Finite Group Databasebeta

← #44558665

Subgroups — #44558665

Order 32128.

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×#22548702C211yesmaximal, characteristic, complemented
2C2×#22548695C211yesmaximal, characteristic, complemented
3C2×#22548686C211yesmaximal, characteristic
4C2×C42×D251C211yesmaximal, characteristic, complemented
5#30631593C211yesmaximal, characteristic, complemented
6#30631580C214yesmaximal, complemented
7#30631563C214yesmaximal, complemented
8#30631552C211yesmaximal, characteristic, complemented
9#390×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C22×C4×D251C2212yes—
13C22×C4×D251C2211yescharacteristic
14#22548702—24no—
15#22548695—24no—
16#22548686—24no—
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2212yes—
19C2×C22⋊Dic251C2214yes—
20C2×C22⋊Dic251C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×C4×Dic251C2212yes—
23C2×C4×Dic251C2211yescharacteristic
24C42×D251—24no—
25C251⋊#95C2214yes—
26C251⋊#95C2214yes—
27C22×D4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2214yes—
30C2×C42×C251C2211yescharacteristic
31C23×D251C2311yescharacteristic
32C2×Dic502—22no—
33C2×Dic502—22no—
34C2×Dic502—22no—
35C2×C22⋊D251—24no—
36C2×C22⋊D251—28no—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C22×Dic251C2311yescharacteristic
41C2×C4×D251—24no—
42C2×C4×D251—24no—
43C2×C4×D251—22no—
44C2×C4×D251—24no—
45C2×C4×D251—22no—
46C2×C4×D251D414yes—
47C4⋊Dic251—24no—
48C4⋊Dic251—22no—
49C22⋊Dic251—24no—
50C22⋊Dic251—28no—
51(C2×D251)⋊C4—24no—
52C4×Dic251—24no—
53C4×Dic251D414yescomplemented
54C24×C251C2312yes—
55C2×D4×C251—22no—
56C2×D4×C251—22no—
57C2×D4×C251—22no—
58C22×C4×C251C2312yes—
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C4⋊C4×C251—22no—
62C22⋊C4×C251—28no—
63C42×C251—22no—
64C22×D251—21no—
65C22×D251—22no—
66C22×D251—21no—
67C22×D251C2×D412yes—
68C22×D251#4114yes—
69Dic502—44no—
70C22⋊D251—48no—
71C2×Dic251—21no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—21no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—24no—
79C2×Dic251—22no—
80C2×Dic251C2×D414yes—
81C2×Dic251C2×D412yes—
82C4×D251—216no—
83C4×D251—28no—
84C23×C251—22no—
85C23×C251—22no—
86C23×C251—24no—
87C23×C251—22no—
88C23×C251—24no—
89C23×C251C2411yescharacteristic
90D4×C251—44no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—24no—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D412yes—
97C2×C4×C251#4114yes—
98C2×C4×C251#4112yes—
99C22×C251—42no—
100C22×C251—44no—
101C22×C251—42no—
102C22×C251—44no—
103C22×C251—42no—
104C22×C251C22×D411yescharacteristic
105C22×C251C22×D411yescharacteristic
106C22×C251C2×#4112yes—
107C22×C251C2×#4112yes—
108C22×C251C2×#4111yescharacteristic
109C2×D251—22no—
110C2×D251—24no—
111C2×D251—24no—
112C2×D251—22no—
113C2×D251#12414yescomplemented
114C4×C251—24no—
115C4×C251—42no—
116C4×C251#12114yes—
117Dic251—22no—
118Dic251—24no—
119Dic251—42no—
120Dic251—42no—
121C2×C251—82no—
122C2×C251C2×#12111yescharacteristic
123C2×C251#52512yes—
124C2×C251C2×#12411yescharacteristic
125C2×C251#52912yes—
126C2×C251#53211yescharacteristic
127D251—24no—
