Finite Group Databasebeta

← C2×#26011680 #36362950

Subgroups — C2×#26011680 #36362950

Order 22144.

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

normal subgroups only

This group has 794 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×#19149724C211yesmaximal, characteristic
2C2×#19149662C212yesmaximal, complemented
3#26011680C218yesmaximal, direct product factor
4C2×#19149594C211yesmaximal, characteristic, complemented
5C2×#19149593C212yesmaximal, complemented
6C2×#123×C173C211yesmaximal, characteristic, complemented
7#19149724—22no—
8#19149724—24no—
9#19149724C2214yes—
10C2×#14583826—21no—
11C2×#14583825—22no—
12C22×C22⋊D173—21no—
13C22×C22⋊D173C2212yes—
14C2×D4×D173—22no—
15#19149662—24no—
16#19149662C2218yescomplemented
17C2×#14583805—22no—
18C2×C22⋊Dic173C2212yescomplemented
19C2×#14583803—22no—
20#19149594C2214yescomplemented
21#19149593C2218yescomplemented
22C22×D4×C173C2211yescharacteristic
23C23×C4×C173C2211yescharacteristic, complemented
24#123×C173—22no—
25#123×C173C2214yescomplemented
26C2×C4⋊C4×C173—21no—
27C2×C22⋊C4×C173—21no—
28C2×C22⋊C4×C173C2211yescharacteristic, complemented
29C23×D173—21no—
30C23×D173D412yes—
31#14583826—24no—
32#14583826—24no—
33#14583825—28no—
34#14583825—28no—
35C2×Dic346—21no—
36C2×D692—21no—
37C2×C22⋊D173—22no—
38C2×C22⋊D173—24no—
39C2×C22⋊D173—22no—
40C2×C22⋊D173—22no—
41C2×C22⋊D173—24no—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44C2×C22⋊D173D414yes—
45C2×C22⋊D173C2314yes—
46D4×D173—28no—
47D4×D173—28no—
48C22×Dic173—21no—
49C22×Dic173—22no—
50C22×Dic173D412yes—
51C2×C4×D173—22no—
52#14583805—28no—
53C22⋊Dic173—22no—
54C22⋊Dic173C2314yescomplemented
55#14583803—28no—
56C24×C173D411yescharacteristic
57C24×C173C2311yescharacteristic
58C2×D4×C173—22no—
59C2×D4×C173—22no—
60C2×D4×C173D412yes—
61C2×D4×C173C2312yes—
62C22×C4×C173—22no—
63C22×C4×C173—22no—
64C22×C4×C173—21no—
65C22×C4×C173—21no—
66C22×C4×C173—21no—
67C22×C4×C173D411yescharacteristic
68C22×C4×C173C2312yescomplemented
69C4⋊C4×C173—24no—
70C22⋊C4×C173—24no—
71C22⋊C4×C173—21no—
72C22⋊C4×C173C2312yescomplemented
73C22×D173—22no—
74C22×D173—24no—
75C22×D173—22no—
76C22×D173—44no—
77C22×D173—82no—
78C22×D173C2×D414yes—
79Dic346—24no—
80D692—24no—
81C22⋊D173—28no—
82C22⋊D173—24no—
83C22⋊D173—24no—
84C22⋊D173—24no—
85C22⋊D173—24no—
86C2×Dic173—24no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—24no—
90C2×Dic173—44no—
91C2×Dic173C2×D414yes—
92C4×D173—28no—
93C23×C173—21no—
94C23×C173—21no—
95C23×C173—42no—
96C23×C173—42no—
97C23×C173C2×D412yes—
98C23×C173C2×D412yes—
99C23×C173C2×D411yescharacteristic
100C23×C173C2×D412yes—
101C23×C173C2×D411yescharacteristic
102C23×C173C2411yescharacteristic
103D4×C173—24no—
104D4×C173—24no—
105C2×C4×C173—21no—
106C2×C4×C173—21no—
107C2×C4×C173—22no—
108C2×C4×C173—22no—
109C2×C4×C173—24no—
110C2×C4×C173—42no—
111C2×C4×C173—41no—
112C2×C4×C173—42no—
113C2×C4×C173—41no—
114C2×C4×C173C2×D412yes—
115C22×C173—22no—
116C22×C173—24no—
117C22×C173—22no—
118C22×C173—42no—
119C22×C173—41no—
120C22×C173—41no—
121C22×C173—42no—
122C22×C173—42no—
123C22×C173—81no—
124C22×C173#12011yescharacteristic
125C22×C173#12012yes—
126C22×C173#12011yescharacteristic
127C22×C173C22×D412yes—
128C22×C173C22×D411yescharacteristic
129C2×D173—24no—
130C2×D173—44no—
131C2×D173—44no—
132C2×D173—82no—
133C2×D173—84no—
134C4×C173—22no—
135C4×C173—42no—
136C4×C173—42no—
137Dic173—24no—
138Dic173—44no—
139C2×C173—22no—
140C2×C173—21no—
141C2×C173—22no—
142C2×C173—21no—
143C2×C173—42no—
144C2×C173#45812yescomplemented
145C2×C173C2×#12011yescharacteristic
146D173—44no—
147C173C2×#45811yescharacteristic, complemented, Sylow
