Finite Group Databasebeta

← #27186389

Subgroups — #27186389

Order 12928.

316 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 68 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#20266148C212yesmaximal, complemented
2#20266022C212yesmaximal, complemented
3#20266021C212yesmaximal, complemented
4C2×C4×D404C211yesmaximal, characteristic
5C2×#13108084—22no—
6#15692312—22no—
7C22⋊#11717581C2214yes—
8C22×D404C411yescharacteristic
9C22×C4×D101C411yescharacteristic
10C22×C4×D101C2211yescharacteristic
11C4×D404—22no—
12C4×D404—22no—
13C2×C4⋊Dic101C2211yescharacteristic
14C2×#13108051C411yescharacteristic
15C2×#13108051C2211yescharacteristic
16C2×C42×C101C411yescharacteristic
17#13108084—44no—
18C2×#11717581—24no—
19C23×D101C2×C411yescharacteristic
20C23×D101C2×C411yescharacteristic
21C2×D404—22no—
22C2×D404—22no—
23C2×D404—22no—
24C22×Dic101C2×C411yescharacteristic
25C22×Dic101C2311yescharacteristic
26C2×C4×D101—21no—
27C2×C4×D101—21no—
28C2×C4×D101—42no—
29C2×C4×D101—42no—
30C2×C4×D101D412yes—
31C2×C4×D101D412yes—
32C4⋊Dic101—22no—
33#13108051—22no—
34#13108051—22no—
35C22×C4×C101C2×C411yescharacteristic
36C22×C4×C101C2×C411yescharacteristic
37C22×C4×C101C2×C411yescharacteristic
38C42×C101—21no—
39C42×C101—21no—
40C23×C101C22×C411yescharacteristic
41C22×D101—21no—
42C22×D101—21no—
43C22×D101—21no—
44C22×D101—21no—
45C22×D101—41no—
46C22×D101—41no—
47C22×D101—41no—
48C22×D101—41no—
49C22×D101C22⋊C412yes—
50C22×D101M1612yescomplemented
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—22no—
54C2×C4×C101—21no—
55C2×C4×C101—41no—
56C2×C4×C101C22⋊C411yescharacteristic
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101M1612yescomplemented
59C2×Dic101—21no—
60C2×Dic101—41no—
61C2×Dic101—41no—
62C2×Dic101C2×D412yes—
63D404—28no—
64C4×D101—24no—
65C4×D101—42no—
66C4×D101—42no—
67C4×D101—42no—
68#11717581—44no—
69C22×C101—21no—
70C22×C101—21no—
71C22×C101C2×C22⋊C411yescharacteristic
72C22×C101C2×M1611yescharacteristic
73C22×C101C2×M1611yescharacteristic
74C2×D101—22no—
75C2×D101—22no—
76C2×D101—22no—
77C2×D101—22no—
78C2×D101—42no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—41no—
82C2×D101—41no—
83C2×D101—42no—
84C4×C101—22no—
85C4×C101—21no—
86C4×C101—41no—
87C4×C101#9912yes—
88Dic101—22no—
89Dic101—41no—
90C2×C101—21no—
91C2×C101—21no—
92C2×C101#40711yescharacteristic
93C2×C101C2×#9911yescharacteristic
94C2×C101#41311yescharacteristic
95D101—42no—
96D101—42no—
97#1465—1011nomaximal, Sylow
98C101#146511yescharacteristic, complemented, Sylow
99C2×C4×D4—1011no—
100#407—1012no—
101#324—1012no—
102#323—1012no—
103C22×D4—1011no—
104C23×C4—1011no—
105C23×C4—1011no—
106C2×M16—2022no—
107C4×D4—2022no—
108C4×D4—2022no—
109C2×C4⋊C4—1011no—
110C2×C22⋊C4—1011no—
111C2×C22⋊C4—1011no—
112C2×C42#8793811yescharacteristic
113C22⋊C8—1014no—
114C22⋊C8—2022no—
115C24—1011no—
116C24—1011no—
117C2×D4—2022no—
118C2×D4—2022no—
119C2×D4—2022no—
120C22×C4—1011no—
121C22×C4—1012no—
122C22×C4—1012no—
123C22×C4—1011no—
124C22×C4—4042no—
125C22×C4—4042no—
126C22×C4—2021no—
127C22×C4—2021no—
128C22×C4C2×#8793811yescharacteristic
129C22×C4C2×#8793811yescharacteristic
130C22×C4C2×#8793811yescharacteristic
131M16—4044no—
132C2×C8—2024no—
133C4⋊C4—2022no—
134C22⋊C4—2022no—
135C22⋊C4—2022no—
136C42—21no—
137C42—21no—
138C23—1012no—
139C23—1012no—
140C23—4041no—
141C23—4041no—
142C23—4041no—
143C23—4041no—
144C23—2021no—
145C23—2021no—
146C23—2021no—
147C23—2021no—
148C23C22×#8793811yescharacteristic
149D4—2028no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—1012no—
156C2×C4—4042no—
157C2×C4—4042no—
158C2×C4—4042no—
159C2×C4—4041no—
160C2×C4—4041no—
161C2×C4—2021no—
162C2×C4—2024no—
163C2×C4C22⋊#8793811yescharacteristic
164C2×C4C22⋊#8793811yescharacteristic
165C2×C4#1310808412yescomplemented
166C8—4044no—
167C22—21no—
168C22—21no—
169C22—4042no—
170C22—4042no—
171C22—4042no—
172C22—4042no—
173C22—4041no—
174C22—4041no—
175C22—2022no—
176C22—2022no—
177C22—2022no—
178C22—2022no—
179C22C2×C22⋊#8793811yescharacteristic
180C22D101⋊M1611yescharacteristic
181C22C2×#1310808411yescharacteristic
182C4—22no—
183C4—21no—
184C4—41no—
185C4—4041no—
186C4—2022no—
187C4#1569231412yes—
188C2—21no—
189C2—21no—
190C2—4042no—
191C2—4042no—
192C2#2026614811yescharacteristic
193C2C2×#1569231411yescharacteristic
194C2#2026616311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C42×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC22×D404C4
cosocleC22×Dic101C23
hypercenterC2×C42#87938
hypocenterC101#1465
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. #20266022C2semidirect
3. #20266021C2semidirect
50. C22×D101M16semidirect
58. C2×C4×C101M16semidirect
98. C101#1465semidirect
165. C2×C4#13108084semidirect

Maximal subgroups

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

subgroupindex
#202661482
#202660222
#202660212
C2×C4×D4042
#1465101