Finite Group Databasebeta

← #27170052

Subgroups — #27170052

Order 12928.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265739C211yesmaximal, characteristic, complemented
2#20265313C212yesmaximal, complemented
3#20265086C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic, complemented
7C22×C22⋊D101—21no—
8C22×C22⋊D101—21no—
9C2×D808—22no—
10#15692239—22no—
11#15692229—21no—
12#15692229C2212yes—
13#15692225—22no—
14C2×#13108051—21no—
15#15692177—22no—
16#15692162C2211yescharacteristic, complemented
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yescomplemented
19C23×D101—21no—
20C23×D101—21no—
21C2×D404—22no—
22C2×D404—21no—
23C2×D404—42no—
24C2×D404D412yescomplemented
25C2×C22⋊D101—22no—
26C2×C22⋊D101—22no—
27C2×C22⋊D101—42no—
28C2×C22⋊D101—42no—
29C2×C22⋊D101—42no—
30C2×C22⋊D101—42no—
31C2×C22⋊D101—42no—
32C22×Dic101—21no—
33C22×Dic101—21no—
34D808—44no—
35C4⋊Dic101—21no—
36C4⋊Dic101D412yescomplemented
37#13108051—22no—
38#13108051—22no—
39#13108051—42no—
40C24×C101D411yescharacteristic
41C22×C4×C101D411yescharacteristic
42C22×C4×C101C2311yescharacteristic
43C2×C8×C101—22no—
44C22⋊C4×C101—22no—
45C23×C101—21no—
46C23×C101—21no—
47C23×C101—41no—
48C23×C101—41no—
49C23×C101D812yescomplemented
50C23×C101C2×D411yescharacteristic
51C22×D101—22no—
52C22×D101—22no—
53C22×D101—41no—
54C22×D101—41no—
55C22×D101—42no—
56C22×D101—41no—
57C22×D101—41no—
58C22×D101—41no—
59C22×D101—41no—
60C22×D101—81no—
61C22×D101—81no—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101C2×D412yes—
66C22⋊D101—44no—
67C22⋊D101—48no—
68C22⋊D101—42no—
69C22⋊D101—42no—
70C2×Dic101—22no—
71C2×Dic101—22no—
72C2×Dic101—41no—
73C2×Dic101—42no—
74C2×Dic101—41no—
75D404—24no—
76D404—42no—
77D404—42no—
78D404—81no—
79C8×C101—42no—
80C22×C101—21no—
81C22×C101—21no—
82C22×C101—22no—
83C22×C101—42no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101—81no—
87C22×C101#12011yescharacteristic
88C22×C101C2×D811yescharacteristic
89C22×C101AGL(1,ℤ/8)11yescharacteristic
90C2×D101—42no—
91C2×D101—42no—
92C2×D101—42no—
93C2×D101—42no—
94C2×D101—42no—
95C2×D101—42no—
96C2×D101—81no—
97C2×D101—81no—
98C2×D101—84no—
99C2×D101—81no—
100C2×D101—81no—
101C4×C101—22no—
102C4×C101—41no—
103Dic101—42no—
104Dic101—42no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—42no—
108C2×C101#44711yescharacteristic
109C2×C101#45311yescharacteristic
110C2×C101B(4,𝔽2)11yescharacteristic
111D101—82no—
112D101—82no—
113#1595—1011nomaximal, Sylow
114C101#159511yescharacteristic, complemented, Sylow
115#534—1011no—
116#447—1012no—
117#392—1011no—
118#327—1012no—
119#323D10111yescharacteristic, complemented
120C22×D4—1011no—
121C22×D4—2021no—
122C22×D4—2021no—
123C2×D8—2022no—
124#123—2022no—
125#121—1012no—
126#121—2021no—
127#120—2022no—
128C2×C22⋊C4—2021no—
129C2×C22⋊C4C2×D10111yescharacteristic
130#102—2022no—
131C22⋊C8C2×D10112yescomplemented
132#95—1011no—
133C24—2021no—
134C24—2021no—
135C24D40411yescharacteristic
136C2×D4—1012no—
137C2×D4—4042no—
138C2×D4—4042no—
139C2×D4—4042no—
140C2×D4—4042no—
141C2×D4—4042no—
142C2×D4—4042no—
143C2×D4—2021no—
144C2×D4—2022no—
145C2×D4—2022no—
146C2×D4—2022no—
147C22×C4—2021no—
148C22×C4—2021no—
149C22×C4D40411yescharacteristic
150C22×C4C22×D10111yescharacteristic
151D8—4044no—
152C2×C8—22no—
153C4⋊C4—1012no—
154C4⋊C4—2021no—
155C22⋊C4—22no—
156C22⋊C4—4042no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—8081no—
164C23—8081no—
165C23—4041no—
166C23—4041no—
167C23—4041no—
168C23—4042no—
169C23—4041no—
170C23—4041no—
171C23—4041no—
172C23—2022no—
173C23—2022no—
174C23D80812yescomplemented
175C23C2×D40411yescharacteristic
176D4—8081no—
177D4—4042no—
178D4—4042no—
179D4—4044no—
180D4—4048no—
181D4—4042no—
182D4—4042no—
183D4—2024no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—4041no—
188C2×C4—4042no—
189C2×C4—4041no—
190C2×C4—2022no—
191C2×C4—2022no—
192C2×C4D4×D10112yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—8081no—
202C22—8081no—
203C22—8084no—
204C22—8081no—
205C22—8081no—
206C22—4042no—
207C22—4042no—
208C22—4042no—
209C22—4042no—
210C22—4042no—
211C22—4042no—
212C22#1569222511yescharacteristic
213C22C2×D80811yescharacteristic
214C22#1569228311yescharacteristic
215C4—22no—
216C4—41no—
217C4—4042no—
218C4—4042no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—8082no—
223C2—8082no—
224C2#2026531311yescharacteristic
225C2#2026534711yescharacteristic
226C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
hypercenter#323D101
hypocenterC101#1595
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #20265739C2semidirect
2. #20265313C2semidirect
3. #20265086C2semidirect
4. #20264912C2semidirect
5. #323×C101C2semidirect
6. C22×D404C22semidirect
16. #15692162C22semidirect
18. C22⋊C8×C101C22semidirect
24. C2×D404D4semidirect
36. C4⋊Dic101D4semidirect
49. C23×C101D8semidirect
114. C101#1595semidirect
119. #323D101semidirect
131. C22⋊C8C2×D101semidirect
174. C23D808semidirect

Maximal subgroups

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

subgroupindex
#202657392
#202653132
#202650862
#202649122
#323×C1012
#1595101