148C2×#458—1731nomaximal, Sylow
149C2×H(5,𝔽2)—1731no—
150C2×#123—1732no—
151C2×#123D17311yescharacteristic, complemented
152#458—1738no—
153C2×#99—1731no—
154C2×#99—1732no—
155H(5,𝔽2)—3464no—
156H(5,𝔽2)—3462no—
157H(5,𝔽2)—1734no—
158C2×#41—3462no—
159C2×#41—3461no—
160C22×D4—3462no—
161C22×D4—3461no—
162C22×D4—1732no—
163C22×D4C2×D17311yescharacteristic
164C23×C4C2×D17311yescharacteristic, complemented
165#123—22no—
166#123—3464no—
167#123—1738no—
168#123C2×D17314yescomplemented
169C2×C4⋊C4—21no—
170C2×C4⋊C4—3462no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—3462no—
173C2×C22⋊C4—1732no—
174C2×C22⋊C4C2×D17311yescharacteristic, complemented
175#99—1734no—
176#99—1738no—
177C24—3461no—
178C24—1732no—
179C24C22⋊D17311yescharacteristic
180C24C22×D17311yescharacteristic
181#41—3464no—
182#41—3468no—
183#41—3468no—
184#41—3464no—
185C2×Q8—3461no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—3461no—
189C2×D4—3468no—
190C2×D4—3468no—
191C2×D4—3462no—
192C2×D4—3464no—
193C2×D4—3462no—
194C2×D4—3462no—
195C2×D4—3462no—
196C2×D4—3464no—
197C2×D4—3462no—
198C2×D4—1734no—
199C2×D4—1734no—
200C2×D4C22⋊D17312yes—
201C2×D4C22×D17312yes—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—3462no—
208C22×C4—3461no—
209C22×C4—3462no—
210C22×C4—1732no—
211C22×C4C22⋊D17311yescharacteristic
212C22×C4C22×D17312yescomplemented
213C4⋊C4—24no—
214C4⋊C4—3468no—
215C22⋊C4—24no—
216C22⋊C4—21no—
217C22⋊C4—3468no—
218C22⋊C4—3462no—
219C22⋊C4—1734no—
220C22⋊C4C22×D17312yescomplemented
221C23—21no—
222C23—21no—
223C23—42no—
224C23—42no—
225C23—3464no—
226C23—3462no—
227C23—3462no—
228C23—13842no—
229C23—1734no—
230C23—6924no—
231C23D4×D17312yes—
232C23D4×D17312yes—
233C23C2×C22⋊D17311yescharacteristic
234C23C2×C22⋊D17312yes—
235C23C2×C22⋊D17311yescharacteristic
236C23C23×D17311yescharacteristic
237Q8—3464no—
238D4—24no—
239D4—24no—
240D4—3464no—
241D4—3468no—
242D4—3464no—
243D4—3464no—
244D4—3464no—
245D4—3464no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—3468no—
256C2×C4—3464no—
257C2×C4—3462no—
258C2×C4—3462no—
259C2×C4—3464no—
260C2×C4—1734no—
261C2×C4—6924no—
262C2×C4C2×C22⋊D17312yes—
263C22—22no—
264C22—24no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—3464no—
273C22—13842no—
274C22—13844no—
275C22—6924no—
276C22—6924no—
277C22#1914964711yescharacteristic
278C22#1914964712yes—
279C22#1914964711yescharacteristic
280C22C2×D4×D17312yes—
281C22C22×C22⋊D17311yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—3464no—
286C4—6924no—
287C2—21no—
288C2—22no—
289C2—21no—
290C2—22no—
291C2—42no—
292C2—6924no—
293C2#2601168012yesdirect product factor
294C2C2×#1914964711yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
FittingC2×#123×C173C2
socleC22×C173#120
prime radicalC2×#19149724C2
hypercenterC2×#123D173
hypocenterC173C2×#458
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#19149662C2semidirect
3. #26011680C2direct (N × H)
4. C2×#19149594C2semidirect
5. C2×#19149593C2semidirect
6. C2×#123×C173C2semidirect
16. #19149662C22semidirect
18. C2×C22⋊Dic173C22semidirect
20. #19149594C22semidirect
21. #19149593C22semidirect
23. C23×C4×C173C22semidirect
25. #123×C173C22semidirect
28. C2×C22⋊C4×C173C22semidirect
54. C22⋊Dic173C23semidirect
68. C22×C4×C173C23semidirect
72. C22⋊C4×C173C23semidirect
144. C2×C173#458semidirect
147. C173C2×#458semidirect
151. C2×#123D173semidirect
164. C23×C4C2×D173semidirect
168. #123C2×D173semidirect
174. C2×C22⋊C4C2×D173semidirect
212. C22×C4C22×D173semidirect
220. C22⋊C4C22×D173semidirect
293. C2#26011680direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497242
C2×#191496622
#260116802
C2×#191495942
C2×#191495932
C2×#123×C1732
C2×#458173