Finite Group Databasebeta

← C22×#16837312 #28754866

Subgroups — C22×#16837312 #28754866

Order 15104.

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×D236C213yesmaximal, complemented
2C23×(C2×D59)⋊C4C213yesmaximal, complemented
3C2×#16837312C2124yesmaximal, direct product factor
4C22×#95×C59C211yesmaximal, characteristic, complemented
5C25×D59C2213yes—
6C23×D236—248no—
7C23×D236—224no—
8C23×D236C22136yescomplemented
9C24×Dic59C2211yescharacteristic
10C22×(C2×D59)⋊C4—224no—
11C22×(C2×D59)⋊C4C22136yescomplemented
12#16837312C22164yesdirect product factor
13C24×C4×C59C2213yes—
14C2×#95×C59C22112yescomplemented
15C24×D59—224no—
16C24×D59—224no—
17C24×D59—212no—
18C24×D59—212no—
19C24×D59D4124yescomplemented
20C24×D59C23118yes—
21C22×D236—2384no—
22C22×D236—2288no—
23C22×D236—2144no—
24C22×D236C23148yescomplemented
25C23×Dic59—212no—
26C23×Dic59C2316yes—
27C2×(C2×D59)⋊C4—2144no—
28C2×(C2×D59)⋊C4C23148yescomplemented
29C25×C59C2311yescharacteristic
30C23×C4×C59—224no—
31C23×C4×C59D4124yescomplemented
32C23×C4×C59C23118yes—
33#95×C59C23116yescomplemented
34C24×C59C2×D4112yes—
35C24×C59C2×D4112yes—
36C24×C59#4114yes—
37C24×C59C2413yes—
38C23×D59—2192no—
39C23×D59—296no—
40C23×D59—236no—
41C23×D59—272no—
42C23×D59—272no—
43C23×D59—236no—
44C23×D59—448no—
45C23×D59—496no—
46C23×D59—448no—
47C23×D59C2×D4172yescomplemented
48C23×D59C24112yes—
49C22×C4×C59—296no—
50C22×C4×C59—272no—
51C22×C4×C59C2×D4172yescomplemented
52C22×C4×C59C24112yes—
53C22×Dic59—236no—
54C22×Dic59—416no—
55C22×Dic59C2414yes—
56C2×D236—21152no—
57C2×D236—2192no—
58C2×D236—296no—
59(C2×D59)⋊C4—296no—
60C23×C59#120148yes—
61C23×C59#121148yes—
62C23×C59#127116yes—
63C23×C59C22×D4118yes—
64C23×C59C22×D4118yes—
65C23×C59C2×#4116yes—
66C23×C59C2511yescharacteristic
67C22×D59—2288no—
68C22×D59—2144no—
69C22×D59—212no—
70C22×D59—224no—
71C22×D59—224no—
72C22×D59—212no—
73C22×D59—472no—
74C22×D59—4144no—
75C22×D59—472no—
76C22×D59—4384no—
77C22×D59C22×D4124yescomplemented
78C2×C4×C59—2144no—
79C2×C4×C59—224no—
80C2×C4×C59C22×D4124yescomplemented
81C2×Dic59—212no—
82C2×Dic59—424no—
83D236—2384no—
84C22×#392—591nomaximal, Sylow
85C22×C59#392164yescomplemented
86C22×C59C2×#120136yes—
87C22×C59C2×#121136yes—
88C22×C59C2×#127112yes—
89C22×C59C23×D413yes—
90C22×C59C23×D413yes—
91C22×C59C22×#4111yescharacteristic
92C2×D59—248no—
93C2×D59—224no—
94C2×D59—4288no—
95C2×D59—412no—
96C2×D59—424no—
97C2×D59—412no—
98C4×C59—224no—
99Dic59—44no—
100C24×D4—593no—
101C23×C22⋊C4—593no—
102C2×#392—5924no—
103C22×#95D5911yescharacteristic, complemented
104C2×C59C2×#392124yescomplemented
105C2×C59C22×#12013yes—
106C2×C59C22×#12113yes—
107C2×C59C22×#12711yescharacteristic
108D59—424no—
109C26—593no—
110C23×D4—5936no—
111C23×D4—11848no—
112C23×D4—11824no—
113C24×C4—591no—
114C24×C4C2×D5913yes—
115C22×C22⋊C4—5936no—
116C22×C22⋊C4—11824no—
117#392—5964no—
118C2×#95C2×D59112yescomplemented
119C59C22×#39211yescharacteristic, complemented, Sylow
120C25—5924no—
