Finite Group Databasebeta

← #27170077

Subgroups — #27170077

Order 12928.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265767C211yesmaximal, characteristic, complemented
2#20265311C211yesmaximal, characteristic, complemented
3#20265085C211yesmaximal, characteristic, complemented
4#20264914C211yesmaximal, characteristic, complemented
5#20264912C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#450×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101—21no—
9C2×#13108059—21no—
10#15692250—21no—
11#15692244—21no—
12#15692240—21no—
13#15692230—21no—
14#15692229C2211yescharacteristic
15#15692225—21no—
16#15692215—21no—
17C2×C22⋊Dic101—21no—
18C2×#13108051C2211yescharacteristic
19#15692176—21no—
20C22⋊#11717579C2211yescharacteristic
21#15692162C2211yescharacteristic
22C22×D4×C101C2211yescharacteristic
23C2×SD16×C101—21no—
24C22⋊Q8×C101C2211yescharacteristic
25#102×C101—21no—
26C22⋊C8×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×D404—21no—
29C2×D404—21no—
30C2×D404D411yescharacteristic, complemented
31C2×C22⋊D101—21no—
32C2×C22⋊D101—42no—
33C2×C22⋊D101—42no—
34C2×C22⋊D101—42no—
35C22×Dic101—21no—
36C22×Dic101D411yescharacteristic
37C2×C4×D101—21no—
38#13108059—42no—
39C2×#11717579—21no—
40C4⋊Dic101D411yescharacteristic, complemented
41#13108053—21no—
42C22⋊Dic101—42no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49C4×Dic101—21no—
50C24×C101—21no—
51C2×Q8×C101D411yescharacteristic, complemented
52C2×D4×C101—21no—
53C2×D4×C101—21no—
54C2×D4×C101—42no—
55C22×C4×C101C2311yescharacteristic
56SD16×C101—42no—
57C2×C8×C101—21no—
58C4⋊C4×C101—21no—
59C4⋊C4×C101D411yescharacteristic, complemented
60C22⋊C4×C101—21no—
61C23×C101—22no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101—41no—
65C23×C101—41no—
66C23×C101—41no—
67C23×C101—41no—
68C23×C101C2×D411yescharacteristic
69C22×D101—21no—
70C22×D101—21no—
71C22×D101—21no—
72C22×D101—21no—
73C22×D101—41no—
74C22×D101—41no—
75Q8×C101—41no—
76D4×C101—22no—
77D4×C101—22no—
78D4×C101—41no—
79D4×C101—41no—
80C2×C4×C101—21no—
81C2×C4×C101—21no—
82C2×C4×C101—41no—
83C2×C4×C101C2×D411yescharacteristic
84C2×C4×C101C2×D411yescharacteristic
85C22⋊D101—48no—
86C22⋊D101—41no—
87C22⋊D101—41no—
88C2×Dic101—22no—
89C2×Dic101—21no—
90C2×Dic101—21no—
91C2×Dic101—41no—
92C2×Dic101—41no—
93C2×Dic101—41no—
94D404—41no—
95D404—81no—
96D404—81no—
97C4×D101—81no—
98C8×C101—41no—
99#11717579—41no—
100C22×C101—21no—
101C22×C101—21no—
102C22×C101—42no—
103C22×C101—42no—
104C22×C101—42no—
105C22×C101—81no—
106C22×C101—81no—
107C22×C101—81no—
108C22×C101—81no—
109C22×C101#12011yescharacteristic
110C22×C101#13511yescharacteristic
111C22×C101AGL(1,ℤ/8)11yescharacteristic
112C2×D101—42no—
113C2×D101—41no—
114C2×D101—41no—
115C2×D101—41no—
116C2×D101—41no—
117C2×D101—41no—
118C2×D101—41no—
119C2×D101—81no—
120C4×C101—21no—
121C4×C101—21no—
122C4×C101—41no—
123C4×C101—41no—
124Dic101—42no—
125Dic101—41no—
126Dic101—41no—
127C2×C101—21no—
128C2×C101—21no—
129C2×C101—82no—
130C2×C101#44911yescharacteristic
131C2×C101#45311yescharacteristic
