Finite Group Databasebeta

← #27187385

Subgroups — #27187385

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#20266080C212yesmaximal, complemented
4#20266078C212yesmaximal, complemented
5#20265689C211yesmaximal, characteristic
6D101⋊M16—21no—
7#15692334—21no—
8#15692316—21no—
9#15692315C2212yes—
10#15692314—21no—
11#15692313C2212yes—
12C2×#13108074C2211yescharacteristic
13C22×C4×D101C411yescharacteristic, complemented
14#15692239C411yescharacteristic, complemented
15#15692239C2211yescharacteristic
16#15692235—21no—
17#15692230—21no—
18#15692225C411yescharacteristic, complemented
19#15692225C2211yescharacteristic
20C4×D404—21no—
21C4×D404—21no—
22#117×C101C411yescharacteristic, complemented
23#13108084—41no—
24#13108083—22no—
25D101⋊C8—41no—
26C4⋊#87938—41no—
27C22⋊#87938—22no—
28C4×#87938—41no—
29C23×D101C2×C411yescharacteristic
30#13108074—41no—
31#13108074—41no—
32C2×Dic202D411yescharacteristic
33C2×D404—21no—
34C2×D404—21no—
35C2×D404D411yescharacteristic
36C2×C22⋊D101C2×C411yescharacteristic
37C2×C22⋊D101C2311yescharacteristic
38C22×Dic101C2×C411yescharacteristic
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—41no—
42C2×C4×D101—41no—
43C2×C4×D101D411yescharacteristic
44C2×C4×D101D411yescharacteristic
45C4⋊Dic101—21no—
46C4⋊Dic101—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51C22×C4×C101C2×C411yescharacteristic
52C4⋊C4×C101—21no—
53C22⋊C4×C101C2×C411yescharacteristic
54C22⋊C4×C101C2×C411yescharacteristic
55C42×C101—21no—
56C23×C101C22×C411yescharacteristic
57C22×D101—21no—
58C22×D101—42no—
59C22×D101—41no—
60C22×D101C22⋊C411yescharacteristic
61C22×D101C2×D411yescharacteristic
62C2×#87938—42no—
63C2×C4×C101—21no—
64C2×C4×C101—21no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101C22⋊C411yescharacteristic
68C2×C4×C101C22⋊C411yescharacteristic
69C22⋊D101—41no—
70C22⋊D101—41no—
71C2×Dic101—21no—
72C2×Dic101—41no—
73C2×Dic101C22⋊C411yescharacteristic
74C2×Dic101C2×D411yescharacteristic
75D404—41no—
76D404—42no—
77D404—42no—
78C4×D101—21no—
79C4×D101—21no—
80C4×D101—21no—
81C4×D101—21no—
82C4×D101—42no—
83C4×D101—41no—
84C4×D101—41no—
85Dic202—41no—
86#11717581—42no—
87C22×C101—21no—
88C22×C101—41no—
89C22×C101C2×C22⋊C411yescharacteristic
90C2×D101—22no—
91C2×D101—21no—
92C2×D101—21no—
93C2×D101—41no—
94C2×D101—42no—
95C2×D101—41no—
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—42no—
99#87938—82no—
100C4×C101—21no—
101C4×C101—41no—
102C4×C101—41no—
103C4×C101#9911yescharacteristic
104C4×C101#9911yescharacteristic
105Dic101—21no—
106Dic101—21no—
107Dic101—41no—
108C2×C101—21no—
109C2×C101—41no—
110C2×C101C2×#9911yescharacteristic
111D101—41no—
112D101—41no—
113D101—42no—
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—1011no—
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—2021no—
133#117#8793811yescharacteristic, complemented
134#101—2021no—
135#100—1012no—
136#99—1012no—
137#99—2021no—
138C24—1011no—
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—1011no—
148C22×C4—1011no—
149C22×C4—1011no—
150C22×C4—4041no—
151C22×C4—4041no—
152C22×C4—2021no—
153C22×C4—2021no—
154C22×C4C2×#8793811yescharacteristic
155M16—4041no—
156M16—2022no—
157C2×C8—4041no—
158C4⋊C4—21no—
159C4⋊C4—4041no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C22⋊C4—2021no—
165C22⋊C4—2021no—
166C22⋊C4—2022no—
167C22⋊C4C2×#8793811yescharacteristic
168C22⋊C4C2×#8793811yescharacteristic
169C42—21no—
170C42—4041no—
171C23—1011no—
172C23—1011no—
173C23—4041no—
174C23—4042no—
175C23—2021no—
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—21no—
187C2×C4—1011no—
188C2×C4—1011no—
189C2×C4—4042no—
190C2×C4—4042no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4—2021no—
199C2×C4C22⋊#8793811yescharacteristic
200C2×C4C22⋊#8793811yescharacteristic
201C8—4042no—
202C22—21no—
203C22—41no—
204C22—4041no—
205C22—4041no—
206C22—4041no—
207C22—4042no—
208C22—4042no—
209C22—4041no—
210C22—2022no—
211C22—2021no—
212C22—2021no—
213C22C2×C22⋊#8793811yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—8082no—
218C4—4041no—
219C4—2021no—
220C4—2021no—
221C4#1569231411yescharacteristic
222C4#1569231411yescharacteristic
223C2—21no—
224C2—41no—
225C2—4041no—
226C2—4042no—
227C2—4041no—
228C2C2×#1569231411yescharacteristic
subgroupG/H
centerC4#15692314
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
Fitting#117×C101C4
socleC2×C101C2×#99
prime radical#20265689C2
cosocleC2×C22⋊D101C23
hypercenter#117#87938
hypocenterC101#2120
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20266163C2semidirect
2. #20266156C2semidirect
3. #20266080C2semidirect
4. #20266078C2semidirect
13. C22×C4×D101C4semidirect
14. #15692239C4semidirect
18. #15692225C4semidirect
22. #117×C101C4semidirect
115. C101#2120semidirect
133. #117#87938semidirect

Maximal subgroups

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

subgroupindex
#202661632
#202661562
#202660802
#202660782
#202656892
#2120101