Finite Group Databasebeta

← #27174389

Subgroups — #27174389

Order 12928.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×C22⋊D101C211yesmaximal, characteristic
2#20265115C211yesmaximal, characteristic, complemented
3#20265112C211yesmaximal, characteristic, complemented
4#20265108C211yesmaximal, characteristic, complemented
5#20265105C212yesmaximal, complemented
6#20265103C211yesmaximal, characteristic, complemented
7#20265092C212yesmaximal
8C4×#13108053C212yesmaximal, complemented
9C4×#13108051C212yesmaximal, complemented
10C4×C22⋊Dic101C211yesmaximal, characteristic, complemented
11C2×#117×C101C211yesmaximal, characteristic, complemented
12C22×C22⋊D101C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C4×C22⋊D101—24no—
15C2×#13108053C2212yes—
16C2×#13108053C2212yes—
17C2×#13108053C2211yescharacteristic
18C2×C22⋊Dic101C2212yes—
19C2×C22⋊Dic101C2211yescharacteristic
20C2×#13108051C2212yes—
21C2×#13108051C2212yes—
22C2×#13108051C2211yescharacteristic
23C2×C4×Dic101C2212yes—
24C2×C4×Dic101C2211yescharacteristic
25#15692162C2211yescharacteristic
26#15692162C2211yescharacteristic
27#15692162C2212yes—
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31C23×C4×C101C2211yescharacteristic
32#117×C101—24no—
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C4⋊C4×C101C2211yescharacteristic
35C2×C22⋊C4×C101C2212yes—
36C2×C42×C101C2212yes—
37C23×D101C2311yescharacteristic
38C2×C22⋊D101—22no—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C22×Dic101C2311yescharacteristic
42C22×Dic101C2312yes—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2311yescharacteristic
45C22×Dic101C2312yes—
46C2×C4×D101—22no—
47C2×C4×D101—24no—
48#13108053—22no—
49#13108053—24no—
50#13108053—24no—
51C22⋊Dic101—22no—
52C22⋊Dic101—24no—
53#13108051—24no—
54#13108051—22no—
55#13108051—24no—
56C4×Dic101—22no—
57C4×Dic101—24no—
58C24×C101C2311yescharacteristic
59C22×C4×C101—24no—
60C22×C4×C101D414yes—
61C22×C4×C101C2312yes—
62C22×C4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2311yescharacteristic
65C4⋊C4×C101—22no—
66C4⋊C4×C101—22no—
67C22⋊C4×C101—24no—
68C42×C101—24no—
69C23×C101—22no—
70C23×C101—22no—
71C23×C101—21no—
72C23×C101—21no—
73C23×C101C2×D412yes—
74C23×C101C2411yescharacteristic
75C22×D101—21no—
76C22×D101—22no—
77C22×D101—22no—
78C22×D101—21no—
79C22×D101#4112yes—
80C2×C4×C101—22no—
81C2×C4×C101—28no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101C2×D412yes—
89C2×C4×C101C2×D412yes—
90C2×C4×C101#4114yes—
91C22⋊D101—28no—
92C2×Dic101—21no—
93C2×Dic101—22no—
94C2×Dic101—22no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97C2×Dic101—21no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101#4114yes—
104C2×Dic101#4112yes—
105C4×D101—44no—
106C22×C101—22no—
107C22×C101—22no—
108C22×C101—22no—
109C22×C101—42no—
110C22×C101—42no—
111C22×C101C22×D411yescharacteristic
112C22×C101C2×#4111yescharacteristic
113C22×C101C2×#4111yescharacteristic
114C22×C101C2×#4112yes—
115C22×C101C2×#4112yes—
116C2×D101—22no—
117C2×D101—24no—
118C2×D101—41no—
119C2×D101—42no—
120C2×D101—41no—
121C4×C101—22no—
122C4×C101—42no—
123C4×C101—42no—
124C4×C101#12314yes—
125Dic101—22no—
126Dic101—24no—
127Dic101—41no—
128Dic101—42no—
129Dic101—41no—
130C2×C101—42no—
131C2×C101C2×#12311yescharacteristic
132C2×C101#52511yescharacteristic
133C2×C101#52912yes—
134C2×C101#52912yes—