132C2×C101B(4,𝔽2)11yescharacteristic
133D101—81no—
134D101—81no—
135#1600—1011nomaximal, Sylow
136C101#160011yescharacteristic, complemented, Sylow
137#538—1011no—
138#450D10111yescharacteristic, complemented
139#447—1011no—
140#392—1011no—
141#328—1011no—
142#327—1011no—
143#323—1011no—
144C22×D4—2021no—
145C22×D4C2×D10111yescharacteristic
146C2×SD16—21no—
147C2×D8—2021no—
148#126—2021no—
149#124—2021no—
150#123—2021no—
151C22⋊Q8C2×D10111yescharacteristic
152#121—1011no—
153#121—2021no—
154#120—2021no—
155C4×D4—2021no—
156C2×C22⋊C4—1011no—
157C2×C22⋊C4—2021no—
158#102—21no—
159#102—2021no—
160C22⋊C8—1011no—
161C22⋊C8C2×D10111yescharacteristic
162#95—1011no—
163C24—21no—
164C24—1011no—
165C2×Q8C22⋊D10111yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—1011no—
170C2×D4—4042no—
171C2×D4—4042no—
172C2×D4—4042no—
173C2×D4—2021no—
174C2×D4—2021no—
175C2×D4—2021no—
176C22×C4—1011no—
177C22×C4—2021no—
178C22×C4—2021no—
179C22×C4C22×D10111yescharacteristic
180SD16—42no—
181D8—4042no—
182C2×C8—21no—
183C2×C8—2021no—
184C4⋊C4—21no—
185C4⋊C4—1011no—
186C4⋊C4—2021no—
187C4⋊C4C22⋊D10111yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—4042no—
190C22⋊C4—2021no—
191C22⋊C4—2021no—
192C22⋊C4—2021no—
193C22⋊C4—2021no—
194C22⋊C4—2021no—
195C22⋊C4—2021no—
196C42—2021no—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—4041no—
205C23—4041no—
206C23—2021no—
207C23—2021no—
208C23—2021no—
209C23—2021no—
210C23D4×D10111yescharacteristic
211Q8—41no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—8081no—
217D4—8081no—
218D4—4041no—
219D4—4048no—
220D4—4041no—
221D4—4041no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—8081no—
226C2×C4—4041no—
227C2×C4—4041no—
228C2×C4—4041no—
229C2×C4—2022no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4D4×D10111yescharacteristic
233C2×C4C2×C22⋊D10111yescharacteristic
234C8—41no—
235C8—4041no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—8081no—
246C22—4042no—
247C22—4041no—
248C22—4041no—
249C22—4041no—
250C22—4041no—
251C22—4041no—
252C22—4041no—
253C22#1569222311yescharacteristic
254C22#1569227611yescharacteristic
255C22#1569228211yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—4042no—
261C4—4041no—
262C4—4041no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2—8081no—
267C2—8081no—
268C2#2026532111yescharacteristic
269C2#2026534311yescharacteristic
270C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#450×C101C2
socleC22×C101#120
hypercenter#450D101
hypocenterC101#1600
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20265767C2semidirect
2. #20265311C2semidirect
3. #20265085C2semidirect
4. #20264914C2semidirect
5. #20264912C2semidirect
6. #20264900C2semidirect
7. #450×C101C2semidirect
30. C2×D404D4semidirect
40. C4⋊Dic101D4semidirect
51. C2×Q8×C101D4semidirect
59. C4⋊C4×C101D4semidirect
136. C101#1600semidirect
138. #450D101semidirect
165. C2×Q8C22⋊D101semidirect
187. C4⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657672
#202653112
#202650852
#202649142
#202649122
#202649002
#450×C1012
#1600101