Finite Group Databasebeta

← #27170117

Subgroups — #27170117

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#20265490C212yesmaximal, complemented
3#20264910C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic, complemented
5#397×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101—21no—
7#15692239—22no—
8#15692229—21no—
9#15692229C2212yes—
10#15692225—22no—
11#15692188—22no—
12#15692175—22no—
13C22⋊#11717579C2212yescomplemented
14C2×C4⋊C4×C101C2211yescharacteristic, complemented
15C2×C22⋊C4×C101—21no—
16C2×C22⋊C4×C101C2211yescharacteristic
17#95×C101—21no—
18#95×C101C2211yescharacteristic, complemented
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—
27C2×#11717579—22no—
28C4⋊Dic101—21no—
29C4⋊Dic101D412yes—
30#13108051—22no—
31#13108051—22no—
32C24×C101D411yescharacteristic
33C22×C4×C101—21no—
34C22×C4×C101—21no—
35C22×C4×C101D411yescharacteristic
36C22×C4×C101C2311yescharacteristic
37C4⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—42no—
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×C101—41no—
54C2×C4×C101—41no—
55C2×C4×C101—41no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101C2×D412yes—
60C22⋊D101—44no—
61C22⋊D101—42no—
62C22⋊D101—42no—
63C2×Dic101—22no—
64C2×Dic101—42no—
65D404—42no—
66D404—81no—
67#11717579—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—
84C4×C101—81no—
85C4×C101—81no—
86Dic101—42no—
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#397D10111yescharacteristic, complemented
99#327—1012no—
100#323—1011no—
101C22×D4—2021no—
102#123—2022no—
103#121—1012no—
104#121—2021no—
105#120—2022no—
106C2×C4⋊C4C2×D10111yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D10111yescharacteristic
109#107—2022no—
110#102—2022no—
111C22⋊C8—1012no—
112#95—21no—
113#95C2×D10111yescharacteristic, complemented
114C24—2021no—
115C24C22⋊D10111yescharacteristic
116C2×D4—1012no—
117C2×D4—4042no—
118C2×D4—4042no—
119C2×D4—2021no—
120C2×D4—2022no—
121C2×D4—2022no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—2021no—
125C22×C4C22⋊D10111yescharacteristic
126C22×C4C22×D10111yescharacteristic
127C2×C8—2022no—
128C4⋊C4—22no—
129C4⋊C4—1012no—
130C4⋊C4—2021no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—2022no—
134C22⋊C4—2022no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—8081no—
140C23—8081no—
141C23—4042no—
142C23—2022no—
143C23#1310805912yes—
144C23C2×C22⋊D10111yescharacteristic
145D4—8081no—
146D4—4042no—
147D4—4044no—
148D4—4042no—
149D4—4042no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—4042no—
160C2×C4—2022no—
161C2×C4D4×D10112yes—
162C8—4042no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—8084no—
171C22—4042no—
172C22—4042no—
173C22—4042no—
174C22#1569222311yescharacteristic
175C22C2×#1310805911yescharacteristic
176C22#1569228011yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—4042no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—8082no—
186C2#2026531011yescharacteristic
187C2#2026535411yescharacteristic
188C2#2026536611yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#397×C101C2
socleC22×C101#120
prime radical#20265739C2
hypercenter#397D101
hypocenterC101#1607
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #20265490C2semidirect
3. #20264910C2semidirect
4. #20264901C2semidirect
5. #397×C101C2semidirect
13. C22⋊#11717579C22semidirect
14. C2×C4⋊C4×C101C22semidirect
18. #95×C101C22semidirect
95. C101#1607semidirect
98. #397D101semidirect
106. C2×C4⋊C4C2×D101semidirect
113. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657392
#202654902
#202649102
#202649012
#397×C1012
#1607101