Finite Group Databasebeta

← #27187146

Subgroups — #27187146

Order 12928.

432 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×D101⋊C8C211yesmaximal, characteristic, complemented
2C2×#15692317C212yesmaximal, complemented
3#20266119C211yesmaximal, characteristic, complemented
4#20266050C212yesmaximal, complemented
5#121×D101C211yesmaximal, characteristic
6C2×D101⋊C8—24no—
7C2×D101⋊C8—21no—
8C2×D101⋊C8C2212yes—
9C22×#11717581—21no—
10C2×C4⋊#87938C2212yes—
11C2×C22⋊#87938—21no—
12#15692317—24no—
13(C2×#87938)⋊C4—21no—
14C2×D4×D101—21no—
15C2×D4×D101C2211yescharacteristic
16C22×C4×D101C2211yescharacteristic
17#15692230C411yescharacteristic, complemented
18#15692229C411yescharacteristic, complemented
19#15692228—21no—
20#15692227—21no—
21#15692226C411yescharacteristic, complemented
22C4⋊C4×D101C2211yescharacteristic
23C22⋊C4×D101—21no—
24#121×C101C411yescharacteristic, complemented
25C22×#87938—22no—
26D101⋊C8—24no—
27D101⋊C8—22no—
28D101⋊C8D414yescomplemented
29C2×#11717581—24no—
30C2×#11717581—22no—
31C4⋊#87938—22no—
32C4⋊#87938—22no—
33C22⋊#87938—42no—
34C23×D101—21no—
35C23×D101D411yescharacteristic
36C2×D404—21no—
37C2×D404C2×C411yescharacteristic
38C2×C22⋊D101—21no—
39C2×C22⋊D101—21no—
40C2×C22⋊D101—21no—
41D4×D101—22no—
42D4×D101—22no—
43D4×D101—42no—
44D4×D101—42no—
45C22×Dic101D411yescharacteristic
46C2×C4×D101—22no—
47C2×C4×D101—22no—
48C2×C4×D101—21no—
49C2×C4×D101C2×C411yescharacteristic
50C2×C4×D101D411yescharacteristic
51C2×C4×D101D411yescharacteristic
52C2×C4×D101C2311yescharacteristic
53C4⋊Dic101C2×C411yescharacteristic
54#13108053—21no—
55C22⋊Dic101—21no—
56#13108051—21no—
57#13108051—21no—
58C2×D4×C101—21no—
59C2×D4×C101C2×C411yescharacteristic
60C22×C4×C101C2×C411yescharacteristic
61C4⋊C4×C101C2×C411yescharacteristic
62C22⋊C4×C101—21no—
63C23×C101—21no—
64C23×C101C22⋊C411yescharacteristic
65C22×D101—22no—
66C22×D101—22no—
67C22×D101—21no—
68C22×D101—42no—
69C22×D101—41no—
70C22×D101—41no—
71C22×D101—42no—
72C22×D101C22⋊C411yescharacteristic
73C22×D101D812yes—
74C22×D101SD1612yes—
75C22×D101C2×D411yescharacteristic
76C2×#87938—42no—
77C2×#87938—42no—
78C2×#87938—42no—
79D4×C101—22no—
80D4×C101—42no—
81C2×C4×C101—22no—
82C2×C4×C101—21no—
83C2×C4×C101C22⋊C411yescharacteristic
84C2×C4×C101C22×C411yescharacteristic
85C22⋊D101—42no—
86C22⋊D101—42no—
87C22⋊D101—41no—
88C22⋊D101—41no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—21no—
92C2×Dic101C22⋊C411yescharacteristic
93C2×Dic101C2×D411yescharacteristic
94D404—22no—
95D404—42no—
96C4×D101—22no—
97C4×D101—22no—
98C4×D101—21no—
99C4×D101—21no—
100C4×D101—41no—
101C4×D101—41no—
102C4×D101C2×D412yes—
103C4×D101#4112yes—
104#11717581—24no—
105C22×C101—21no—
106C22×C101—21no—
107C22×C101—41no—
108C22×C101—41no—
109C22×C101—41no—
110C22×C101#10212yescomplemented
111C22×C101C2×C22⋊C411yescharacteristic
112C2×D101—21no—
113C2×D101—21no—
114C2×D101—24no—
115C2×D101—24no—
116C2×D101—41no—
117C2×D101—41no—
118C2×D101—41no—
119C2×D101—82no—
120C2×D101—82no—
121C2×D101#10212yescomplemented
122C2×D101#12112yes—
123C2×D101C2×D811yescharacteristic
124C2×D101C2×SD1611yescharacteristic
125C2×D101#13512yes—
126#87938—82no—
127C4×C101—21no—
128C4×C101—41no—
129C4×C101C4×D412yes—
130Dic101—21no—
131Dic101—41no—
132Dic101#12011yescharacteristic
133Dic101#12311yescharacteristic
134C2×C101—22no—
135C2×C101—81no—
136C2×C101#38611yescharacteristic
