Finite Group Databasebeta

← C22×#16839672 #29846190

Subgroups — C22×#16839672 #29846190

Order 15616.

13646 conjugacy classes of subgroups; only the 1539 normal subgroups are shown.

normal subgroups only

This group has 13646 conjugacy classes of subgroups — too many to draw. Only its 1539 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
1C24×D244C213yesmaximal, complemented
2C23×#11758198C213yesmaximal, complemented
3C2×#16839672C2124yesmaximal, direct product factor
4C22×#95×C61C211yesmaximal, characteristic, complemented
5C25×D61C2213yes—
6C23×D244—224no—
7C23×D244—248no—
8C23×D244C22136yescomplemented
9C24×Dic61C2211yescharacteristic
10C22×#11758198—224no—
11C22×#11758198C22136yescomplemented
12#16839672C22164yesdirect product factor
13C24×C4×C61C2213yes—
14C2×#95×C61C22112yescomplemented
15C24×D61—212no—
16C24×D61—224no—
17C24×D61—224no—
18C24×D61—212no—
19C24×D61D4124yescomplemented
20C24×D61C23118yes—
21C22×D244—2384no—
22C22×D244—2288no—
23C22×D244—2144no—
24C22×D244C23148yescomplemented
25C23×Dic61—212no—
26C23×Dic61C2316yes—
27C2×#11758198—2144no—
28C2×#11758198C23148yescomplemented
29C25×C61C2311yescharacteristic
30C23×C4×C61—224no—
31C23×C4×C61D4124yescomplemented
32C23×C4×C61C23118yes—
33#95×C61C23116yescomplemented
34C24×C61C2×D4112yes—
35C24×C61C2×D4112yes—
36C24×C61#4114yes—
37C24×C61C2413yes—
38C23×D61—2192no—
39C23×D61—296no—
40C23×D61—272no—
41C23×D61—272no—
42C23×D61—236no—
43C23×D61—236no—
44C23×D61—448no—
45C23×D61—496no—
46C23×D61—448no—
47C23×D61C2×D4172yescomplemented
48C23×D61C24112yes—
49C22×C4×C61—296no—
50C22×C4×C61—272no—
51C22×C4×C61C2×D4172yescomplemented
52C22×C4×C61C24112yes—
53C22×Dic61—236no—
54C22×Dic61—416no—
55C22×Dic61C2414yes—
56C2×D244—21152no—
57C2×D244—2192no—
58C2×D244—296no—
59#11758198—296no—
60C23×C61#120148yes—
61C23×C61#121148yes—
62C23×C61#127116yes—
63C23×C61C22×D4118yes—
64C23×C61C22×D4118yes—
65C23×C61C2×#4116yes—
66C23×C61C2511yescharacteristic
67C22×D61—2288no—
68C22×D61—2144no—
69C22×D61—212no—
70C22×D61—224no—
71C22×D61—224no—
72C22×D61—212no—
73C22×D61—472no—
74C22×D61—4144no—
75C22×D61—472no—
76C22×D61—4384no—
77C22×D61C22×D4124yescomplemented
78C2×C4×C61—2144no—
79C2×C4×C61—224no—
80C2×C4×C61C22×D4124yescomplemented
81C2×Dic61—212no—
82C2×Dic61—424no—
83D244—2384no—
84C22×#392—611nomaximal, Sylow
85C22×C61#392164yescomplemented
86C22×C61C2×#120136yes—
87C22×C61C2×#121136yes—
88C22×C61C2×#127112yes—
89C22×C61C23×D413yes—
90C22×C61C23×D413yes—
91C22×C61C22×#4111yescharacteristic
92C2×D61—248no—
93C2×D61—224no—
94C2×D61—4288no—
95C2×D61—412no—
96C2×D61—424no—
97C2×D61—412no—
98C4×C61—224no—
99Dic61—44no—
100C24×D4—613no—
101C23×C22⋊C4—613no—
102C2×#392—6124no—
103C22×#95D6111yescharacteristic, complemented
104C2×C61C2×#392124yescomplemented
105C2×C61C22×#12013yes—
106C2×C61C22×#12113yes—
107C2×C61C22×#12711yescharacteristic
108D61—424no—
109C26—613no—
110C23×D4—6136no—
111C23×D4—12248no—
112C23×D4—12224no—
113C24×C4—611no—
114C24×C4C2×D6113yes—
115C22×C22⋊C4—6136no—
116C22×C22⋊C4—12224no—
117#392—6164no—
