Finite Group Databasebeta

← C2×#26011651 #36362770

Subgroups — C2×#26011651 #36362770

Order 22144.

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×#14583826C211yesmaximal, characteristic
2C22×#14583816C211yesmaximal, characteristic, complemented
3C22×#14583813C211yesmaximal, characteristic, complemented
4#26011651C218yesmaximal, direct product factor
5C2×#19149602C211yesmaximal, characteristic, complemented
6C2×#19149599C211yesmaximal, characteristic, complemented
7C2×C22⋊#13063436C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C173C211yesmaximal, characteristic, complemented
9C2×#14583826—22no—
10C2×#14583826—24no—
11C2×#14583826—24no—
12C2×#14583826C2214yes—
13C22×Dic346C2211yescharacteristic
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C22×C4×D173—21no—
17C2×#14583816—24no—
18C2×#14583816C2214yescomplemented
19C2×#14583813—24no—
20C2×#14583813C2214yescomplemented
21C22×#13063436C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149602C2214yescomplemented
24#19149599C2214yescomplemented
25C22⋊#13063436C2214yescomplemented
26C22×Q8×C173C2211yescharacteristic
27C23×C4×C173C2211yescharacteristic
28C22⋊Q8×C173—22no—
29C22⋊Q8×C173C2214yescomplemented
30C2×C4⋊C4×C173—21no—
31C2×C4⋊C4×C173C2211yescharacteristic
32C2×C22⋊C4×C173—21no—
33C23×D173—21no—
34#14583826—216no—
35#14583826—28no—
36#14583826—28no—
37C2×Dic346—22no—
38C2×Dic346—21no—
39C2×Dic346D414yes—
40C2×Dic346C2312yes—
41C2×D692—22no—
42C2×D692—21no—
43C2×D692D414yes—
44C2×D692C2312yes—
45C2×C22⋊D173—28no—
46C2×C22⋊D173—24no—
47C22×Dic173—21no—
48C2×C4×D173—28no—
49C2×C4×D173—24no—
50#14583816—28no—
51#14583813—28no—
52C2×#13063436—22no—
53C2×#13063436—22no—
54C2×#13063436C2312yes—
55C2×#13063436C2312yes—
56C24×C173D411yescharacteristic
57C2×Q8×C173—21no—
58C2×Q8×C173—22no—
59C2×Q8×C173—22no—
60C2×Q8×C173C2312yes—
61C22×C4×C173—22no—
62C22×C4×C173—22no—
63C22×C4×C173—21no—
64C22×C4×C173—21no—
65C22×C4×C173D411yescharacteristic
66C22×C4×C173D412yes—
67C22×C4×C173C2312yes—
68C22×C4×C173C2311yescharacteristic
69C4⋊C4×C173—24no—
70C4⋊C4×C173—21no—
71C4⋊C4×C173C2312yes—
72C22⋊C4×C173—24no—
73C22×D173—24no—
74C22×D173—22no—
75C22×D173—42no—
76C22×D173—42no—
77Dic346—24no—
78Dic346—22no—
79Dic346C2×D414yes—
80D692—24no—
81D692—22no—
82D692C2×D414yes—
83C22⋊D173—216no—
84C2×Dic173—24no—
85C2×Dic173—22no—
86C4×D173—216no—
87#13063436—22no—
88#13063436—22no—
89C23×C173—22no—
90C23×C173—22no—
91C23×C173—22no—
92C23×C173C2×D412yes—
93C23×C173C2×D411yescharacteristic
94Q8×C173—22no—
95Q8×C173—22no—
96Q8×C173—42no—
97C2×C4×C173—22no—
98C2×C4×C173—24no—
99C2×C4×C173—22no—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—42no—
104C2×C4×C173—42no—
105C2×C4×C173C2×D412yes—
106C2×C4×C173C2×D412yes—
107C2×C4×C173C2×D412yes—
108C2×C4×C173C2×D411yescharacteristic
109C2×C4×C173C2411yescharacteristic
110C22×C173—22no—
111C22×C173—22no—
112C22×C173—22no—
113C22×C173—44no—
114C22×C173#12012yes—
115C22×C173#13512yes—
116C22×C173#13712yes—
117C22×C173C22×D411yescharacteristic
118C2×D173—24no—
119C2×D173—42no—
120C2×D173—42no—
121C2×D173—48no—
122C4×C173—22no—
123C4×C173—42no—
124C4×C173—42no—
125C4×C173#12012yes—
126C4×C173C22×D411yescharacteristic
127C4×C173C22×D411yescharacteristic
128Dic173—24no—
129C2×C173—42no—
130C2×C173#45214yescomplemented
131C2×C173C2×#12011yescharacteristic
132C2×C173C2×#13511yescharacteristic
133C2×C173C2×#13711yescharacteristic
134D173—44no—
135C173C2×#45211yescharacteristic, complemented, Sylow
