Finite Group Databasebeta

← #27186409

Subgroups — #27186409

Order 12928.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#20266148C211yesmaximal, characteristic, complemented
2D101⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
3#20266021C214yesmaximal, complemented
4C2×#15692230C211yesmaximal, characteristic
5C2×#13108084—21no—
6C2×D101⋊C8—21no—
7#15692312—21no—
8#15692312—21no—
9C22⋊#11717581C2212yes—
10C22⋊#11717581C2212yes—
11C22×D404C412yes—
12C22×D404C411yescharacteristic
13C22×C4×D101C2211yescharacteristic
14#15692230—21no—
15#15692230—21no—
16#15692230—21no—
17#15692230—21no—
18C2×#13108051C2212yes—
19C2×C4⋊C4×C101C411yescharacteristic
20#13108084—42no—
21D101⋊C8—42no—
22C2×#11717581—22no—
23C2×#11717581—22no—
24C23×D101C2×C411yescharacteristic
25C23×D101C2×C412yes—
26C2×D404—22no—
27C2×D404—22no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—42no—
31C2×D404—44no—
32C22×Dic101C2311yescharacteristic
33C2×C4×D101—42no—
34C2×C4×D101D411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38#13108051—22no—
39#13108051—22no—
40C22×C4×C101C2×C412yes—
41C22×C4×C101C2×C411yescharacteristic
42C4⋊C4×C101—21no—
43C4⋊C4×C101—21no—
44C23×C101C22×C411yescharacteristic
45C22×D101—22no—
46C22×D101—22no—
47C22×D101—22no—
48C22×D101—24no—
49C22×D101—41no—
50C22×D101—42no—
51C22×D101—42no—
52C22×D101C22⋊C411yescharacteristic
53C22×D101C22⋊C411yescharacteristic
54C2×C4×C101—22no—
55C2×C4×C101—22no—
56C2×C4×C101—42no—
57C2×C4×C101C22⋊C411yescharacteristic
58C2×C4×C101C22⋊C411yescharacteristic
59C2×Dic101—41no—
60C2×Dic101C2×D411yescharacteristic
61C2×Dic101C2×D411yescharacteristic
62D404—42no—
63D404—44no—
64D404—44no—
65D404—42no—
66C4×D101—22no—
67C4×D101—22no—
68C4×D101—41no—
69C4×D101—41no—
70#11717581—42no—
71#11717581—42no—
72C22×C101—21no—
73C22×C101—21no—
74C22×C101C2×C22⋊C411yescharacteristic
75C22×C101#13112yes—
76C2×D101—21no—
77C2×D101—21no—
78C2×D101—21no—
79C2×D101—21no—
80C2×D101—21no—
81C2×D101—44no—
82C2×D101—42no—
83C2×D101—42no—
84C2×D101—42no—
85C2×D101—84no—
86C2×D101#10012yes—
87C4×C101—21no—
88C4×C101—42no—
89C4×C101#9912yes—
90Dic101—21no—
91Dic101—21no—
92C2×C101—22no—
93C2×C101#40811yescharacteristic
94C2×C101C2×#9911yescharacteristic
95C2×C101C2×#10011yescharacteristic
96D101—22no—
97D101—41no—
98D101—82no—
99#1471—1011nomaximal, Sylow
100C101#147111yescharacteristic, complemented, Sylow
101C2×#121—1011no—
102#407—1011no—
103C2×C22⋊C8—1011no—
104#323—1014no—
105C22×D4—1012no—
106C22×D4—1011no—
107C23×C4—1011no—
108C2×M16—2021no—
109C22×C8—2021no—
110#121—2021no—
111#121—2021no—
112#121—2021no—
113#121—2021no—
114C2×C4⋊C4#8793811yescharacteristic
115C2×C22⋊C4—1012no—
116C22⋊C8—1012no—
117C22⋊C8—1012no—
118C22⋊C8—2021no—
119C22⋊C8—2021no—
120C24—1011no—
121C24—1012no—
122C2×D4—4042no—
123C2×D4—4044no—
124C2×D4—2022no—
125C2×D4—2022no—
126C2×D4—2021no—
127C2×D4—2021no—
128C22×C4—1011no—
129C22×C4—1011no—
130C22×C4—1011no—
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—4042no—
134C22×C4C2×#8793812yes—
135C22×C4C2×#8793811yescharacteristic
136M16—4042no—
137C2×C8—4042no—
138C2×C8—2022no—
139C2×C8—2022no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C22⋊C4—2022no—
143C22⋊C4—2022no—
144C23—1011no—
145C23—1011no—
146C23—4042no—
147C23—4042no—
148C23—4041no—
149C23—2022no—
150C23—2024no—
151C23—2022no—
152C23—2022no—
153C23C22×#8793811yescharacteristic
154D4—4042no—
155D4—4044no—
156D4—4044no—
157D4—4042no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—1011no—
162C2×C4—1011no—
163C2×C4—4041no—
164C2×C4—4041no—
165C2×C4—4041no—
166C2×C4—2022no—
167C2×C4—2022no—
168C2×C4C22⋊#8793811yescharacteristic
169C2×C4C22⋊#8793811yescharacteristic
170C8—4042no—
171C8—4042no—
172C22—21no—
173C22—21no—
174C22—8084no—
175C22—1012no—
176C22—4044no—
177C22—4042no—
178C22—4042no—
179C22—4042no—
180C22—2021no—
181C22—2021no—
182C22—2021no—
183C22—2021no—
184C22—2021no—
185C22C2×C22⋊#8793811yescharacteristic
186C22#1569234312yes—
187C4—21no—
188C4—42no—
189C4—2021no—
190C4—2021no—
191C4#1569231412yes—
192C2—22no—
193C2—8082no—
194C2—4041no—
195C2—2022no—
196C2#2026615211yescharacteristic
197C2C2×#1569231411yescharacteristic
198C2D101⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC2×C4⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×#15692230C2
cosocleC22×Dic101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1471
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. D101⋊(C22⋊C8)C2semidirect
3. #20266021C2semidirect
100. C101#1471semidirect

Maximal subgroups

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

subgroupindex
#202661482
D101⋊(C22⋊C8)2
#202660212
C2×#156922302
#1471101