137C2×C101C2×#10211yescharacteristic
138C2×C101#41611yescharacteristic
139D101—22no—
140D101—81no—
141D101#46512yes—
142D101#46612yes—
143#1893—1011nomaximal, Sylow
144C101#189311yescharacteristic, complemented, Sylow
145C23×C8—1011no—
146C2×#121—1011no—
147C2×#102—1012no—
148#380—1011no—
149#340—1012no—
150C22×D4—1011no—
151C22×D4—2021no—
152C23×C4—1011no—
153C22×C8—1012no—
154C22×C8—2024no—
155C22×C8—2021no—
156C22×C8—2021no—
157#121—1011no—
158#121—1011no—
159#121—1011no—
160#121—2021no—
161#121—2021no—
162#121#8793811yescharacteristic, complemented
163C2×C4⋊C4—1011no—
164C2×C4⋊C4—1012no—
165C2×C22⋊C4—2021no—
166C2×C22⋊C4—2021no—
167#102—2024no—
168#95—2021no—
169C24—1011no—
170C24—2021no—
171C2×D4—21no—
172C2×D4—1011no—
173C2×D4—4042no—
174C2×D4—4042no—
175C2×D4—2021no—
176C2×D4—2022no—
177C2×D4—2022no—
178C2×D4—2021no—
179C2×D4—2021no—
180C2×D4—2021no—
181C2×D4C2×#8793811yescharacteristic
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—1011no—
187C22×C4—2022no—
188C22×C4—2022no—
189C22×C4—2021no—
190C22×C4—2022no—
191C22×C4C2×#8793811yescharacteristic
192C2×C8—1014no—
193C2×C8—2022no—
194C2×C8—2022no—
195C2×C8—2024no—
196C2×C8—2024no—
197C4⋊C4—1011no—
198C4⋊C4—2021no—
199C4⋊C4—2022no—
200C4⋊C4—2022no—
201C4⋊C4C2×#8793811yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—4042no—
204C22⋊C4—2021no—
205C22⋊C4—2021no—
206C22⋊C4—2021no—
207C23—21no—
208C23—1011no—
209C23—1012no—
210C23—1012no—
211C23—1011no—
212C23—4041no—
213C23—4041no—
214C23—4042no—
215C23—4042no—
216C23—2022no—
217C23—2022no—
218C23—2021no—
219C23C22⋊#8793811yescharacteristic
220D4—22no—
221D4—42no—
222D4—4042no—
223D4—4041no—
224D4—4041no—
225D4—4042no—
226D4—4042no—
227D4—2022no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—1011no—
231C2×C4—1012no—
232C2×C4—1012no—
233C2×C4—1011no—
234C2×C4—4041no—
235C2×C4—4041no—
236C2×C4—4042no—
237C2×C4—4042no—
238C2×C4—4042no—
239C2×C4—2021no—
240C2×C4—2021no—
241C2×C4—2022no—
242C2×C4—2021no—
243C2×C4—2021no—
244C2×C4—2021no—
245C2×C4—2022no—
246C2×C4C22⋊#8793811yescharacteristic
247C2×C4C22×#8793811yescharacteristic
248C8—2024no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—8082no—
255C22—8082no—
256C22—1012no—
257C22—1011no—
258C22—1011no—
259C22—1012no—
260C22—1012no—
261C22—4041no—
262C22—4041no—
263C22—4041no—
264C22—2024no—
265C22—2024no—
266C22—2021no—
267C22—2021no—
268C22#1569231712yescomplemented
269C22C2×C22⋊#8793811yescharacteristic
270C4—21no—
271C4—41no—
272C4—8082no—
273C4—1011no—
274C4—1011no—
275C4—4041no—
276C4—2021no—
277C4D4×#8793812yes—
278C2—22no—
279C2—81no—
280C2—8081no—
281C2—1012no—
282C2—1012no—
283C2—2022no—
284C2#2026612511yescharacteristic
285C2C2×#1569231711yescharacteristic
286C2#2026616711yescharacteristic
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#1893
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×D101⋊C8C2semidirect
2. C2×#15692317C2semidirect
3. #20266119C2semidirect
4. #20266050C2semidirect
17. #15692230C4semidirect
18. #15692229C4semidirect
21. #15692226C4semidirect
24. #121×C101C4semidirect
28. D101⋊C8D4semidirect
110. C22×C101#102semidirect
121. C2×D101#102semidirect
144. C101#1893semidirect
162. #121#87938semidirect
268. C22#15692317semidirect

Maximal subgroups

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

subgroupindex
C22×D101⋊C82
C2×#156923172
#202661192
#202660502
#121×D1012
#1893101