136C2×#452—1731nomaximal, Sylow
137C22×#41—1731no—
138C22×Q16—1731no—
139C22×SD16—1731no—
140C2×C22⋊Q8D17311yescharacteristic, complemented
141#452—1738no—
142C2×#103—1731no—
143C2×#102—1731no—
144C2×C22⋊C8—1731no—
145C2×#41—3464no—
146C2×#41—3464no—
147C2×#41—3462no—
148C2×#41—1734no—
149C22×Q8—1731no—
150C22×Q8C2×D17311yescharacteristic
151C22×D4—3461no—
152C22×D4—1731no—
153C23×C4—3461no—
154C23×C4C2×D17311yescharacteristic
155C2×Q16—3464no—
156C2×Q16—1734no—
157C2×SD16—3464no—
158C2×SD16—1734no—
159C22×C8—1731no—
160C22×C8—1731no—
161C22⋊Q8—22no—
162C22⋊Q8C2×D17314yescomplemented
163C2×C4⋊C4—21no—
164C2×C4⋊C4C2×D17311yescharacteristic
165C2×C22⋊C4—21no—
166#103—1734no—
167#102—1734no—
168C22⋊C8—1734no—
169C24—3461no—
170C24C22⋊D17311yescharacteristic
171#41—34616no—
172#41—3468no—
173#41—3468no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—3462no—
178C2×Q8—3461no—
179C2×Q8—1734no—
180C2×Q8—1732no—
181C2×Q8C22×D17312yes—
182C2×D4—3468no—
183C2×D4—3464no—
184C2×D4—3462no—
185C2×D4—3461no—
186C2×D4—1734no—
187C2×D4—1732no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—3468no—
193C22×C4—3461no—
194C22×C4—3464no—
195C22×C4C22⋊D17312yes—
196C22×C4C22⋊D17311yescharacteristic
197C22×C4C22×D17312yes—
198C22×C4C22×D17311yescharacteristic
199Q16—3468no—
200SD16—3468no—
201C2×C8—3462no—
202C2×C8—3462no—
203C2×C8—1732no—
204C2×C8—1732no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4C22×D17312yes—
208C22⋊C4—24no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—3464no—
213C23—3462no—
214C23—6922no—
215C23—6922no—
216C23C2×C22⋊D17312yes—
217C23C2×C22⋊D17311yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—42no—
221Q8—3464no—
222Q8—3462no—
223Q8—1734no—
224D4—34616no—
225D4—3464no—
226D4—3462no—
227D4—1734no—
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—34616no—
237C2×C4—3464no—
238C2×C4—3462no—
239C2×C4D4×D17312yes—
240C2×C4D4×D17312yes—
241C2×C4C2×C22⋊D17312yes—
242C2×C4C2×C22⋊D17311yescharacteristic
243C2×C4C23×D17311yescharacteristic
244C8—3462no—
245C8—3462no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—44no—
250C22—3464no—
251C22—6928no—
252C22—6922no—
253C22—6922no—
254C22#1914964712yes—
255C22#1914969912yes—
256C22#1914971012yes—
257C22C22×C22⋊D17311yescharacteristic
258C4—22no—
259C4—42no—
260C4—42no—
261C4—3464no—
262C4#1914964712yes—
263C4C2×D4×D17311yescharacteristic
264C4C2×D4×D17311yescharacteristic
265C2—42no—
266C2—6924no—
267C2#2601165114yesdirect product factor
268C2C2×#1914964711yescharacteristic
269C2C2×#1914969911yescharacteristic
270C2C2×#1914971011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×C22⋊Q8×C173C2
socleC23×C173C2×D4
prime radicalC22×#14583826C2
hypercenterC2×C22⋊Q8D173
hypocenterC173C2×#452
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×#14583816C2semidirect
3. C22×#14583813C2semidirect
4. #26011651C2direct (N × H)
5. C2×#19149602C2semidirect
6. C2×#19149599C2semidirect
7. C2×C22⋊#13063436C2semidirect
8. C2×C22⋊Q8×C173C2semidirect
18. C2×#14583816C22semidirect
20. C2×#14583813C22semidirect
23. #19149602C22semidirect
24. #19149599C22semidirect
25. C22⋊#13063436C22semidirect
29. C22⋊Q8×C173C22semidirect
130. C2×C173#452semidirect
135. C173C2×#452semidirect
140. C2×C22⋊Q8D173semidirect
162. C22⋊Q8C2×D173semidirect
267. C2#26011651direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145838262
C22×#145838162
C22×#145838132
#260116512
C2×#191496022
C2×#191495992
C2×C22⋊#130634362
C2×C22⋊Q8×C1732
C2×#452173