Finite Group Databasebeta

← C2×#26011634 #36362762

Subgroups — C2×#26011634 #36362762

Order 22144.

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

normal subgroups only

This group has 812 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, complemented
2C22×#14583814C211yesmaximal, characteristic, complemented
3C22×#14583811C211yesmaximal, characteristic, complemented
4#26011634C218yesmaximal, direct product factor
5C2×#19149602C211yesmaximal, characteristic, complemented
6C2×#19149599C211yesmaximal, characteristic, complemented
7C2×C22⋊#13063436C211yesmaximal, characteristic, complemented
8C2×#121×C173C211yesmaximal, characteristic, complemented
9C2×#14583826—22no—
10C2×#14583826—24no—
11C2×#14583826—24no—
12C2×#14583826C2214yescomplemented
13C22×Dic346C2211yescharacteristic
14C22×D692C2211yescharacteristic
15C22×C22⋊D173—21no—
16C22×C4×D173—21no—
17C2×#14583814—24no—
18C2×#14583814C2214yescomplemented
19C2×#14583811—24no—
20C2×#14583811C2214yescomplemented
21C22×#13063436C2211yescharacteristic
22C22×#13063436C2211yescharacteristic
23#19149602C2214yescomplemented
24#19149599C2214yescomplemented
25C22⋊#13063436C2214yescomplemented
26C22×D4×C173—21no—
27C22×D4×C173C2211yescharacteristic
28C23×C4×C173C2211yescharacteristic
29#121×C173—22no—
30#121×C173C2214yescomplemented
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×Dic346D414yescomplemented
40C2×Dic346C2312yes—
41C2×D692—22no—
42C2×D692—21no—
43C2×D692D414yescomplemented
44C2×D692C2312yes—
45C2×C22⋊D173—28no—
46C2×C22⋊D173—24no—
47C22×Dic173—21no—
48C2×C4×D173—28no—
49C2×C4×D173—24no—
50#14583814—28no—
51#14583811—28no—
52C2×#13063436—22no—
53C2×#13063436—22no—
54C2×#13063436C2312yes—
55C2×#13063436C2312yes—
56C24×C173—21no—
57C24×C173D411yescharacteristic
58C2×D4×C173—24no—
59C2×D4×C173—22no—
60C2×D4×C173—22no—
61C2×D4×C173—21no—
62C2×D4×C173—44no—
63C2×D4×C173C2312yes—
64C22×C4×C173—22no—
65C22×C4×C173—22no—
66C22×C4×C173—21no—
67C22×C4×C173D411yescharacteristic
68C22×C4×C173D412yes—
69C22×C4×C173C2312yes—
70C22×C4×C173C2311yescharacteristic
71C4⋊C4×C173—21no—
72C4⋊C4×C173C2312yes—
73C22⋊C4×C173—24no—
74C22×D173—24no—
75C22×D173—22no—
76C22×D173—42no—
77C22×D173—42no—
78Dic346—24no—
79Dic346—22no—
80Dic346C2×D414yescomplemented
81D692—24no—
82D692—22no—
83D692C2×D414yescomplemented
84C22⋊D173—216no—
85C2×Dic173—24no—
86C2×Dic173—22no—
87C4×D173—216no—
88#13063436—22no—
89#13063436—22no—
90C23×C173—22no—
91C23×C173—22no—
92C23×C173—22no—
93C23×C173—22no—
94C23×C173—42no—
95C23×C173—42no—
96C23×C173—42no—
97C23×C173C2×D412yes—
98C23×C173C2×D411yescharacteristic
99D4×C173—22no—
100D4×C173—22no—
101D4×C173—48no—
102D4×C173—42no—
103C2×C4×C173—22no—
104C2×C4×C173—24no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—22no—
108C2×C4×C173—42no—
109C2×C4×C173C2×D412yes—
110C2×C4×C173C2×D412yes—
111C2×C4×C173C2×D412yes—
112C2×C4×C173C2×D411yescharacteristic
113C2×C4×C173C2411yescharacteristic
114C22×C173—22no—
115C22×C173—22no—
116C22×C173—22no—
117C22×C173—44no—
118C22×C173—42no—
119C22×C173—42no—
120C22×C173—42no—
121C22×C173—84no—
122C22×C173#12012yes—
123C22×C173#13512yes—
124C22×C173AGL(1,ℤ/8)12yes—
125C22×C173C22×D411yescharacteristic
126C2×D173—24no—
127C2×D173—42no—
128C2×D173—42no—
129C2×D173—48no—
130C4×C173—22no—
131C4×C173—42no—
132C4×C173#12012yes—
133C4×C173C22×D411yescharacteristic
134C4×C173C22×D411yescharacteristic
135Dic173—24no—
136C2×C173—42no—
137C2×C173—82no—
138C2×C173#44914yescomplemented
139C2×C173C2×#12011yescharacteristic
140C2×C173C2×#13511yescharacteristic
141C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
142D173—44no—
143C173C2×#44911yescharacteristic, complemented, Sylow