128C251#285011yescharacteristic, complemented, Sylow
129#2850—2511nomaximal, Sylow
130C2×#124—2511no—
131C2×C22⋊Q8—2511no—
132C2×#121—2511no—
133C22×C42—2511no—
134#391—2511no—
135#390D25111yescharacteristic, complemented
136#388—2514no—
137#386—2514no—
138#382—2511no—
139C22×Q8—2511no—
140C22×D4—2512no—
141C22×D4C2×D25111yescharacteristic
142C23×C4—2512no—
143C23×C4—2511no—
144#124—5024no—
145C22⋊Q8—5024no—
146#121—5024no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4—2512no—
149C2×C4⋊C4C2×D25111yescharacteristic
150C2×C22⋊C4—2512no—
151C2×C22⋊C4—2514no—
152C2×C22⋊C4—2512no—
153C2×C22⋊C4C2×D25114yes—
154C2×C42—5024no—
155C2×C42—2512no—
156C2×C42—2511no—
157C2×C42C2×D25111yescharacteristic
158#95—2514no—
159#95—2514no—
160C24—2511no—
161C24C22×D25112yes—
162C2×Q8—5022no—
163C2×Q8—5022no—
164C2×Q8—5022no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—5024no—
169C2×D4—5028no—
170C22×C4—5024no—
171C22×C4—5024no—
172C22×C4—5022no—
173C22×C4—5024no—
174C22×C4—5022no—
175C22×C4—2512no—
176C22×C4—2512no—
177C22×C4—2512no—
178C22×C4—2511no—
179C22×C4—2514no—
180C22×C4C22×D25112yes—
181C22×C4C22×D25112yes—
182C22×C4C22×D25111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—5024no—
185C4⋊C4—5022no—
186C22⋊C4—28no—
187C22⋊C4—5024no—
188C22⋊C4—5024no—
189C22⋊C4—5028no—
190C42—22no—
191C42—5024no—
192C42—2514no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—5021no—
199C23—5022no—
200C23—5021no—
201C23—2512no—
202C23—2514no—
203C23C23×D25111yescharacteristic
204Q8—10044no—
205D4—44no—
206D4—10048no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—50216no—
212C2×C4—5022no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5024no—
216C2×C4—5022no—
217C2×C4—5021no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—5021no—
221C2×C4—5028no—
222C2×C4—2514no—
223C2×C4—2512no—
224C2×C4D4×D25112yes—
225C2×C4D4×D25112yes—
226C2×C4#1686080914yes—
227C2×C4#1686081012yes—
228C22—44no—
229C22—42no—
230C22—44no—
231C22—42no—
232C22—42no—
233C22—5024no—
234C22—5024no—
235C22—5022no—
236C22—5022no—
237C22—2514no—
238C22C2×D4×D25111yescharacteristic
239C22C2×D4×D25111yescharacteristic
240C22#41×D25112yes—
241C22C2×#1686080912yes—
242C22C2×#1686081011yescharacteristic
243C4—24no—
244C4—42no—
245C4—10042no—
246C4—10042no—
247C4—5024no—
248C4—5022no—
249C4#2254868814yes—
250C2—82no—
251C2—5024no—
252C2C2×#2254868811yescharacteristic
253C2#3063219712yes—
254C2#124×D25111yescharacteristic
255C2#3063221912yes—
256C2#3063223511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#390×C251C2
prime radicalC2×#22548686C2
hypercenter#390D251
hypocenterC251#2850
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#22548702C2semidirect
2. C2×#22548695C2semidirect
4. C2×C42×D251C2semidirect
5. #30631593C2semidirect
6. #30631580C2semidirect
7. #30631563C2semidirect
8. #30631552C2semidirect
9. #390×C251C2semidirect
53. C4×Dic251D4semidirect
113. C2×D251#124semidirect
128. C251#2850semidirect
135. #390D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487022
C2×#225486952
C2×#225486862
C2×C42×D2512
#306315932
#306315802
#306315632
#306315522
#390×C2512
#2850251