Finite Group Databasebeta

← #44558669

Subgroups — #44558669

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×#22548704C211yesmaximal, characteristic, complemented
2C2×#22548691C211yesmaximal, characteristic, complemented
3C2×#22548690C211yesmaximal, characteristic
4C2×C42×D251C211yesmaximal, characteristic, complemented
5#30631589C211yesmaximal, characteristic, complemented
6#30631576C214yesmaximal, complemented
7#30631565C214yesmaximal, complemented
8#30631552C211yesmaximal, characteristic, complemented
9#391×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×D1004C2212yes—
12C22×C4×D251C2212yes—
13C22×C4×D251C2211yescharacteristic
14#22548704—24no—
15#22548691—24no—
16#22548690—24no—
17C2×C4⋊Dic251C2211yescharacteristic
18C2×Dic251⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2214yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2214yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×C4×Dic251C2212yes—
24C2×C4×Dic251C2211yescharacteristic
25C42×D251—24no—
26C251⋊#95C2214yes—
27C22×Q8×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C42×C251C2211yescharacteristic
30#95×C251C2214yes—
31C23×D251C2312yes—
32C23×D251C2311yescharacteristic
33C2×D1004—24no—
34C2×D1004—28no—
35C2×D1004—22no—
36C2×D1004—22no—
37C2×D1004—22no—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C22×Dic251C2311yescharacteristic
41C2×C4×D251—24no—
42C2×C4×D251—24no—
43C2×C4×D251—22no—
44C2×C4×D251—22no—
45C2×C4×D251—24no—
46C2×C4×D251D414yes—
47C4⋊Dic251—22no—
48Dic251⋊C4—24no—
49(C2×D251)⋊C4—24no—
50(C2×D251)⋊C4—24no—
51(C2×D251)⋊C4—28no—
52(C2×D251)⋊C4—28no—
53C4×Dic251—24no—
54C4×Dic251D414yescomplemented
55C2×Q8×C251—22no—
56C2×Q8×C251—22no—
57C2×Q8×C251—22no—
58C22×C4×C251C2312yes—
59C22×C4×C251C2312yes—
60C22×C4×C251C2312yes—
61C22×C4×C251C2311yescharacteristic
62C4⋊C4×C251—22no—
63C42×C251—22no—
64C22×D251—21no—
65C22×D251—21no—
66C22×D251—22no—
67C22×D251—22no—
68C22×D251—22no—
69C22×D251—24no—
70C22×D251—24no—
71C22×D251—22no—
72C22×D251C2×D412yes—
73C22×D251#4114yes—
74D1004—48no—
75D1004—44no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—24no—
82C2×Dic251C2×D414yes—
83C2×Dic251C2×D412yes—
84C4×D251—216no—
85C4×D251—28no—
86C23×C251C2411yescharacteristic
87Q8×C251—44no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
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#4112yes—
98C2×C4×C251#4114yes—
99C22×C251C22×D411yescharacteristic
100C22×C251C22×D411yescharacteristic
101C22×C251C2×#4112yes—
102C22×C251C2×#4111yescharacteristic
103C22×C251C2×#4112yes—
104C2×D251—22no—
105C2×D251—24no—
106C2×D251—24no—
107C2×D251—22no—
108C2×D251—42no—
109C2×D251—44no—
110C2×D251—42no—
111C2×D251—44no—
112C2×D251—42no—
113C2×D251#12414yescomplemented
114C4×C251—24no—
115C4×C251—42no—
116C4×C251—42no—
117C4×C251#12114yes—
118Dic251—22no—
119Dic251—24no—
120Dic251—42no—
121C2×C251C2×#12111yescharacteristic
122C2×C251#52512yes—
123C2×C251C2×#12411yescharacteristic
124C2×C251#52912yes—
125C2×C251#53211yescharacteristic
126D251—24no—
127D251—82no—
128C251#285011yescharacteristic, complemented, Sylow
129#2850—2511nomaximal, Sylow
130C2×#124—2511no—
131C2×C22⋊Q8—2511no—
132C2×#121—2511no—
133C22×C42—2511no—
134#391D25111yescharacteristic, complemented
135#390—2511no—
136#388—2514no—
137#386—2514no—
138#382—2511no—
139C22×Q8C2×D25111yescharacteristic
140C22×D4—2511no—
141C22×D4—2512no—
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—2512no—
152C2×C22⋊C4—2514no—
153C2×C22⋊C4—2514no—
154C2×C42—5024no—
155C2×C42—2512no—
156C2×C42—2511no—
157C2×C42C2×D25111yescharacteristic
158#95—2514no—
159#95C2×D25114yes—
160C24—2512no—
161C24—2511no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—5024no—
166C2×D4—5028no—
167C2×D4—5022no—
168C2×D4—5022no—
169C2×D4—5022no—
170C22×C4—5024no—
171C22×C4—5024no—
172C22×C4—5022no—
173C22×C4—5022no—
174C22×C4—5024no—
175C22×C4—2512no—
176C22×C4—2512no—
177C22×C4—2511no—
178C22×C4—2514no—
179C22×C4C22×D25112yes—
180C22×C4C22×D25112yes—
181C22×C4C22×D25112yes—
182C22×C4C22×D25111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—5024no—
185C4⋊C4—5022no—
186C22⋊C4—5024no—
187C22⋊C4—5024no—
188C22⋊C4—5028no—
189C22⋊C4—5028no—
190C42—22no—
191C42—5024no—
192C42—2514no—
193C23—5022no—
194C23—5022no—
195C23—5024no—
196C23—5022no—
197C23—5024no—
198C23—5021no—
199C23—5021no—
200C23—5022no—
201C23—2512no—
202C23—2514no—
203C23C23×D25111yescharacteristic
204Q8—44no—
205D4—10048no—
206D4—10044no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—50216no—
215C2×C4—5022no—
216C2×C4—5024no—
217C2×C4—5021no—
218C2×C4—5022no—
219C2×C4—5021no—
220C2×C4—5022no—
221C2×C4—5028no—
222C2×C4—2514no—
223C2×C4—2512no—
224C2×C4D4×D25112yes—
225C2×C4D4×D25112yes—
226C2×C4#1686081012yes—
227C2×C4#1686081114yes—
228C22—10044no—
229C22—10042no—
230C22—10044no—
231C22—10042no—
232C22—10042no—
233C22—5024no—
234C22—5024no—
235C22—5022no—
236C22—5022no—
237C22—2514no—
238C22C2×D4×D25111yescharacteristic
239C22C2×D4×D25111yescharacteristic
240C22#41×D25112yes—
241C22C2×#1686081011yescharacteristic
242C22C2×#1686081112yes—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—10042no—
247C4—5024no—
248C4—5022no—
249C4#2254868814yes—
250C2—20082no—
251C2—5024no—
252C2C2×#2254868811yescharacteristic
253C2#3063220012yes—
254C2#124×D25111yescharacteristic
255C2#3063222412yes—
256C2#3063223811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#391×C251C2
prime radicalC2×#22548690C2
hypercenter#391D251
hypocenterC251#2850
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22548704C2semidirect
2. C2×#22548691C2semidirect
4. C2×C42×D251C2semidirect
5. #30631589C2semidirect
6. #30631576C2semidirect
7. #30631565C2semidirect
8. #30631552C2semidirect
9. #391×C251C2semidirect
54. C4×Dic251D4semidirect
113. C2×D251#124semidirect
128. C251#2850semidirect
134. #391D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487042
C2×#225486912
C2×#225486902
C2×C42×D2512
#306315892
#306315762
#306315652
#306315522
#391×C2512
#2850251