144C2×#449—1731nomaximal, Sylow
145C22×#41—1731no—
146C22×SD16—1731no—
147C22×D8—1731no—
148C2×#121D17311yescharacteristic, complemented
149#449—1738no—
150C2×#103—1731no—
151C2×#102—1731no—
152C2×C22⋊C8—1731no—
153C2×#41—3464no—
154C2×#41—3464no—
155C2×#41—3462no—
156C2×#41—1734no—
157C22×Q8—1731no—
158C22×D4—21no—
159C22×D4—3461no—
160C22×D4—1731no—
161C22×D4C2×D17311yescharacteristic
162C23×C4—3461no—
163C23×C4C2×D17311yescharacteristic
164C2×SD16—3464no—
165C2×SD16—1734no—
166C2×D8—3464no—
167C2×D8—1734no—
168C22×C8—1731no—
169C22×C8—1731no—
170#121—22no—
171#121C2×D17314yescomplemented
172C2×C4⋊C4C2×D17311yescharacteristic
173C2×C22⋊C4—21no—
174#103—1734no—
175#102—1734no—
176C22⋊C8—1734no—
177C24—21no—
178C24—3461no—
179C24C22⋊D17311yescharacteristic
180#41—34616no—
181#41—3468no—
182#41—3468no—
183C2×Q8—3462no—
184C2×Q8—3461no—
185C2×Q8—1734no—
186C2×Q8—1732no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—44no—
192C2×D4—3468no—
193C2×D4—3464no—
194C2×D4—3462no—
195C2×D4—3461no—
196C2×D4—1734no—
197C2×D4—1732no—
198C2×D4C22×D17312yes—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—3468no—
203C22×C4—3461no—
204C22×C4—3464no—
205C22×C4C22⋊D17312yes—
206C22×C4C22⋊D17311yescharacteristic
207C22×C4C22×D17312yes—
208C22×C4C22×D17311yescharacteristic
209SD16—3468no—
210D8—3468no—
211C2×C8—3462no—
212C2×C8—3462no—
213C2×C8—1732no—
214C2×C8—1732no—
215C4⋊C4—21no—
216C4⋊C4C22×D17312yes—
217C22⋊C4—24no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—3464no—
226C23—3462no—
227C23—6922no—
228C23—6922no—
229C23C2×C22⋊D17312yes—
230C23C2×C22⋊D17311yescharacteristic
231Q8—3464no—
232Q8—3462no—
233Q8—1734no—
234D4—22no—
235D4—22no—
236D4—48no—
237D4—42no—
238D4—34616no—
239D4—3464no—
240D4—3462no—
241D4—1734no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—34616no—
249C2×C4—3464no—
250C2×C4—3462no—
251C2×C4D4×D17312yes—
252C2×C4D4×D17312yes—
253C2×C4C2×C22⋊D17312yes—
254C2×C4C2×C22⋊D17311yescharacteristic
255C2×C4C23×D17311yescharacteristic
256C8—3462no—
257C8—3462no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—44no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—84no—
266C22—3464no—
267C22—6922no—
268C22—6922no—
269C22—6928no—
270C22#1914964712yes—
271C22#1914969912yes—
272C22#1914970512yes—
273C22C22×C22⋊D17311yescharacteristic
274C4—22no—
275C4—42no—
276C4—3464no—
277C4#1914964712yes—
278C4C2×D4×D17311yescharacteristic
279C4C2×D4×D17311yescharacteristic
280C2—42no—
281C2—82no—
282C2—6924no—
283C2#2601163414yesdirect product factor
284C2C2×#1914964711yescharacteristic
285C2C2×#1914969911yescharacteristic
286C2C2×#1914970511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#121×C173C2
socleC23×C173C2×D4
hypercenterC2×#121D173
hypocenterC173C2×#449
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14583826C2semidirect
2. C22×#14583814C2semidirect
3. C22×#14583811C2semidirect
4. #26011634C2direct (N × H)
5. C2×#19149602C2semidirect
6. C2×#19149599C2semidirect
7. C2×C22⋊#13063436C2semidirect
8. C2×#121×C173C2semidirect
12. C2×#14583826C22semidirect
18. C2×#14583814C22semidirect
20. C2×#14583811C22semidirect
23. #19149602C22semidirect
24. #19149599C22semidirect
25. C22⋊#13063436C22semidirect
30. #121×C173C22semidirect
39. C2×Dic346D4semidirect
43. C2×D692D4semidirect
80. Dic346C2×D4semidirect
83. D692C2×D4semidirect
138. C2×C173#449semidirect
143. C173C2×#449semidirect
148. C2×#121D173semidirect
171. #121C2×D173semidirect
283. C2#26011634direct (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×#145838142
C22×#145838112
#260116342
C2×#191496022
C2×#191495992
C2×C22⋊#130634362
C2×#121×C1732
C2×#449173