Finite Group Databasebeta

← #27170146

Subgroups — #27170146

Order 12928.

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

normal subgroups only

This group has 232 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#20265505C211yesmaximal, characteristic, complemented
2#20265488C211yesmaximal, characteristic, complemented
3#20265102C211yesmaximal, characteristic, complemented
4#20264914C211yesmaximal, characteristic, complemented
5#20264911C211yesmaximal, characteristic, complemented
6#20264901C211yesmaximal, characteristic, complemented
7#538×C101C211yesmaximal, characteristic
8C2×C4⋊Dic101C2211yescharacteristic
9C2×C22⋊Dic101—21no—
10#15692188—21no—
11#15692186—21no—
12#15692179—21no—
13#15692176—21no—
14C22⋊#11717579C2211yescharacteristic
15C22⋊#11717579C2211yescharacteristic
16#15692162—21no—
17#15692162C2211yescharacteristic
18#126×C101—21no—
19#124×C101—21no—
20#123×C101—21no—
21C22⋊Q8×C101C2211yescharacteristic
22#121×C101—21no—
23#121×C101C2211yescharacteristic
24#120×C101—21no—
25C4×D4×C101—21no—
26C2×C22⋊C4×C101C2211yescharacteristic
27C22×Dic101—21no—
28C22×Dic101—21no—
29C2×#11717579—21no—
30C2×#11717579—21no—
31C4⋊Dic101—21no—
32C4⋊Dic101—21no—
33C22⋊Dic101—42no—
34C24×C101D411yescharacteristic
35C2×Q8×C101D411yescharacteristic
36C2×D4×C101—21no—
37C2×D4×C101—21no—
38C2×D4×C101—21no—
39C2×D4×C101D411yescharacteristic
40C22×C4×C101—21no—
41C22×C4×C101D411yescharacteristic
42C22×C4×C101C2311yescharacteristic
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D411yescharacteristic
46C4⋊C4×C101D411yescharacteristic
47C22⋊C4×C101—21no—
48C22⋊C4×C101—21no—
49C22⋊C4×C101—21no—
50C22⋊C4×C101—21no—
51C22⋊C4×C101—21no—
52C22⋊C4×C101—21no—
53C22⋊C4×C101—21no—
54C42×C101—21no—
55C23×C101—21no—
56C23×C101—21no—
57C23×C101—21no—
58C23×C101—21no—
59C23×C101—41no—
60C23×C101—41no—
61C23×C101C2×D411yescharacteristic
62Q8×C101—41no—
63D4×C101—41no—
64D4×C101—41no—
65D4×C101—41no—
66D4×C101—81no—
67D4×C101—81no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—21no—
71C2×C4×C101—21no—
72C2×C4×C101—41no—
73C2×C4×C101—41no—
74C2×C4×C101—81no—
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83#11717579—41no—
84#11717579—41no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—42no—
88C22×C101—41no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101—81no—
95C22×C101#12011yescharacteristic
96C22×C101#13511yescharacteristic
97C22×C101AGL(1,ℤ/8)11yescharacteristic
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—41no—
101C4×C101—41no—
102C4×C101—41no—
103C4×C101—41no—
104Dic101—81no—
105Dic101—81no—
106C2×C101—21no—
107C2×C101—21no—
108C2×C101—81no—
109C2×C101—81no—
110C2×C101#44911yescharacteristic
111C2×C101#45511yescharacteristic
112C2×C101#45811yescharacteristic
113#1612—1011nomaximal, Sylow
114C101#161211yescharacteristic, complemented, Sylow
115#538D10111yescharacteristic
116#483—1011no—
117#480—1011no—
118#397—1011no—
119#328—1011no—
120#327—1011no—
121#323—1011no—
122#126—21no—
123#124—21no—
124#123—21no—
125C22⋊Q8C2×D10111yescharacteristic
126#121—21no—
127#121C2×D10111yescharacteristic
128#120—21no—
129C4×D4—21no—
130C2×C4⋊C4—1011no—
131C2×C22⋊C4—2021no—
132C2×C22⋊C4C2×D10111yescharacteristic
133#107—2021no—
134#106—2021no—
135#103—2021no—
136#102—2021no—
137C22⋊C8—1011no—
138C22⋊C8—1011no—
139#95—1011no—
140#95—2021no—
141C24C22⋊D10111yescharacteristic
142C2×Q8C22⋊D10111yescharacteristic
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C22⋊D10111yescharacteristic
147C22×C4—21no—
148C22×C4—2021no—
149C22×C4—2021no—
150C22×C4C22⋊D10111yescharacteristic
151C22×C4C22×D10111yescharacteristic
152C2×C8—2021no—
153C2×C8—2021no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2021no—
157C4⋊C4—2021no—
158C4⋊C4C22⋊D10111yescharacteristic
159C4⋊C4C22⋊D10111yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—4042no—
168C42—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊D10111yescharacteristic
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—4041no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—4041no—
195C2×C4C2×C22⋊D10111yescharacteristic
196C2×C4C2×C22⋊D10111yescharacteristic
197C8—4041no—
198C8—4041no—
199C22—21no—
200C22—21no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—81no—
209C22#1569222411yescharacteristic
210C22#1569227511yescharacteristic
211C22#1569228011yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—8081no—
219C4—8081no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—81no—
224C2#2026532211yescharacteristic
225C2#2026535511yescharacteristic
226C2#2026536711yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting, prime radical#538×C101C2
socleC22×C101#120
hypercenter#538D101
hypocenterC101#1612
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #20265505C2semidirect
2. #20265488C2semidirect
3. #20265102C2semidirect
4. #20264914C2semidirect
5. #20264911C2semidirect
6. #20264901C2semidirect
114. C101#1612semidirect

Maximal subgroups

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

subgroupindex
#202655052
#202654882
#202651022
#202649142
#202649112
#202649012
#538×C1012
#1612101