Finite Group Databasebeta

← #27173121

Subgroups — #27173121

Order 12928.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#20265857C211yesmaximal, characteristic, complemented
2#20265367C211yesmaximal, characteristic, complemented
3#20265364C211yesmaximal, characteristic
4#20264972C211yesmaximal, characteristic, complemented
5#20264968C211yesmaximal, characteristic, complemented
6#20264965C211yesmaximal, characteristic, complemented
7#455×C101C211yesmaximal, characteristic, complemented
8#15692248—21no—
9#15692237—21no—
10#15692236—21no—
11#15692236C2211yescharacteristic
12#15692231—21no—
13#15692223—21no—
14#15692223C2211yescharacteristic
15#15692215—21no—
16#15692212—21no—
17C22⋊C4×D101—21no—
18#15692170—21no—
19#15692170—21no—
20#15692170C2211yescharacteristic
21#15692170C2211yescharacteristic
22H(5,𝔽2)×C101C2211yescharacteristic
23#137×C101—21no—
24#124×C101C2211yescharacteristic
25#104×C101—21no—
26#100×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×C22⋊D101—21no—
29C2×C22⋊D101—41no—
30C22×Dic101D411yescharacteristic, complemented
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C4⋊Dic101—21no—
34#13108053—21no—
35#13108053—21no—
36#13108053—21no—
37#13108053—41no—
38C22⋊Dic101—21no—
39C22⋊Dic101—21no—
40C22⋊Dic101—21no—
41C22⋊Dic101—41no—
42C22⋊Dic101D411yescharacteristic
43C22⋊Dic101D411yescharacteristic, complemented
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—41no—
48C4×Dic101—21no—
49#41×C101—21no—
50#41×C101—41no—
51C2×Q8×C101D411yescharacteristic, complemented
52C2×D4×C101—21no—
53C2×D4×C101—21no—
54C2×D4×C101—41no—
55C2×D4×C101C2311yescharacteristic
56Q16×C101—41no—
57SD16×C101—41no—
58M16×C101—21no—
59C22⋊C4×C101—21no—
60C22⋊C4×C101—21no—
61C42×C101D411yescharacteristic, complemented
62C23×C101—21no—
63C23×C101—21no—
64C23×C101C2×D411yescharacteristic
65C23×C101C2×D411yescharacteristic
66C22×D101—21no—
67C22×D101—42no—
68C22×D101—41no—
69Q8×C101—21no—
70Q8×C101—41no—
71D4×C101—21no—
72D4×C101—41no—
73D4×C101—41no—
74D4×C101—41no—
75D4×C101—41no—
76C2×C4×C101—21no—
77C2×C4×C101—21no—
78C2×C4×C101—21no—
79C2×C4×C101—21no—
80C2×C4×C101—41no—
81C2×C4×C101C2×D411yescharacteristic
82C22⋊D101—42no—
83C22⋊D101—82no—
84C2×Dic101—21no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89C2×Dic101—41no—
90C4×D101—42no—
91C4×D101—81no—
92C8×C101—41no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—81no—
99C22×C101—81no—
100C22×C101#12011yescharacteristic
101C2×D101—22no—
102C2×D101—41no—
103C2×D101—41no—
104C2×D101—42no—
105C2×D101—81no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—41no—
110Dic101—41no—
111Dic101—41no—
112Dic101—41no—
113Dic101—81no—
114Dic101—81no—
115C2×C101—21no—
116C2×C101—41no—
117C2×C101—41no—
118C2×C101—81no—
119C2×C101B(4,𝔽2)11yescharacteristic
120D101—42no—
121#2197—1011nomaximal, Sylow
122C101#219711yescharacteristic, complemented, Sylow
123#551—1011no—
124#458—1011no—
125B(4,𝔽2)—1011no—
126#455D10111yescharacteristic, complemented
127#354—1011no—
128#352—1011no—
129AΣL(1,𝔽16)—1011no—
130H(5,𝔽2)C2×D10111yescharacteristic
131#137—21no—
132#126—2021no—
133#124C2×D10111yescharacteristic
134#123—1011no—
135#123—2021no—
136#123—2021no—
137C22⋊Q8—2021no—
138#120—1011no—
139#120—2021no—
140C4×D4—2021no—
141#117—2021no—
142C2×C22⋊C4—2021no—
143#104—21no—
144#100C2×D10111yescharacteristic
145#99—1011no—
146#99—1011no—
147#99—2021no—
148#99—2021no—
149C24—1011no—
150#41—21no—
151#41—41no—
152C2×Q8C22⋊D10111yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—4041no—
157C2×D4—2021no—
158C2×D4C22×D10111yescharacteristic
159C22×C4—1011no—
160C22×C4—2021no—
161C22×C4—2021no—
162Q16—41no—
163SD16—41no—
164M16—21no—
165C4⋊C4—4041no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C4⋊C4—2021no—
169C4⋊C4—2021no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—1011no—
173C22⋊C4—1011no—
174C22⋊C4—4041no—
175C22⋊C4—4041no—
176C22⋊C4—2021no—
177C22⋊C4—2021no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C42—2021no—
183C42C22⋊D10111yescharacteristic, complemented
184C23—21no—
185C23—21no—
186C23—4041no—
187C23—4042no—
188C23—2021no—
189C23D4×D10111yescharacteristic
190C23D4×D10111yescharacteristic
191Q8—21no—
192Q8—41no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—8082no—
199D4—4042no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—8081no—
206C2×C4—4042no—
207C2×C4—4041no—
208C2×C4—4041no—
209C2×C4—4041no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4C2×C22⋊D10111yescharacteristic
214C8—41no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—8081no—
223C22—4041no—
224C22—4042no—
225C22—4041no—
226C22—2022no—
227C22#1569222311yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—8081no—
233C4—8081no—
234C4—4041no—
235C4—4041no—
236C4—4041no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—4042no—
242C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#455×C101C2
socleC2×C101B(4,𝔽2)
prime radical#20265364C2
hypercenter#455D101
hypocenterC101#2197
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #20265857C2semidirect
2. #20265367C2semidirect
4. #20264972C2semidirect
5. #20264968C2semidirect
6. #20264965C2semidirect
7. #455×C101C2semidirect
30. C22×Dic101D4semidirect
43. C22⋊Dic101D4semidirect
51. C2×Q8×C101D4semidirect
61. C42×C101D4semidirect
122. C101#2197semidirect
126. #455D101semidirect
152. C2×Q8C22⋊D101semidirect
183. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202658572
#202653672
#202653642
#202649722
#202649682
#202649652
#455×C1012
#2197101