Finite Group Databasebeta

← #27186640

Subgroups — #27186640

Order 12928.

344 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 70 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#20266195C211yesmaximal, characteristic, complemented
2#20266182C211yesmaximal, characteristic, complemented
3#20266035C214yesmaximal, complemented
4#127×D101C211yesmaximal, characteristic
5C2×#13108084—21no—
6#15692322—21no—
7#15692322C2212yescomplemented
8#15692322C2212yescomplemented
9#15692312—21no—
10C4×#11717581—21no—
11C2×D4×D101—21no—
12C2×D4×D101—21no—
13C2×D4×D101C412yes—
14#15692252C411yescharacteristic
15#15692251—21no—
16#15692251—21no—
17#15692251C2212yes—
18C42×D101C2211yescharacteristic
19#127×C101C411yescharacteristic
20#13108084—42no—
21C2×#11717581—22no—
22C2×#11717581—22no—
23C23×D101—22no—
24C2×D404—21no—
25C2×D404—21no—
26C2×D404C2×C412yes—
27C2×C22⋊D101—22no—
28C2×C22⋊D101—22no—
29C2×C22⋊D101—22no—
30D4×D101—44no—
31D4×D101—44no—
32D4×D101—42no—
33D4×D101—42no—
34C2×C4×D101C2×C411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C4×Dic101D411yescharacteristic
38C4×Dic101D411yescharacteristic
39C4×Dic101C2311yescharacteristic
40C2×D4×C101—21no—
41C2×D4×C101—21no—
42C2×D4×C101C2×C412yes—
43C42×C101C2×C411yescharacteristic
44C23×C101—22no—
45C22×D101—22no—
46C22×D101—44no—
47C22×D101—44no—
48C22×D101—44no—
49C22×D101C22⋊C411yescharacteristic
50D4×C101—42no—
51D4×C101—42no—
52D4×C101—42no—
53C2×C4×C101C22⋊C411yescharacteristic
54C2×C4×C101C22⋊C411yescharacteristic
55C2×C4×C101C22×C411yescharacteristic
56C22⋊D101—44no—
57C22⋊D101—42no—
58C22⋊D101—42no—
59C22⋊D101—44no—
60C2×Dic101C22⋊C411yescharacteristic
61C2×Dic101C2×D411yescharacteristic
62C2×Dic101C2×D411yescharacteristic
63D404—42no—
64D404—42no—
65D404—42no—
66C4×D101—22no—
67C4×D101—22no—
68C4×D101D812yes—
69C4×D101SD1612yes—
70#11717581—42no—
71#11717581—42no—
72C22×C101—42no—
73C22×C101—42no—
74C22×C101—42no—
75C22×C101C2×C22⋊C411yescharacteristic
76C2×D101—21no—
77C2×D101—21no—
78C2×D101—42no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—88no—
82C2×D101#10212yes—
83C4×C101—21no—
84C4×C101—21no—
85C4×C101#10212yes—
86Dic101#10012yes—
87Dic101C2×D811yescharacteristic
88Dic101C2×SD1611yescharacteristic
89Dic101AGL(1,ℤ/8)12yes—
90C2×C101—82no—
91C2×C101C2×#10011yescharacteristic
92C2×C101C2×#10211yescharacteristic
93C2×C101#41811yescharacteristic
94D101—22no—
95D101—82no—
96#1545—1011nomaximal, Sylow
97C101#154511yescharacteristic, complemented, Sylow
98C2×#127—1011no—
99#431—1011no—
100#423—1011no—
101#331—1014no—
102C22×D4—1012no—
103C22×D4—2021no—
104C22×D4—2021no—
105C2×M16—2021no—
106#127—1012no—
107#127—1011no—
108#127—2021no—
109#127—2021no—
110#127#8793811yescharacteristic
111C2×C42—1011no—
112C4⋊C8—1012no—
113C4⋊C8—1012no—
114C4⋊C8—2021no—
115C22⋊C8—2021no—
116C4×C8—2021no—
117C24—2022no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—1012no—
121C2×D4—4042no—
122C2×D4—4042no—
123C2×D4—4044no—
124C2×D4—4044no—
125C2×D4—2022no—
126C2×D4—2022no—
127C2×D4—2022no—
128C2×D4—2021no—
129C2×D4—2021no—
130C2×D4C2×#8793812yes—
131C22×C4—1011no—
132C22×C4—1011no—
133C22×C4—1011no—
134M16—4042no—
135C2×C8—2022no—
136C2×C8—2022no—
137C42—1011no—
138C42—1011no—
139C42—1011no—
140C42C2×#8793811yescharacteristic
141C23—22no—
142C23—1011no—
143C23—4044no—
144C23—4044no—
145C23—4044no—
146C23—2022no—
147D4—42no—
148D4—42no—
149D4—42no—
150D4—4042no—
151D4—4042no—
152D4—4042no—
153D4—4042no—
154D4—4044no—
155D4—4044no—
156D4—4042no—
157C2×C4—1011no—
158C2×C4—1011no—
159C2×C4—1011no—
160C2×C4—1012no—
161C2×C4—1012no—
162C2×C4—2022no—
163C2×C4—2022no—
164C2×C4C22⋊#8793811yescharacteristic
165C2×C4C22⋊#8793811yescharacteristic
166C2×C4C22×#8793811yescharacteristic
167C8—4042no—
168C8—4042no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—8088no—
173C22—1012no—
174C22—4042no—
175C22—4042no—
176C22—4042no—
177C22—2021no—
178C22—2021no—
179C22C2×C22⋊#8793811yescharacteristic
180C4—21no—
181C4—21no—
182C4—1012no—
183C4—1011no—
184C4—1011no—
185C4—1012no—
186C4#1569231712yes—
187C2—82no—
188C2—8082no—
189C2—2022no—
190C2D101⋊#10011yescharacteristic
191C2C2×#1569231711yescharacteristic
192C2#2026616911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#127×C101C4
socleC22×C101C2×C22⋊C4
prime radical#127×D101C2
cosocleC4×Dic101C23
hypercenter#127#87938
hypocenterC101#1545
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20266195C2semidirect
2. #20266182C2semidirect
3. #20266035C2semidirect
7. #15692322C22semidirect
8. #15692322C22semidirect
97. C101#1545semidirect

Maximal subgroups

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

subgroupindex
#202661952
#202661822
#202660352
#127×D1012
#1545101