26041439direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156684492
C22×#156684392
C22×#156684362
#260414392
C2×#191537392
C2×#191537362
C2×C22⋊#130896782
C2×C22⋊Q8×C1812
C2×#452181