121C25—5918no—
122C25—11812no—
123C25—11824no—
124C25—11824no—
125C25—11812no—
126C25C22×D5911yescharacteristic
127C22×D4—5948no—
128C22×D4—118288no—
129C22×D4—118384no—
130C22×D4—118144no—
131C23×C4—224no—
132C23×C4—596no—
133C23×C4—11812no—
134C23×C4D236124yescomplemented
135C23×C4C22×D59118yes—
136C2×C22⋊C4—5948no—
137C2×C22⋊C4—118144no—
138#95C22×D59116yescomplemented
139C24—5972no—
140C24—5912no—
141C24—118192no—
142C24—11896no—
143C24—11872no—
144C24—11836no—
145C24—11872no—
146C24—11836no—
147C24—23648no—
148C24—23696no—
149C24—23648no—
150C24C2×D236112yes—
151C24D4×D59112yes—
152C24#1174568914yes—
153C24C23×D5913yes—
154C2×D4—118192no—
155C2×D4—1181152no—
156C2×D4—11896no—
157C22×C4—296no—
158C22×C4—272no—
159C22×C4—594no—
160C22×C4—11836no—
161C22×C4—23616no—
162C22×C4C2×D236172yescomplemented
163C22×C4C23×D59112yes—
164C22⋊C4—11896no—
165C23—5924no—
166C23—118288no—
167C23—118144no—
168C23—11824no—
169C23—11812no—
170C23—11824no—
171C23—11812no—
172C23—23672no—
173C23—236144no—
174C23—23672no—
175C23—236384no—
176C23#14273697148yes—
177C23#14273702148yes—
178C23#14273724116yes—
179C23C2×D4×D59118yes—
180C23C22×D236118yes—
181C23C2×#1174568916yes—
182C23C24×D5911yescharacteristic
183D4—118384no—
184C2×C4—2144no—
185C2×C4—224no—
186C2×C4—11812no—
187C2×C4—23624no—
188C2×C4C22×D236124yescomplemented
189C22—11848no—
190C22—11824no—
191C22—236288no—
192C22—23612no—
193C22—23624no—
194C22—23612no—
195C22#16837312164yesdirect product factor
196C22C2×#14273697136yes—
197C22C2×#14273702136yes—
198C22C2×#14273724112yes—
199C22C22×D4×D5913yes—
200C22C23×D23613yes—
201C22C22×#1174568911yescharacteristic
202C4—224no—
203C4—2364no—
204C2—23624no—
205C2C2×#16837312124yesdirect product factor
206C2C22×#1427369713yes—
207C2C22×#1427370213yes—
208C2C22×#1427372411yescharacteristic
subgroupG/H
centerC25C22×D59
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC22×#95×C59C2
socleC25×C59C23
hypercenterC22×#95D59
hypocenterC59C22×#392
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C25 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C24×D236C2semidirect
2. C23×(C2×D59)⋊C4C2semidirect
3. C2×#16837312C2direct (N × H)
4. C22×#95×C59C2semidirect
8. C23×D236C22semidirect
11. C22×(C2×D59)⋊C4C22semidirect
12. #16837312C22direct (N × H)
14. C2×#95×C59C22semidirect
19. C24×D59D4semidirect
24. C22×D236C23semidirect
28. C2×(C2×D59)⋊C4C23semidirect
31. C23×C4×C59D4semidirect
33. #95×C59C23semidirect
47. C23×D59C2×D4semidirect
51. C22×C4×C59C2×D4semidirect
77. C22×D59C22×D4semidirect
80. C2×C4×C59C22×D4semidirect
85. C22×C59#392semidirect
103. C22×#95D59semidirect
104. C2×C59C2×#392semidirect
118. C2×#95C2×D59semidirect
119. C59C22×#392semidirect
134. C23×C4D236semidirect
138. #95C22×D59semidirect
162. C22×C4C2×D236semidirect
188. C2×C4C22×D236semidirect
195. C22#16837312direct (N × H)
205. C2C2×#16837312direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D2362
C23×(C2×D59)⋊C42
C2×#168373122
C22×#95×C592
C22×#39259