Finite Group Databasebeta

← #27179432

Subgroups — #27179432

Order 12928.

696 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 165 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×#15692242C211yesmaximal, characteristic, complemented
2C2×#15692235C211yesmaximal, characteristic, complemented
3C2×#15692226C211yesmaximal, characteristic
4C2×C42×D101C211yesmaximal, characteristic, complemented
5#20265083C211yesmaximal, characteristic, complemented
6#20265070C214yesmaximal, complemented
7#20265053C214yesmaximal, complemented
8#20265042C211yesmaximal, characteristic, complemented
9#390×C101C211yesmaximal, characteristic, complemented
10C22×Dic202C2211yescharacteristic
11C22×C22⋊D101C2212yes—
12C22×C4×D101C2212yes—
13C22×C4×D101C2211yescharacteristic
14#15692242—24no—
15#15692235—24no—
16#15692226—24no—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×C4⋊Dic101C2212yes—
19C2×C22⋊Dic101C2214yes—
20C2×C22⋊Dic101C2212yes—
21C2×#13108051C2212yes—
22C2×C4×Dic101C2212yes—
23C2×C4×Dic101C2211yescharacteristic
24C42×D101—24no—
25#15692162C2214yes—
26#15692162C2214yes—
27C22×D4×C101C2211yescharacteristic
28C2×C4⋊C4×C101C2211yescharacteristic
29C2×C22⋊C4×C101C2214yes—
30C2×C42×C101C2211yescharacteristic
31C23×D101C2311yescharacteristic
32C2×Dic202—22no—
33C2×Dic202—22no—
34C2×Dic202—22no—
35C2×C22⋊D101—24no—
36C2×C22⋊D101—28no—
37C22×Dic101C2312yes—
38C22×Dic101C2312yes—
39C22×Dic101C2312yes—
40C22×Dic101C2311yescharacteristic
41C2×C4×D101—24no—
42C2×C4×D101—22no—
43C2×C4×D101—24no—
44C2×C4×D101—22no—
45C2×C4×D101—24no—
46C2×C4×D101D414yes—
47C4⋊Dic101—22no—
48C4⋊Dic101—24no—
49C22⋊Dic101—24no—
50C22⋊Dic101—28no—
51#13108051—24no—
52C4×Dic101—24no—
53C4×Dic101D414yescomplemented
54C24×C101C2312yes—
55C2×D4×C101—22no—
56C2×D4×C101—22no—
57C2×D4×C101—22no—
58C22×C4×C101C2312yes—
59C22×C4×C101C2312yes—
60C22×C4×C101C2311yescharacteristic
61C4⋊C4×C101—22no—
62C22⋊C4×C101—28no—
63C42×C101—22no—
64C23×C101—22no—
65C23×C101—22no—
66C23×C101—24no—
67C23×C101—22no—
68C23×C101—24no—
69C23×C101C2411yescharacteristic
70C22×D101—21no—
71C22×D101—22no—
72C22×D101—21no—
73C22×D101C2×D412yes—
74C22×D101#4114yes—
75D4×C101—44no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—24no—
80C2×C4×C101C2×D412yes—
81C2×C4×C101C2×D412yes—
82C2×C4×C101#4114yes—
83C2×C4×C101#4112yes—
84C22⋊D101—48no—
85C2×Dic101—21no—
86C2×Dic101—22no—
87C2×Dic101—22no—
88C2×Dic101—21no—
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—22no—
92C2×Dic101—24no—
93C2×Dic101—22no—
94C2×Dic101C2×D414yes—
95C2×Dic101C2×D412yes—
96C4×D101—216no—
97C4×D101—28no—
98Dic202—44no—
99C22×C101—42no—
100C22×C101—44no—
101C22×C101—42no—
102C22×C101—44no—
103C22×C101—42no—
104C22×C101C22×D411yescharacteristic
105C22×C101C22×D411yescharacteristic
106C22×C101C2×#4112yes—
107C22×C101C2×#4112yes—
108C22×C101C2×#4111yescharacteristic
109C2×D101—22no—
110C2×D101—24no—
111C2×D101—24no—
112C2×D101—22no—
113C2×D101#12414yescomplemented
114C4×C101—24no—
115C4×C101—42no—
116C4×C101#12114yes—
117Dic101—22no—
118Dic101—24no—
119Dic101—42no—
120Dic101—42no—
121C2×C101—82no—
122C2×C101C2×#12111yescharacteristic
123C2×C101#52512yes—
124C2×C101C2×#12411yescharacteristic
