Finite Group Databasebeta

← #27172261

Subgroups — #27172261

Order 12928.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#15692236C212yesmaximal, complemented
2C2×#15692170C212yesmaximal, complemented
3#20265038C211yesmaximal, characteristic, complemented
4#20264944C211yesmaximal, characteristic, complemented
5#534×C101C211yesmaximal, characteristic
6C23×Dic101C2211yescharacteristic
7#15692236—24no—
8#15692236—24no—
9C2×#13108053—22no—
10C2×C22⋊Dic101—22no—
11C2×C22⋊Dic101C2211yescharacteristic
12C2×C22⋊Dic101C2212yes—
13#15692170—24no—
14#15692162—22no—
15C22×D4×C101C2212yes—
16#123×C101—22no—
17#121×C101—22no—
18#120×C101—22no—
19C2×C22⋊C4×C101C2211yescharacteristic
20C22×Dic101—21no—
21C22×Dic101—22no—
22C22×Dic101—22no—
23C22×Dic101—22no—
24C22×Dic101—21no—
25C22×Dic101—21no—
26C22×Dic101D412yescomplemented
27#13108053—28no—
28C22⋊Dic101—28no—
29C22⋊Dic101—22no—
30C22⋊Dic101—22no—
31C22⋊Dic101D414yescomplemented
32C24×C101D412yes—
33C24×C101C2311yescharacteristic
34C2×D4×C101—22no—
35C2×D4×C101—22no—
36C2×D4×C101—22no—
37C2×D4×C101—22no—
38C2×D4×C101—22no—
39C2×D4×C101—22no—
40C2×D4×C101—22no—
41C22×C4×C101D412yes—
42C22×C4×C101D412yes—
43C4⋊C4×C101—22no—
44C22⋊C4×C101—24no—
45C22⋊C4×C101—22no—
46C23×C101—22no—
47C23×C101—22no—
48C23×C101—22no—
49C23×C101—42no—
50C23×C101—44no—
51C23×C101C2×D411yescharacteristic
52C23×C101C2×D412yes—
53C23×C101C2×D412yes—
54C23×C101C2×D411yescharacteristic
55C23×C101#4111yescharacteristic
56D4×C101—42no—
57D4×C101—42no—
58D4×C101—42no—
59D4×C101—42no—
60D4×C101—82no—
61C2×C4×C101—22no—
62C2×C4×C101—22no—
63C2×C4×C101—22no—
64C2×C4×C101—22no—
65C2×C4×C101—42no—
66C2×Dic101—24no—
67C2×Dic101—24no—
68C2×Dic101—22no—
69C2×Dic101—44no—
70C2×Dic101—41no—
71C2×Dic101—42no—
72C2×Dic101—41no—
73C2×Dic101—42no—
74C2×Dic101—41no—
75C22×C101—22no—
76C22×C101—24no—
77C22×C101—41no—
78C22×C101—42no—
79C22×C101—41no—
80C22×C101—44no—
81C22×C101—42no—
82C22×C101—44no—
83C22×C101—82no—
84C22×C101#12012yes—
85C22×C101#12011yescharacteristic
86C22×C101#12111yescharacteristic
87C22×C101#12112yes—
88C22×C101#12711yescharacteristic
89C4×C101—42no—
90C4×C101—42no—
91Dic101—44no—
92Dic101—42no—
93Dic101—81no—
94C2×C101—22no—
95C2×C101—21no—
96C2×C101—22no—
97C2×C101—21no—
98C2×C101—82no—
99C2×C101#39211yescharacteristic
100C2×C101#45812yes—
101#2022—1011nomaximal, Sylow
102C101#202211yescharacteristic, complemented, Sylow
103#534D10111yescharacteristic
104C2×#123—1012no—
105C2×#99—1012no—
106#381—1011no—
107#342—1011no—
108C22×D4C2×D10112yes—
109C23×C4—1011no—
110#123—22no—
111#123—2024no—
112#123—2024no—
113#121—22no—
114#120—22no—
115C2×C4⋊C4—2022no—
116C2×C22⋊C4—1012no—
117C2×C22⋊C4—1011no—
118C2×C22⋊C4—2022no—
119C2×C22⋊C4C2×D10111yescharacteristic
120#99—2024no—
121#95—2022no—
122C24C22⋊D10112yes—
123C24C22×D10111yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C22×C4—1012no—
132C22×C4—2021no—
133C22×C4—2022no—
134C22×C4—2021no—
135C22×C4—2022no—
136C22×C4—2022no—
137C22×C4—2021no—
138C22×C4C22⋊D10112yes—
139C22×C4C22⋊D10112yes—
140C4⋊C4—22no—
141C4⋊C4—2028no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—1014no—
145C22⋊C4—2022no—
146C22⋊C4—2028no—
147C22⋊C4—2022no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—42no—
152C23—44no—
153C23D4×D10111yescharacteristic
154C23D4×D10112yes—
155C23C2×C22⋊D10112yes—
156C23C2×C22⋊D10111yescharacteristic
157C23#1310807311yescharacteristic
158D4—42no—
159D4—42no—
160D4—42no—
161D4—42no—
162D4—82no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—4042no—
169C2×C4—4041no—
170C2×C4—4044no—
171C2×C4—4041no—
172C2×C4—4042no—
173C2×C4—4041no—
174C2×C4—2024no—
175C2×C4—2024no—
176C2×C4—2022no—
177C22—22no—
178C22—24no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—44no—
183C22—42no—
184C22—44no—
185C22—82no—
186C22#1569222312yes—
187C22#1569222411yescharacteristic
188C22#1569222611yescharacteristic
189C22#1569222712yes—
190C22#1569225111yescharacteristic
191C4—42no—
192C4—42no—
193C4—8081no—
194C4—4044no—
195C4—4042no—
196C2—21no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2#2026508511yescharacteristic
202C2#2026536712yes—
subgroupG/H
centerC22#15692224
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
Fitting, prime radical#534×C101C2
socleC22×C101#120
hypercenter#534D101
hypocenterC101#2022
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#15692236C2semidirect
2. C2×#15692170C2semidirect
3. #20265038C2semidirect
4. #20264944C2semidirect
26. C22×Dic101D4semidirect
31. C22⋊Dic101D4semidirect
102. C101#2022semidirect

Maximal subgroups

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

subgroupindex
C2×#156922362
C2×#156921702
#202650382
#202649442
#534×C1012
#2022101