135C2×C101#53211yescharacteristic
136D101—42no—
137#2395—1011nomaximal, Sylow
138C101#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—1011no—
140C2×#117D10111yescharacteristic, complemented
141#400—1011no—
142#399—1011no—
143#398—1011no—
144#397—1011no—
145#397—1012no—
146#394—1012no—
147C4×C4⋊C4—1012no—
148C4×C22⋊C4—1011no—
149C4×C22⋊C4—1012no—
150C22×D4—1011no—
151C23×C4—1011no—
152C23×C4C2×D10111yescharacteristic
153C4×D4—2024no—
154#117—24no—
155C2×C4⋊C4—1011no—
156C2×C4⋊C4—1012no—
157C2×C4⋊C4—1012no—
158C2×C4⋊C4C2×D10111yescharacteristic
159C2×C4⋊C4C2×D10111yescharacteristic
160C2×C22⋊C4—1011no—
161C2×C22⋊C4—1012no—
162C2×C22⋊C4—1011no—
163C2×C22⋊C4—1012no—
164C2×C22⋊C4—1012no—
165C2×C22⋊C4C2×D10112yes—
166C2×C42—1011no—
167C2×C42—1012no—
168C2×C42C2×D10112yes—
169#95—1012no—
170#95—1011no—
171#95—1012no—
172#95—1011no—
173#95—1012no—
174#95—1012no—
175C24—1011no—
176C24C22×D10111yescharacteristic
177C2×D4—2022no—
178C2×D4—2022no—
179C2×D4—2022no—
180C22×C4—24no—
181C22×C4—1012no—
182C22×C4—1011no—
183C22×C4—1012no—
184C22×C4—1011no—
185C22×C4—1011no—
186C22×C4—2022no—
187C22×C4—2024no—
188C22×C4C22⋊D10114yes—
189C22×C4C22×D10112yes—
190C22×C4C22×D10112yes—
191C22×C4C22×D10111yescharacteristic
192C22×C4C22×D10111yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—2024no—
196C4⋊C4—2024no—
197C4⋊C4—2022no—
198C22⋊C4—24no—
199C22⋊C4—2024no—
200C22⋊C4—2024no—
201C22⋊C4—2024no—
202C22⋊C4—2022no—
203C22⋊C4—2022no—
204C42—24no—
205C42—2022no—
206C42—2024no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—1012no—
212C23—2021no—
213C23—2022no—
214C23—2021no—
215C23—2022no—
216C23C2×C22⋊D10112yes—
217C23C23×D10111yescharacteristic
218D4—2028no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—1014no—
228C2×C4—1012no—
229C2×C4—4044no—
230C2×C4—2021no—
231C2×C4—2021no—
232C2×C4—2022no—
233C2×C4—2022no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4—2022no—
237C2×C4—2021no—
238C2×C4—2022no—
239C2×C4—2021no—
240C2×C4—2022no—
241C2×C4C2×C22⋊D10112yes—
242C2×C4C2×C22⋊D10112yes—
243C2×C4#1310807414yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—4041no—
250C22—4042no—
251C22—4041no—
252C22—2022no—
253C22—2024no—
254C22C22×C22⋊D10111yescharacteristic
255C22#41×D10111yescharacteristic
256C22#41×D10111yescharacteristic
257C22#41×D10112yes—
258C22C2×#1310807412yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—4041no—
263C4—4042no—
264C4—4041no—
265C4—2022no—
266C4—2024no—
267C4#1569223814yes—
268C2—42no—
269C2—4042no—
270C2C2×#1569223811yescharacteristic
271C2#2026568411yescharacteristic
272C2#2026570512yes—
273C2#2026570512yes—
274C2#2026572311yescharacteristic
subgroupG/H
centerC22×C4C22×D101
commutator, socle, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC2×#117×C101C2
prime radicalC22×C22⋊D101C22
hypercenterC2×#117D101
hypocenterC101#2395
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20265115C2semidirect
3. #20265112C2semidirect
4. #20265108C2semidirect
5. #20265105C2semidirect
6. #20265103C2semidirect
8. C4×#13108053C2semidirect
9. C4×#13108051C2semidirect
10. C4×C22⋊Dic101C2semidirect
11. C2×#117×C101C2semidirect
138. C101#2395semidirect
140. C2×#117D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1012
#202651152
#202651122
#202651082
#202651052
#202651032
#202650922
C4×#131080532
C4×#131080512
C4×C22⋊Dic1012
C2×#117×C1012
#2395101