Finite Group Databasebeta

← #27173728

Subgroups — #27173728

Order 12928.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×C22⋊Dic101C211yesmaximal, characteristic
2#20265071C214yesmaximal, complemented
3#20265065C214yesmaximal, complemented
4#20265035C212yesmaximal, complemented
5#20265033C211yesmaximal, characteristic
6C22⋊C4×Dic101C212yesmaximal, complemented
7#379×C101C211yesmaximal, characteristic
8C23×Dic101C2211yescharacteristic
9C23×Dic101C2211yescharacteristic
10C2×#13108053C2214yescomplemented
11C2×C22⋊Dic101C418yescomplemented
12C2×C22⋊Dic101C2212yes—
13C2×C22⋊Dic101C2212yes—
14C2×C22⋊Dic101C2212yes—
15C2×C22⋊Dic101C2212yes—
16C2×C22⋊Dic101C2212yes—
17C2×C4×Dic101C2214yescomplemented
18#15692162C2214yescomplemented
19#15692162C2214yescomplemented
20C25×C101C2211yescharacteristic
21C2×C22⋊C4×C101C2212yes—
22C2×C22⋊C4×C101C2212yes—
23C2×C22⋊C4×C101C2212yes—
24C22×Dic101—42no—
25C22×Dic101—42no—
26C22×Dic101C2×C414yes—
27C22×Dic101C2×C414yes—
28C22×Dic101C2314yes—
29C22×Dic101C2312yes—
30C22×Dic101C2312yes—
31#13108053—44no—
32C22⋊Dic101—24no—
33C22⋊Dic101—42no—
34C22⋊Dic101—42no—
35C22⋊Dic101C2×C418yes—
36C22⋊Dic101C2×C418yes—
37C4×Dic101—44no—
38C24×C101—24no—
39C24×C101—42no—
40C24×C101—42no—
41C24×C101C2×C414yes—
42C24×C101C2311yescharacteristic
43C24×C101C2311yescharacteristic
44C24×C101C2311yescharacteristic
45C22×C4×C101C2314yes—
46C22⋊C4×C101—24no—
47C22⋊C4×C101—42no—
48C22⋊C4×C101—42no—
49C23×C101—22no—
50C23×C101—22no—
51C23×C101—28no—
52C23×C101—22no—
53C23×C101—22no—
54C23×C101—22no—
55C23×C101—22no—
56C23×C101—22no—
57C23×C101—22no—
58C23×C101—44no—
59C23×C101—44no—
60C23×C101—416no—
61C23×C101C22×C412yes—
62C23×C101C22×C412yes—
63C23×C101C22×C412yes—
64C23×C101#4114yes—
65C23×C101C2411yescharacteristic
66C2×C4×C101—24no—
67C2×C4×C101—24no—
68C2×C4×C101—24no—
69C2×Dic101—24no—
70C2×Dic101—24no—
71C2×Dic101—24no—
72C2×Dic101—24no—
73C2×Dic101—24no—
74C2×Dic101—44no—
75C2×Dic101—44no—
76C2×Dic101C22×C414yes—
77C2×Dic101C22×C414yes—
78C22×C101—22no—
79C22×C101—22no—
80C22×C101—28no—
81C22×C101—22no—
82C22×C101—22no—
83C22×C101—22no—
84C22×C101—22no—
85C22×C101—22no—
86C22×C101—22no—
87C22×C101—416no—
88C22×C101—44no—
89C22×C101—44no—
90C22×C101#11714yes—
91C22×C101C23×C411yescharacteristic
92C22×C101C2×#4112yes—
93C22×C101H(5,𝔽2)12yes—
94C22×C101H(5,𝔽2)12yes—
95C4×C101—44no—
96Dic101—24no—
97Dic101—24no—
98Dic101—44no—
99C2×C101—24no—
100C2×C101—42no—
101C2×C101—42no—
102C2×C101C2×#11711yescharacteristic
103C2×C101#51811yescharacteristic
104C2×C101#51811yescharacteristic
105C2×C101#53814yes—
106#2312—1011nomaximal, Sylow
107C101#231211yescharacteristic, complemented, Sylow
108C22×C22⋊C4—1011no—
109#388—1014no—
110#388—1014no—
111#380—1012no—
112#379—1011no—
113#379D10111yescharacteristic
114C4×C22⋊C4—1012no—
115C25C2×D10111yescharacteristic
116C23×C4—1011no—
117C23×C4—1011no—
118C2×C4⋊C4—1014no—
119C2×C22⋊C4—1018no—
120C2×C22⋊C4—1012no—
121C2×C22⋊C4—1012no—
122C2×C22⋊C4—1012no—
123C2×C22⋊C4—1012no—
124C2×C22⋊C4—1012no—
125C2×C22⋊C4C2×D10112yes—
126C2×C22⋊C4C2×D10112yes—
127C2×C22⋊C4C2×D10112yes—
128C2×C42—1014no—
129#95—1014no—
130#95—1014no—
131C24—24no—
132C24—42no—
133C24—42no—
134C24C4×D10114yes—
135C24C22×D10111yescharacteristic
136C24C22×D10111yescharacteristic
137C24C22×D10111yescharacteristic
138C22×C4—1014no—
139C22×C4—1014no—
140C22×C4—1012no—
141C22×C4—1012no—
142C22×C4—1014no—
143C22×C4—4042no—
144C22×C4—4042no—
145C22×C4C22×D10114yes—
146C4⋊C4—4044no—
147C22⋊C4—24no—
148C22⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—1018no—
151C22⋊C4—1018no—
152C22⋊C4—4042no—
153C22⋊C4—4042no—
154C22⋊C4—2024no—
155C42—4044no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—28no—
165C23—44no—
166C23—44no—
167C23—416no—
168C23C2×C4×D10112yes—
169C23C2×C4×D10112yes—
170C23C2×C4×D10112yes—
171C23#1310807314yes—
172C23C23×D10111yescharacteristic
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—1014no—
177C2×C4—1014no—
178C2×C4—4044no—
179C2×C4—4044no—
180C2×C4—2024no—
181C2×C4—2024no—
182C2×C4—2024no—
183C2×C4—2024no—
184C2×C4—2024no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—22no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—416no—
195C22—44no—
196C22—44no—
197C22#1569221114yes—
198C22C22×C4×D10111yescharacteristic
199C22C2×#1310807312yes—
200C22#1569230012yes—
201C22#1569230012yes—
202C4—44no—
203C4—4044no—
204C4—2024no—
205C4—2024no—
206C2—24no—
207C2—42no—
208C2—42no—
209C2C2×#1569221111yescharacteristic
210C2#2026565011yescharacteristic
211C2#2026565011yescharacteristic
212C2#2026576214yes—
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#379×C101C2
socle, cosocleC23×C101C24
prime radicalC25×C101C22
hypercenter#379D101
hypocenterC101#2312
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #20265071C2semidirect
3. #20265065C2semidirect
4. #20265035C2semidirect
6. C22⋊C4×Dic101C2semidirect
10. C2×#13108053C22semidirect
11. C2×C22⋊Dic101C4semidirect
17. C2×C4×Dic101C22semidirect
18. #15692162C22semidirect
19. #15692162C22semidirect
107. C101#2312semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic1012
#202650712
#202650652
#202650352
#202650332
C22⋊C4×Dic1012
#379×C1012
#2312101