Finite Group Databasebeta

← #27187383

Subgroups — #27187383

Order 12928.

264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 54 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#20266163C211yesmaximal, characteristic, complemented
2#20266156C211yesmaximal, characteristic, complemented
3#20266079C212yesmaximal, complemented
4#20266077C212yesmaximal, complemented
5#20265688C211yesmaximal, characteristic
6D101⋊M16—21no—
7#15692334—21no—
8#15692316—21no—
9#15692315C2212yes—
10#15692314—21no—
11#15692313C2212yes—
12C2×#13108074C2211yescharacteristic
13#15692238C411yescharacteristic, complemented
14#15692238C2211yescharacteristic
15#15692234—21no—
16#15692229—21no—
17#15692224C411yescharacteristic, complemented
18#15692224C2211yescharacteristic
19C4×C22⋊D101—21no—
20C4×C22⋊D101—21no—
21#15692214C411yescharacteristic, complemented
22C23×C4×C101C411yescharacteristic, complemented
23#13108084—41no—
24#13108083—22no—
25D101⋊C8—41no—
26C4⋊#87938—41no—
27C22⋊#87938—22no—
28C4×#87938—41no—
29#13108074—41no—
30#13108074—41no—
31C2×Dic202D411yescharacteristic
32C2×D404D411yescharacteristic
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101C2×C411yescharacteristic
36C2×C22⋊D101C2311yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C4⋊Dic101—21no—
40#13108053—21no—
41#13108053—21no—
42C22⋊Dic101—21no—
43C22⋊Dic101—21no—
44C22⋊Dic101C2×C411yescharacteristic
45C22⋊Dic101C2×C411yescharacteristic
46#13108051—21no—
47#13108051—21no—
48C4×Dic101—21no—
49C24×C101C2×C411yescharacteristic
50C22×C4×C101—21no—
51C22×C4×C101—21no—
52C22×C4×C101—41no—
53C22×C4×C101—41no—
54C22×C4×C101C2×C411yescharacteristic
55C22×C4×C101C2×C411yescharacteristic
56C23×C101—21no—
57C23×C101—41no—
58C23×C101—42no—
59C23×C101C22×C411yescharacteristic
60C22×D101C22⋊C411yescharacteristic
61C22×D101C2×D411yescharacteristic
62C2×#87938—42no—
63C2×C4×C101—21no—
64C2×C4×C101—21no—
65C2×C4×C101—21no—
66C2×C4×C101—42no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C22⋊C411yescharacteristic
71C2×C4×C101C22⋊C411yescharacteristic
72C22⋊D101—41no—
73C22⋊D101—41no—
74C22⋊D101—42no—
75C22⋊D101—42no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C2×D411yescharacteristic
80D404—41no—
81C4×D101—21no—
82C4×D101—21no—
83C4×D101—21no—
84C4×D101—21no—
85Dic202—41no—
86#11717581—42no—
87C22×C101—22no—
88C22×C101—21no—
89C22×C101—41no—
90C22×C101—41no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101—41no—
94C22×C101C2×C22⋊C411yescharacteristic
95C2×D101—21no—
96C2×D101—21no—
97C2×D101—41no—
98C2×D101—41no—
99#87938—82no—
100C4×C101—21no—
101C4×C101—41no—
102C4×C101#9911yescharacteristic
103C4×C101#9911yescharacteristic
104Dic101—21no—
105Dic101—21no—
106Dic101—41no—
107Dic101—41no—
108C2×C101—21no—
109C2×C101—42no—
110C2×C101—41no—
111C2×C101C2×#9911yescharacteristic
112D101—41no—
113D101—41no—
114#2120—1011nomaximal, Sylow
115C101#212011yescharacteristic, complemented, Sylow
116#525—1011no—
117#413—1011no—
118#410—1011no—
119#352—1012no—
120AΣL(1,𝔽16)—1012no—
121C2×#41—1011no—
122C23×C4#8793811yescharacteristic, complemented
123C2×M16—2021no—
124#123—1011no—
125#123—1011no—
126C22⋊Q8—2021no—
127#121—2021no—
128#120—1011no—
129#120—1011no—
130C4×D4—2021no—
131C4×D4—2021no—
132#117—1011no—
133#117—2021no—
134#101—2021no—
135#100—1012no—
136#99—1012no—
137#99—2021no—
138C24C2×#8793811yescharacteristic
139#41—4041no—
140#41—4041no—
141C2×Q8—1011no—
142C2×D4—1011no—
143C2×D4—1011no—
144C2×D4—1011no—
145C2×D4—2021no—
146C2×D4—2021no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4C2×#8793811yescharacteristic
154C22×C4C2×#8793811yescharacteristic
155M16—4041no—
156M16—2022no—
157C2×C8—4041no—
158C4⋊C4—4041no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C22⋊C4—1011no—
163C22⋊C4—1011no—
164C22⋊C4—2021no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2022no—
169C42—4041no—
170C42—2021no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1011no—
175C23—1011no—
176C23C22×#8793811yescharacteristic
177Q8—4041no—
178D4—4041no—
179D4—4042no—
180D4—4042no—
181D4—4041no—
182D4—4041no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—1011no—
191C2×C4—1011no—
192C2×C4—4042no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4—2021no—
199C2×C4C22⋊#8793811yescharacteristic
200C2×C4C22⋊#8793811yescharacteristic
201C8—4042no—
202C22—22no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—42no—
207C22—42no—
208C22—41no—
209C22—4041no—
210C22—4041no—
211C22—2021no—
212C22—2021no—
213C22C2×C22⋊#8793811yescharacteristic
214C4—21no—
215C4—41no—
216C4—8082no—
217C4—4041no—
218C4—4041no—
219C4—2021no—
220C4—2021no—
221C4#1569231311yescharacteristic
222C4#1569231311yescharacteristic
223C2—21no—
224C2—42no—
225C2—41no—
226C2—4041no—
227C2—4041no—
228C2C2×#1569231311yescharacteristic
subgroupG/H
centerC4#15692313
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC23×C4×C101C4
socleC2×C101C2×#99
prime radical#20265688C2
cosocleC2×C22⋊D101C23
hypercenterC23×C4#87938
hypocenterC101#2120
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #20266163C2semidirect
2. #20266156C2semidirect
3. #20266079C2semidirect
4. #20266077C2semidirect
13. #15692238C4semidirect
17. #15692224C4semidirect
21. #15692214C4semidirect
22. C23×C4×C101C4semidirect
115. C101#2120semidirect
122. C23×C4#87938semidirect

Maximal subgroups

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

subgroupindex
#202661632
#202661562
#202660792
#202660772
#202656882
#2120101