Finite Group Databasebeta

← #27170050

Subgroups — #27170050

Order 12928.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265310C212yesmaximal, complemented
2#20265086C211yesmaximal, characteristic, complemented
3#20264911C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic, complemented
5#534×C101C211yesmaximal, characteristic, complemented
6C22×D404C2211yescharacteristic, complemented
7C22×C22⋊D101—21no—
8C2×#13108059—22no—
9C2×#13108051—21no—
10#15692175—22no—
11C22⋊#11717579C2212yescomplemented
12#15692162C2211yescharacteristic, complemented
13C22×D4×C101—21no—
14#123×C101—22no—
15#121×C101—21no—
16#121×C101C2212yes—
17#120×C101—22no—
18C2×C22⋊C4×C101C2211yescharacteristic
19C23×D101—21no—
20C2×D404—22no—
21C2×D404—42no—
22C2×C22⋊D101—42no—
23C2×C22⋊D101—42no—
24C2×C22⋊D101—42no—
25C22×Dic101—21no—
26#13108059—44no—
27C2×#11717579—22no—
28#13108051—42no—
29C24×C101—21no—
30C24×C101D411yescharacteristic
31C2×D4×C101—22no—
32C2×D4×C101—22no—
33C2×D4×C101—21no—
34C2×D4×C101—42no—
35C2×D4×C101—42no—
36C2×D4×C101D412yescomplemented
37C22×C4×C101—21no—
38C22×C4×C101D411yescharacteristic
39C22×C4×C101C2311yescharacteristic
40C4⋊C4×C101—21no—
41C4⋊C4×C101D412yescomplemented
42C22⋊C4×C101—22no—
43C22⋊C4×C101—22no—
44C22⋊C4×C101—22no—
45C23×C101—21no—
46C23×C101—21no—
47C23×C101—22no—
48C23×C101—41no—
49C23×C101—41no—
50C23×C101—42no—
51C23×C101—81no—
52C23×C101—81no—
53C23×C101D812yescomplemented
54C23×C101C2×D411yescharacteristic
55C22×D101—22no—
56C22×D101—41no—
57C22×D101—41no—
58C22×D101—41no—
59C22×D101—41no—
60C22×D101—41no—
61C22×D101—41no—
62D4×C101—42no—
63D4×C101—44no—
64D4×C101—42no—
65D4×C101—42no—
66D4×C101—81no—
67C2×C4×C101—21no—
68C2×C4×C101—22no—
69C2×C4×C101—41no—
70C2×C4×C101—42no—
71C2×C4×C101—41no—
72C2×C4×C101C2×D412yes—
73C22⋊D101—48no—
74C2×Dic101—22no—
75C2×Dic101—41no—
76C2×Dic101—41no—
77D404—24no—
78D404—42no—
79#11717579—42no—
80C22×C101—21no—
81C22×C101—21no—
82C22×C101—22no—
83C22×C101—42no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101—42no—
87C22×C101—42no—
88C22×C101—42no—
89C22×C101—81no—
90C22×C101—84no—
91C22×C101#12011yescharacteristic
92C22×C101C2×D811yescharacteristic
93C22×C101AGL(1,ℤ/8)11yescharacteristic
94C2×D101—42no—
95C2×D101—42no—
96C2×D101—42no—
97C2×D101—81no—
98C2×D101—81no—
99C2×D101—81no—
100C2×D101—81no—
101C4×C101—22no—
102C4×C101—42no—
103C4×C101—41no—
104Dic101—42no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—42no—
108C2×C101—82no—
109C2×C101#44711yescharacteristic
110C2×C101#45311yescharacteristic
111C2×C101B(4,𝔽2)11yescharacteristic
112D101—82no—
113#1595—1011nomaximal, Sylow
114C101#159511yescharacteristic, complemented, Sylow
115#534D10111yescharacteristic, complemented
116#447—1012no—
117#392—1011no—
118#327—1012no—
119#323—1011no—
120C22×D4—21no—
121C22×D4—1011no—
122C22×D4—2021no—
123C2×D8—2022no—
124#123—22no—
125#121—21no—
126#121C2×D10112yes—
127#120—22no—
128C2×C22⋊C4—2021no—
129C2×C22⋊C4C2×D10111yescharacteristic
130#102—2022no—
131C22⋊C8—1012no—
132#95—1011no—
133C24—21no—
134C24—2021no—
135C24C22⋊D10111yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—4042no—
142C2×D4—4042no—
143C2×D4—4042no—
144C2×D4—4042no—
145C2×D4—2022no—
146C2×D4C22⋊D10112yescomplemented
147C22×C4—21no—
148C22×C4—2021no—
149C22×C4C22⋊D10111yescharacteristic
150C22×C4C22×D10111yescharacteristic
151D8—4044no—
152C2×C8—2022no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D10112yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—4042no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—41no—
163C23—41no—
164C23—42no—
165C23—81no—
166C23—81no—
167C23—4041no—
168C23—4041no—
169C23—4041no—
170C23—4041no—
171C23—4041no—
172C23—4041no—
173C23—2022no—
174C23#1310805912yescomplemented
175C23C2×C22⋊D10111yescharacteristic
176D4—42no—
177D4—44no—
178D4—42no—
179D4—42no—
180D4—81no—
181D4—4042no—
182D4—4048no—
183D4—2024no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—4041no—
190C2×C4—4041no—
191C2×C4—2022no—
192C2×C4C2×C22⋊D10112yes—
193C8—4042no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—84no—
205C22—8081no—
206C22—8081no—
207C22—8081no—
208C22—8081no—
209C22—4042no—
210C22—4042no—
211C22—4042no—
212C22#1569222411yescharacteristic
213C22C2×#1310805911yescharacteristic
214C22#1569228111yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—4042no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—8082no—
224C2#2026531111yescharacteristic
225C2#2026534511yescharacteristic
226C2#2026536411yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#534×C101C2
socleC22×C101#120
hypercenter#534D101
hypocenterC101#1595
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20265310C2semidirect
2. #20265086C2semidirect
3. #20264911C2semidirect
4. #20264901C2semidirect
5. #534×C101C2semidirect
6. C22×D404C22semidirect
11. C22⋊#11717579C22semidirect
12. #15692162C22semidirect
36. C2×D4×C101D4semidirect
41. C4⋊C4×C101D4semidirect
53. C23×C101D8semidirect
114. C101#1595semidirect
115. #534D101semidirect
146. C2×D4C22⋊D101semidirect
154. C4⋊C4C22⋊D101semidirect
174. C23#13108059semidirect

Maximal subgroups

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

subgroupindex
#202653102
#202650862
#202649112
#202649012
#534×C1012
#1595101