Finite Group Databasebeta

← #27170120

Subgroups — #27170120

Order 12928.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265739C211yesmaximal, characteristic
2#20265491C212yesmaximal, complemented
3#20265102C211yesmaximal, characteristic, complemented
4#20264912C212yesmaximal, complemented
5#323×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101—21no—
7#15692239—22no—
8#15692229—21no—
9#15692229C2212yes—
10#15692225—22no—
11C2×C4⋊Dic101C2211yescharacteristic, complemented
12C2×C22⋊Dic101—21no—
13#15692189—22no—
14#15692177—22no—
15#15692162—21no—
16#15692162C2211yescharacteristic, complemented
17C2×C22⋊C4×C101C2211yescharacteristic
18C22⋊C8×C101C2212yescomplemented
19C23×D101—21no—
20C2×D404—21no—
21C2×D404D412yes—
22C2×C22⋊D101—22no—
23C2×C22⋊D101—22no—
24C2×C22⋊D101—42no—
25C2×C22⋊D101—42no—
26C22×Dic101—21no—
27C22×Dic101—21no—
28C22×Dic101—21no—
29C4⋊Dic101—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101D412yes—
32C22⋊Dic101—42no—
33#13108051—22no—
34#13108051—22no—
35C24×C101D411yescharacteristic
36C22×C4×C101D411yescharacteristic
37C22×C4×C101C2311yescharacteristic
38C2×C8×C101—22no—
39C22⋊C4×C101—22no—
40C23×C101—21no—
41C23×C101—21no—
42C23×C101—41no—
43C23×C101—41no—
44C23×C101D812yes—
45C23×C101C2×D411yescharacteristic
46C22×D101—22no—
47C22×D101—42no—
48C22×D101—81no—
49C22×D101—81no—
50C2×C4×C101—21no—
51C2×C4×C101—41no—
52C2×C4×C101—41no—
53C2×C4×C101C2×D412yes—
54C22⋊D101—44no—
55C22⋊D101—42no—
56C22⋊D101—42no—
57C2×Dic101—22no—
58C2×Dic101—42no—
59C2×Dic101—41no—
60C2×Dic101—41no—
61C2×Dic101—41no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65D404—42no—
66D404—81no—
67C8×C101—42no—
68C22×C101—21no—
69C22×C101—21no—
70C22×C101—22no—
71C22×C101—42no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—81no—
75C22×C101#12011yescharacteristic
76C22×C101C2×D811yescharacteristic
77C22×C101AGL(1,ℤ/8)11yescharacteristic
78C2×D101—42no—
79C2×D101—42no—
80C2×D101—42no—
81C2×D101—84no—
82C4×C101—22no—
83C4×C101—41no—
84Dic101—42no—
85Dic101—81no—
86Dic101—81no—
87C2×C101—21no—
88C2×C101—21no—
89C2×C101—42no—
90C2×C101#44711yescharacteristic
91C2×C101#45511yescharacteristic
92C2×C101#45811yescharacteristic
93D101—82no—
94#1607—1011nomaximal, Sylow
95C101#160711yescharacteristic, complemented, Sylow
96#534—1011no—
97#480—1012no—
98#397—1011no—
99#327—1012no—
100#323D10111yescharacteristic, complemented
101C22×D4—2021no—
102#123—2022no—
103#121—1012no—
104#121—2021no—
105#120—2022no—
106C2×C4⋊C4—1011no—
107C2×C22⋊C4—2021no—
108C2×C22⋊C4C2×D10111yescharacteristic
109#107—2022no—
110#102—2022no—
111C22⋊C8C2×D10112yescomplemented
112#95—1011no—
113#95—2021no—
114C24—2021no—
115C24D40411yescharacteristic
116C2×D4—1012no—
117C2×D4—4042no—
118C2×D4—4042no—
119C2×D4—2021no—
120C2×D4—2022no—
121C2×D4—2022no—
122C22×C4—2021no—
123C22×C4—2021no—
124C22×C4—2021no—
125C22×C4D40411yescharacteristic
126C22×C4C22×D10111yescharacteristic
127C2×C8—22no—
128C4⋊C4—1012no—
129C4⋊C4—2021no—
130C4⋊C4—2022no—
131C22⋊C4—22no—
132C22⋊C4—4042no—
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—8081no—
140C23—8081no—
141C23—4042no—
142C23—2022no—
143C23D80812yes—
144C23C2×D40411yescharacteristic
145D4—8081no—
146D4—4042no—
147D4—4044no—
148D4—4042no—
149D4—4042no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—4041no—
154C2×C4—4041no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—4041no—
159C2×C4—4042no—
160C2×C4—2022no—
161C2×C4D4×D10112yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—8084no—
171C22—4042no—
172C22—4042no—
173C22—4042no—
174C22#1569222511yescharacteristic
175C22C2×D80811yescharacteristic
176C22#1569228311yescharacteristic
177C4—22no—
178C4—41no—
179C4—8081no—
180C4—8081no—
181C4—4042no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—8082no—
186C2#2026531311yescharacteristic
187C2#2026535711yescharacteristic
188C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#323×C101C2
socleC22×C101#120
prime radical#20265739C2
hypercenter#323D101
hypocenterC101#1607
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20265491C2semidirect
3. #20265102C2semidirect
4. #20264912C2semidirect
5. #323×C101C2semidirect
11. C2×C4⋊Dic101C22semidirect
16. #15692162C22semidirect
18. C22⋊C8×C101C22semidirect
95. C101#1607semidirect
100. #323D101semidirect
111. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657392
#202654912
#202651022
#202649122
#323×C1012
#1607101