Finite Group Databasebeta

← #27186567

Subgroups — #27186567

Order 12928.

340 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#15692312C211yesmaximal, characteristic, complemented
2#20266119C211yesmaximal, characteristic, complemented
3#20266029C212yesmaximal, complemented
4#20266028C212yesmaximal, complemented
5#121×D101C211yesmaximal, characteristic
6C22×#11717581—21no—
7C2×C4⋊#87938—21no—
8C2×C22⋊#87938—21no—
9#15692312—21no—
10#15692312C2212yescomplemented
11(C2×#87938)⋊C4C2212yescomplemented
12C2×D4×D101—21no—
13C2×D4×D101C411yescharacteristic, complemented
14C22×C4×D101C2211yescharacteristic
15#15692230C2211yescharacteristic
16#15692229C411yescharacteristic, complemented
17#15692228—21no—
18#15692227—21no—
19#15692226C2211yescharacteristic
20C4⋊C4×D101C411yescharacteristic, complemented
21C22⋊C4×D101—21no—
22#121×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24C2×#11717581—22no—
25C2×#11717581—42no—
26C4⋊#87938—42no—
27C22⋊#87938—42no—
28C23×D101—21no—
29C23×D101D411yescharacteristic
30C2×D404—21no—
31C2×D404C2×C411yescharacteristic
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35D4×D101—42no—
36D4×D101—42no—
37D4×D101—42no—
38C22×Dic101D411yescharacteristic
39C2×C4×D101—21no—
40C2×C4×D101—42no—
41C2×C4×D101C2×C411yescharacteristic
42C2×C4×D101D411yescharacteristic
43C2×C4×D101D411yescharacteristic
44C2×C4×D101C2311yescharacteristic
45C4⋊Dic101C2×C411yescharacteristic
46#13108053—21no—
47C22⋊Dic101—21no—
48#13108051—21no—
49#13108051—21no—
50C2×D4×C101—21no—
51C2×D4×C101C2×C411yescharacteristic
52C22×C4×C101C2×C411yescharacteristic
53C4⋊C4×C101C2×C411yescharacteristic
54C22⋊C4×C101—21no—
55C23×C101—21no—
56C23×C101C22⋊C411yescharacteristic
57C22×D101—21no—
58C22×D101—41no—
59C22×D101—41no—
60C22×D101—42no—
61C22×D101—42no—
62C22×D101—42no—
63C22×D101C22⋊C411yescharacteristic
64C22×D101D812yes—
65C22×D101SD1612yes—
66C22×D101C2×D411yescharacteristic
67C2×#87938—42no—
68C2×#87938—42no—
69C2×#87938—42no—
70D4×C101—41no—
71D4×C101—42no—
72C2×C4×C101—21no—
73C2×C4×C101—41no—
74C2×C4×C101C22⋊C411yescharacteristic
75C2×C4×C101C22×C411yescharacteristic
76C22⋊D101—42no—
77C22⋊D101—41no—
78C22⋊D101—41no—
79C22⋊D101—42no—
80C2×Dic101—22no—
81C2×Dic101—21no—
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C2×D411yescharacteristic
84D404—41no—
85D404—41no—
86D404—41no—
87C4×D101—22no—
88C4×D101—22no—
89C4×D101—41no—
90C4×D101—41no—
91C4×D101—42no—
92#11717581—42no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101#10212yes—
99C22×C101C2×C22⋊C411yescharacteristic
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—21no—
103C2×D101—21no—
104C2×D101—42no—
105C2×D101—42no—
106C2×D101—41no—
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—84no—
110C2×D101#10212yes—
111C2×D101C2×D811yescharacteristic
112C2×D101C2×SD1611yescharacteristic
113#87938—82no—
114C4×C101—21no—
115C4×C101—21no—
116C4×C101—41no—
117Dic101—21no—
118Dic101—41no—
119Dic101#9912yes—
120C2×C101—22no—
121C2×C101—81no—
122C2×C101C2×#9911yescharacteristic
123C2×C101C2×#10211yescharacteristic
124C2×C101#42111yescharacteristic