125C2×C101#52912yes—
126C2×C101#53211yescharacteristic
127D101—24no—
128#2850—1011nomaximal, Sylow
129C101#285011yescharacteristic, complemented, Sylow
130C2×#124—1011no—
131C2×C22⋊Q8—1011no—
132C2×#121—1011no—
133C22×C42—1011no—
134#391—1011no—
135#390D10111yescharacteristic, complemented
136#388—1014no—
137#386—1014no—
138#382—1011no—
139C22×Q8—1011no—
140C22×D4—1012no—
141C22×D4C2×D10111yescharacteristic
142C23×C4—1011no—
143C23×C4—1012no—
144#124—2024no—
145C22⋊Q8—2024no—
146#121—2024no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1012no—
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—1012no—
151C2×C22⋊C4—1014no—
152C2×C22⋊C4—1012no—
153C2×C22⋊C4C2×D10114yes—
154C2×C42—1011no—
155C2×C42—1012no—
156C2×C42—2024no—
157C2×C42C2×D10111yescharacteristic
158#95—1014no—
159#95—1014no—
160C24—1011no—
161C24C22×D10112yes—
162C2×Q8—2022no—
163C2×Q8—2022no—
164C2×Q8—2022no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—2024no—
169C2×D4—2028no—
170C22×C4—1012no—
171C22×C4—1011no—
172C22×C4—1014no—
173C22×C4—1012no—
174C22×C4—1012no—
175C22×C4—2024no—
176C22×C4—2024no—
177C22×C4—2022no—
178C22×C4—2024no—
179C22×C4—2022no—
180C22×C4C22×D10112yes—
181C22×C4C22×D10112yes—
182C22×C4C22×D10111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—2024no—
185C4⋊C4—2022no—
186C22⋊C4—28no—
187C22⋊C4—2024no—
188C22⋊C4—2024no—
189C22⋊C4—2028no—
190C42—22no—
191C42—1014no—
192C42—2024no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—1012no—
199C23—1014no—
200C23—2021no—
201C23—2022no—
202C23—2021no—
203C23C23×D10111yescharacteristic
204Q8—4044no—
205D4—44no—
206D4—4048no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—1014no—
212C2×C4—1012no—
213C2×C4—20216no—
214C2×C4—2022no—
215C2×C4—2022no—
216C2×C4—2022no—
217C2×C4—2024no—
218C2×C4—2022no—
219C2×C4—2021no—
220C2×C4—2022no—
221C2×C4—2022no—
222C2×C4—2021no—
223C2×C4—2028no—
224C2×C4D4×D10112yes—
225C2×C4D4×D10112yes—
226C2×C4#1310807314yes—
227C2×C4#1310807412yes—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—1014no—
234C22—2022no—
235C22—2024no—
236C22—2024no—
237C22—2022no—
238C22C2×D4×D10111yescharacteristic
239C22C2×D4×D10111yescharacteristic
240C22#41×D10112yes—
241C22C2×#1310807312yes—
242C22C2×#1310807411yescharacteristic
243C4—24no—
244C4—42no—
245C4—4042no—
246C4—4042no—
247C4—2022no—
248C4—2024no—
249C4#1569222814yes—
250C2—82no—
251C2—2024no—
252C2C2×#1569222811yescharacteristic
253C2#2026568712yes—
254C2#124×D10111yescharacteristic
255C2#2026570912yes—
256C2#2026572511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#390×C101C2
prime radicalC2×#15692226C2
hypercenter#390D101
hypocenterC101#2850
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15692242C2semidirect
2. C2×#15692235C2semidirect
4. C2×C42×D101C2semidirect
5. #20265083C2semidirect
6. #20265070C2semidirect
7. #20265053C2semidirect
8. #20265042C2semidirect
9. #390×C101C2semidirect
53. C4×Dic101D4semidirect
113. C2×D101#124semidirect
129. C101#2850semidirect
135. #390D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922422
C2×#156922352
C2×#156922262
C2×C42×D1012
#202650832
#202650702
#202650532
#202650422
#390×C1012
#2850101