Finite Group Databasebeta

← #27170118

Subgroups — #27170118

Order 12928.

280 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265488C212yesmaximal, complemented
2#20265102C211yesmaximal, characteristic, complemented
3#20264911C212yesmaximal, complemented
4#20264901C211yesmaximal, characteristic, complemented
5#534×C101C211yesmaximal, characteristic
6C2×C4⋊Dic101C2211yescharacteristic, complemented
7C2×C22⋊Dic101—21no—
8#15692188—22no—
9#15692176—22no—
10C22⋊#11717579C2212yescomplemented
11#15692162—21no—
12#15692162C2211yescharacteristic, complemented
13C22×D4×C101—21no—
14#123×C101—22no—
15#121×C101—21no—
16#121×C101C2212yes—
17#120×C101—22no—
18C2×C22⋊C4×C101C2211yescharacteristic
19C22×Dic101—21no—
20C22×Dic101—21no—
21C2×#11717579—22no—
22C4⋊Dic101—22no—
23C22⋊Dic101—42no—
24C24×C101—21no—
25C24×C101D411yescharacteristic
26C2×D4×C101—22no—
27C2×D4×C101—22no—
28C2×D4×C101—21no—
29C2×D4×C101—42no—
30C2×D4×C101—42no—
31C2×D4×C101D412yes—
32C22×C4×C101—21no—
33C22×C4×C101D411yescharacteristic
34C22×C4×C101C2311yescharacteristic
35C4⋊C4×C101—21no—
36C4⋊C4×C101D412yes—
37C22⋊C4×C101—22no—
38C22⋊C4×C101—22no—
39C22⋊C4×C101—22no—
40C23×C101—21no—
41C23×C101—21no—
42C23×C101—22no—
43C23×C101—41no—
44C23×C101—41no—
45C23×C101—42no—
46C23×C101—81no—
47C23×C101—81no—
48C23×C101D812yes—
49C23×C101C2×D411yescharacteristic
50D4×C101—42no—
51D4×C101—44no—
52D4×C101—42no—
53D4×C101—42no—
54D4×C101—81no—
55C2×C4×C101—21no—
56C2×C4×C101—22no—
57C2×C4×C101—41no—
58C2×C4×C101—42no—
59C2×C4×C101—41no—
60C2×C4×C101C2×D412yes—
61C2×Dic101—41no—
62C2×Dic101—41no—
63C2×Dic101—41no—
64C2×Dic101—41no—
65C2×Dic101—41no—
66C2×Dic101—41no—
67#11717579—42no—
68C22×C101—21no—
69C22×C101—21no—
70C22×C101—22no—
71C22×C101—42no—
72C22×C101—41no—
73C22×C101—41no—
74C22×C101—42no—
75C22×C101—42no—
76C22×C101—42no—
77C22×C101—81no—
78C22×C101—84no—
79C22×C101#12011yescharacteristic
80C22×C101C2×D811yescharacteristic
81C22×C101AGL(1,ℤ/8)11yescharacteristic
82C4×C101—22no—
83C4×C101—42no—
84C4×C101—41no—
85Dic101—81no—
86Dic101—81no—
87C2×C101—21no—
88C2×C101—21no—
89C2×C101—42no—
90C2×C101—82no—
91C2×C101#44711yescharacteristic
92C2×C101#45511yescharacteristic
93C2×C101#45811yescharacteristic
94#1607—1011nomaximal, Sylow
95C101#160711yescharacteristic, complemented, Sylow
96#534D10111yescharacteristic
97#480—1012no—
98#397—1011no—
99#327—1012no—
100#323—1011no—
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C2×D10112yes—
105#120—22no—
106C2×C4⋊C4—1011no—
107C2×C22⋊C4—2021no—
108C2×C22⋊C4C2×D10111yescharacteristic
109#107—2022no—
110#102—2022no—
111C22⋊C8—1012no—
112#95—1011no—
113#95—2021no—
114C24—21no—
115C24C22⋊D10111yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4C22⋊D10112yes—
122C22×C4—21no—
123C22×C4—2021no—
124C22×C4—2021no—
125C22×C4C22⋊D10111yescharacteristic
126C22×C4C22×D10111yescharacteristic
127C2×C8—2022no—
128C4⋊C4—21no—
129C4⋊C4—2022no—
130C4⋊C4C22⋊D10112yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—4042no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23#1310805912yes—
144C23C2×C22⋊D10111yescharacteristic
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—4041no—
156C2×C4—4041no—
157C2×C4—4041no—
158C2×C4—4041no—
159C2×C4—4041no—
160C2×C4—4041no—
161C2×C4C2×C22⋊D10112yes—
162C8—4042no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—84no—
174C22#1569222411yescharacteristic
175C22C2×#1310805911yescharacteristic
176C22#1569228111yescharacteristic
177C4—22no—
178C4—42no—
179C4—41no—
180C4—8081no—
181C4—8081no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#2026531111yescharacteristic
187C2#2026535511yescharacteristic
188C2#2026536711yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#534×C101C2
socleC22×C101#120
hypercenter#534D101
hypocenterC101#1607
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20265488C2semidirect
2. #20265102C2semidirect
3. #20264911C2semidirect
4. #20264901C2semidirect
6. C2×C4⋊Dic101C22semidirect
10. C22⋊#11717579C22semidirect
12. #15692162C22semidirect
95. C101#1607semidirect

Maximal subgroups

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

subgroupindex
#202654882
#202651022
#202649112
#202649012
#534×C1012
#1607101