125D101—21no—
126D101—22no—
127D101—21no—
128D101—81no—
129#1519—1011nomaximal, Sylow
130C101#151911yescharacteristic, complemented, Sylow
131C2×#121—1011no—
132C2×C22⋊C8—1011no—
133#380—1011no—
134#327—1012no—
135#327—1012no—
136C22×D4—1011no—
137C22×D4—2021no—
138C23×C4—1011no—
139C22×C8—2021no—
140#121—1011no—
141#121—1011no—
142#121—1011no—
143#121—2021no—
144#121—2021no—
145#121#8793811yescharacteristic, complemented
146C2×C4⋊C4—1011no—
147C2×C4⋊C4—2021no—
148C2×C22⋊C4—2021no—
149C2×C22⋊C4—2021no—
150C22⋊C8—1012no—
151C22⋊C8—2021no—
152#95—1012no—
153C24—1011no—
154C24—2021no—
155C2×D4—21no—
156C2×D4—1011no—
157C2×D4—4042no—
158C2×D4—4042no—
159C2×D4—4042no—
160C2×D4—2021no—
161C2×D4—2021no—
162C2×D4—2021no—
163C2×D4—2021no—
164C2×D4C2×#8793811yescharacteristic
165C22×C4—1011no—
166C22×C4—1011no—
167C22×C4—1011no—
168C22×C4—1011no—
169C22×C4—1011no—
170C22×C4—4042no—
171C22×C4—2021no—
172C22×C4—2022no—
173C22×C4C2×#8793811yescharacteristic
174C2×C8—4042no—
175C2×C8—2022no—
176C4⋊C4—1011no—
177C4⋊C4—4042no—
178C4⋊C4—2021no—
179C4⋊C4C2×#8793811yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—4042no—
182C22⋊C4—2021no—
183C22⋊C4—2021no—
184C22⋊C4—2021no—
185C23—21no—
186C23—1012no—
187C23—1012no—
188C23—1011no—
189C23—1011no—
190C23—4041no—
191C23—4041no—
192C23—4042no—
193C23—4042no—
194C23—4042no—
195C23—2021no—
196C23C22⋊#8793811yescharacteristic
197D4—41no—
198D4—42no—
199D4—4041no—
200D4—4041no—
201D4—4041no—
202D4—4042no—
203D4—4041no—
204D4—4041no—
205D4—4042no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—1011no—
209C2×C4—1011no—
210C2×C4—4041no—
211C2×C4—4041no—
212C2×C4—4042no—
213C2×C4—4042no—
214C2×C4—4042no—
215C2×C4—4042no—
216C2×C4—2022no—
217C2×C4—2022no—
218C2×C4—2022no—
219C2×C4—2021no—
220C2×C4C22⋊#8793811yescharacteristic
221C2×C4C22×#8793811yescharacteristic
222C8—4042no—
223C22—21no—
224C22—21no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—8084no—
229C22—1011no—
230C22—1012no—
231C22—1011no—
232C22—4042no—
233C22—4042no—
234C22—4041no—
235C22—4041no—
236C22—4041no—
237C22—2021no—
238C22—2021no—
239C22—2021no—
240C22—2021no—
241C22#1569231712yes—
242C22C2×C22⋊#8793811yescharacteristic
243C4—21no—
244C4—21no—
245C4—41no—
246C4—8082no—
247C4—1012no—
248C4—4041no—
249C4—2021no—
250C2—22no—
251C2—81no—
252C2—8081no—
253C2—2021no—
254C2—2022no—
255C2—2021no—
256C2D101⋊#9911yescharacteristic
257C2C2×#1569231711yescharacteristic
258C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#121×D101C2
cosocleC2×C4×D101C23
hypercenter#121#87938
hypocenterC101#1519
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#15692312C2semidirect
2. #20266119C2semidirect
3. #20266029C2semidirect
4. #20266028C2semidirect
10. #15692312C22semidirect
11. (C2×#87938)⋊C4C22semidirect
13. C2×D4×D101C4semidirect
16. #15692229C4semidirect
20. C4⋊C4×D101C4semidirect
22. #121×C101C4semidirect
130. C101#1519semidirect
145. #121#87938semidirect

Maximal subgroups

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

subgroupindex
C2×#156923122
#202661192
#202660292
#202660282
#121×D1012
#1519101