Finite Group Databasebeta

← #44558739

Subgroups — #44558739

Order 32128.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#22548706C211yesmaximal, characteristic, complemented
2C2×#22548704C211yesmaximal, characteristic
3#30631594C211yesmaximal, characteristic, complemented
4#30631589C211yesmaximal, characteristic
5#30631588C211yesmaximal, characteristic
6#30631579C214yesmaximal, complemented
7#30631576C214yesmaximal, complemented
8C42×Dic251C211yesmaximal, characteristic, complemented
9#391×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11#22548706—24no—
12#22548704—24no—
13C2×C4⋊Dic251C2212yes—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×Dic251⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2214yes—
19C2×(C2×D251)⋊C4C2214yes—
20C2×C4×Dic251C2212yes—
21C2×C4×Dic251C2212yes—
22C2×C4×Dic251C2211yescharacteristic
23C2×C4×Dic251C2211yescharacteristic
24C251⋊#95C2214yes—
25C22×Q8×C251C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C42×C251C2211yescharacteristic
28#95×C251C2214yes—
29C23×D251C2312yes—
30C2×D1004—22no—
31C2×D1004—22no—
32C2×D1004—22no—
33C22×Dic251C2312yes—
34C22×Dic251C2311yescharacteristic
35C22×Dic251C2311yescharacteristic
36C22×Dic251C2312yes—
37C4⋊Dic251—24no—
38C4⋊Dic251—22no—
39Dic251⋊C4—24no—
40(C2×D251)⋊C4—24no—
41(C2×D251)⋊C4—24no—
42(C2×D251)⋊C4—28no—
43(C2×D251)⋊C4—28no—
44C4×Dic251—24no—
45C4×Dic251—22no—
46C4×Dic251—24no—
47C4×Dic251D414yes—
48C2×Q8×C251—22no—
49C2×Q8×C251—22no—
50C2×Q8×C251—22no—
51C22×C4×C251C2312yes—
52C22×C4×C251C2312yes—
53C22×C4×C251C2312yes—
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—22no—
56C42×C251—22no—
57C22×D251—22no—
58C22×D251—24no—
59C22×D251—24no—
60C22×D251—22no—
61C22×D251—22no—
62D1004—44no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—24no—
69C2×Dic251C2×D412yes—
70C2×Dic251C2×D412yes—
71C2×Dic251#4114yes—
72C23×C251C2411yescharacteristic
73Q8×C251—44no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—24no—
81C2×C4×C251C2×D412yes—
82C2×C4×C251#4112yes—
83C2×C4×C251#4112yes—
84C2×C4×C251#4114yes—
85C22×C251C22×D411yescharacteristic
86C22×C251C2×#4112yes—
87C22×C251C2×#4111yescharacteristic
88C22×C251C2×#4111yescharacteristic
89C22×C251C2×#4112yes—
90C2×D251—42no—
91C2×D251—44no—
92C2×D251—44no—
93C2×D251—42no—
94C2×D251—42no—
95C4×C251—24no—
96C4×C251—42no—
97C4×C251—42no—
98C4×C251#12414yes—
99Dic251—24no—
100Dic251—22no—
101Dic251—22no—
102Dic251—42no—
103C2×C251C2×#12411yescharacteristic
104C2×C251#52912yes—
105C2×C251#52912yes—
106C2×C251#53211yescharacteristic
107C2×C251#53211yescharacteristic
108D251—82no—
109C251#286111yescharacteristic, complemented, Sylow
110#2861—2511nomaximal, Sylow
111C2×#125—2511no—
112C2×#124—2511no—
113#391—2511no—
114#391D25111yescharacteristic, complemented
115#390—2511no—
116#390—2511no—
117#388—2514no—
118#388—2514no—
119C43—2511no—
120C22×Q8C2×D25111yescharacteristic
121C22×D4—2511no—
122#125—5024no—
123#124—5024no—
124C2×C4⋊C4—2512no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2511no—
127C2×C4⋊C4C2×D25111yescharacteristic
128C2×C22⋊C4—2512no—
129C2×C22⋊C4—2512no—
130C2×C22⋊C4—2514no—
131C2×C22⋊C4—2514no—
132C2×C42—2512no—
133C2×C42—2512no—
134C2×C42—2511no—
135C2×C42—2511no—
136C2×C42C2×D25111yescharacteristic
137#95—2514no—
138#95C2×D25114yes—
139C24—2512no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×D4—5022no—
144C2×D4—5022no—
145C2×D4—5022no—
146C22×C4—2512no—
147C22×C4—2512no—
148C22×C4—2511no—
149C22×C4—2511no—
150C22×C4C22×D25112yes—
151C22×C4C22×D25112yes—
152C22×C4C22×D25112yes—
153C22×C4C22×D25111yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—5024no—
156C4⋊C4—5024no—
157C4⋊C4—5022no—
158C22⋊C4—5028no—
159C22⋊C4—5024no—
160C22⋊C4—5024no—
161C22⋊C4—5028no—
162C42—22no—
163C42—5024no—
164C42—5022no—
165C42—5024no—
166C42—2514no—
167C23—5022no—
168C23—5024no—
169C23—5022no—
170C23—5022no—
171C23—5024no—
172C23C23×D25111yescharacteristic
173Q8—44no—
174D4—10044no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—5022no—
183C2×C4—5024no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—2512no—
189C2×C4—2512no—
190C2×C4—2514no—
191C2×C4D4×D25112yes—
192C2×C4#1686080912yes—
193C2×C4#1686081012yes—
194C2×C4#1686081114yes—
195C22—10042no—
196C22—10044no—
197C22—10044no—
198C22—10042no—
199C22—10042no—
200C22C2×D4×D25111yescharacteristic
201C22#41×D25112yes—
202C22C2×#1686080911yescharacteristic
203C22C2×#1686081011yescharacteristic
204C22C2×#1686081112yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—10042no—
209C4—5024no—
210C4—5022no—
211C4—5022no—
212C4#2254870114yes—
213C2—20082no—
214C2C2×#2254870111yescharacteristic
215C2#3063222312yes—
216C2#3063222412yes—
217C2#3063223611yescharacteristic
218C2#3063223811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#391×C251C2
prime radicalC22×D1004C22
hypercenter#391D251
hypocenterC251#2861
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22548706C2semidirect
3. #30631594C2semidirect
6. #30631579C2semidirect
7. #30631576C2semidirect
8. C42×Dic251C2semidirect
9. #391×C251C2semidirect
109. C251#2861semidirect
114. #391D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487062
C2×#225487042
#306315942
#306315892
#306315882
#306315792
#306315762
C42×Dic2512
#391×C2512
#2861251