118C2×#95C2×D61112yescomplemented
119C61C22×#39211yescharacteristic, complemented, Sylow
120C25—6124no—
121C25—6118no—
122C25—12212no—
123C25—12224no—
124C25—12224no—
125C25—12212no—
126C25C22×D6111yescharacteristic
127C22×D4—6148no—
128C22×D4—122288no—
129C22×D4—122384no—
130C22×D4—122144no—
131C23×C4—224no—
132C23×C4—616no—
133C23×C4—12212no—
134C23×C4D244124yescomplemented
135C23×C4C22×D61118yes—
136C2×C22⋊C4—6148no—
137C2×C22⋊C4—122144no—
138#95C22×D61116yescomplemented
139C24—6172no—
140C24—6112no—
141C24—12272no—
142C24—12236no—
143C24—12272no—
144C24—12236no—
145C24—122192no—
146C24—12296no—
147C24—24448no—
148C24—24496no—
149C24—24448no—
150C24C2×D244112yes—
151C24D4×D61112yes—
152C24#1175822614yes—
153C24C23×D6113yes—
154C2×D4—122192no—
155C2×D4—1221152no—
156C2×D4—12296no—
157C22×C4—272no—
158C22×C4—296no—
159C22×C4—614no—
160C22×C4—12236no—
161C22×C4—24416no—
162C22×C4C2×D244172yescomplemented
163C22×C4C23×D61112yes—
164C22⋊C4—12296no—
165C23—6124no—
166C23—122288no—
167C23—122144no—
168C23—12224no—
169C23—12212no—
170C23—12224no—
171C23—12212no—
172C23—24472no—
173C23—244144no—
174C23—24472no—
175C23—244384no—
176C23#14519747148yes—
177C23#14519752148yes—
178C23#14519774116yes—
179C23C2×D4×D61118yes—
180C23C22×D244118yes—
181C23C2×#1175822616yes—
182C23C24×D6111yescharacteristic
183D4—122384no—
184C2×C4—2144no—
185C2×C4—224no—
186C2×C4—12212no—
187C2×C4—24424no—
188C2×C4C22×D244124yescomplemented
189C22—12248no—
190C22—12224no—
191C22—244288no—
192C22—24412no—
193C22—24424no—
194C22—24412no—
195C22#16839672164yesdirect product factor
196C22C2×#14519747136yes—
197C22C2×#14519752136yes—
198C22C2×#14519774112yes—
199C22C22×D4×D6113yes—
200C22C23×D24413yes—
201C22C22×#1175822611yescharacteristic
202C4—224no—
203C4—2444no—
204C2—24424no—
205C2C2×#16839672124yesdirect product factor
206C2C22×#1451974713yes—
207C2C22×#1451975213yes—
208C2C22×#1451977411yescharacteristic
subgroupG/H
centerC25C22×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×#95×C61C2
socleC25×C61C23
hypercenterC22×#95D61
hypocenterC61C22×#392
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C25 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C24×D244C2semidirect
2. C23×#11758198C2semidirect
3. C2×#16839672C2direct (N × H)
4. C22×#95×C61C2semidirect
8. C23×D244C22semidirect
11. C22×#11758198C22semidirect
12. #16839672C22direct (N × H)
14. C2×#95×C61C22semidirect
19. C24×D61D4semidirect
24. C22×D244C23semidirect
28. C2×#11758198C23semidirect
31. C23×C4×C61D4semidirect
33. #95×C61C23semidirect
47. C23×D61C2×D4semidirect
51. C22×C4×C61C2×D4semidirect
77. C22×D61C22×D4semidirect
80. C2×C4×C61C22×D4semidirect
85. C22×C61#392semidirect
103. C22×#95D61semidirect
104. C2×C61C2×#392semidirect
118. C2×#95C2×D61semidirect
119. C61C22×#392semidirect
134. C23×C4D244semidirect
138. #95C22×D61semidirect
162. C22×C4C2×D244semidirect
188. C2×C4C22×D244semidirect
195. C22#16839672direct (N × H)
205. C2C2×#16839672direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D2442
C23×#117581982
C2×#168396722
C22×#95×C612
C22×#39261