Finite Group Databasebeta

← #44553453

Subgroups — #44553453

Order 32128.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548713C211yesmaximal, characteristic
2#30631594C211yesmaximal, characteristic
3#30631591C212yesmaximal
4#30631584C212yesmaximal
5#30631582C212yesmaximal
6C4×Dic251⋊C4C212yesmaximal
7#30631538C212yesmaximal
8#30631538C212yesmaximal
9#391×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11#22548713C418yescomplemented
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×C4×Dic251C2212yes—
19C2×C4×Dic251C2212yes—
20C2×C4×Dic251C2211yescharacteristic
21C2×C4×Dic251C2212yes—
22C2×C4×Dic251C2211yescharacteristic
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2212yes—
27C22×Q8×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C42×C251C2211yescharacteristic
30#95×C251C2212yes—
31#95×C251C2212yes—
32C2×Dic502—41no—
33C2×Dic502—41no—
34C2×Dic502C2×C414yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C4⋊Dic251—41no—
40Dic251⋊C4—24no—
41Dic251⋊C4—42no—
42Dic251⋊C4C2×C418yes—
43Dic251⋊C4C2×C418yes—
44Dic251⋊C4Q818yescomplemented
45C4×Dic251—24no—
46C4×Dic251—24no—
47C4×Dic251—42no—
48C4×Dic251—41no—
49C4×Dic251C2×C414yes—
50C2×Q8×C251—41no—
51C2×Q8×C251—41no—
52C2×Q8×C251C2×C414yes—
53C22×C4×C251C2312yes—
54C22×C4×C251C2311yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C22×C4×C251C2311yescharacteristic
57C22×C4×C251C2312yes—
58C4⋊C4×C251—41no—
59C42×C251—41no—
60Dic502—44no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251C22×C414yes—
69C2×Dic251C22×C414yes—
70C2×Dic251C2×Q814yes—
71C2×Dic251C2×Q814yes—
72C2×Dic251#4114yes—
73C23×C251C2411yescharacteristic
74Q8×C251—44no—
75C2×C4×C251—22no—
76C2×C4×C251—21no—
77C2×C4×C251—22no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—21no—
82C2×C4×C251—21no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251C22×C412yes—
87C2×C4×C251C22×C412yes—
88C2×C4×C251C22×C412yes—
89C2×C4×C251C2×Q814yes—
90C22×C251C23×C411yescharacteristic
91C22×C251C22×Q812yes—
92C22×C251C2×#4112yes—
93C22×C251H(5,𝔽2)11yescharacteristic
94C22×C251H(5,𝔽2)11yescharacteristic
95C4×C251—22no—
96C4×C251—22no—
97C4×C251—22no—
98C4×C251—24no—
99C4×C251—42no—
100Dic251—24no—
101Dic251—24no—
102Dic251—42no—
103Dic251C4×Q818yes—
104C2×C251C2×C4×Q812yes—
105C2×C251#51811yescharacteristic
106C2×C251#55412yes—
107C2×C251#55511yescharacteristic
108C2×C251Q8211yescharacteristic
109C251#236911yescharacteristic, complemented, Sylow
110#2369—2511nomaximal, Sylow
111C2×C4⋊Q8—2511no—
112#391—2511no—
113#391—2512no—
114#391D25111yescharacteristic
115#389—2512no—
116#389—2512no—
117C4×C4⋊C4—2512no—
118C4×C4⋊C4—2512no—
119C4×C4⋊C4—2512no—
120C22×Q8—2511no—
121C22×Q8C2×D25111yescharacteristic
122C4⋊Q8—2518no—
123C2×C4⋊C4—2512no—
124C2×C4⋊C4—2511no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2512no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4C2×D25111yescharacteristic
130C2×C42—2511no—
131C2×C42—2512no—
132C2×C42—2512no—
133C2×C42—2512no—
134C2×C42—2511no—
135C2×C42C2×D25111yescharacteristic
136#95—2512no—
137#95—2512no—
138#95—2512no—
139#95—2512no—
140#95C2×D25112yes—
141#95C2×D25112yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—10041no—
145C2×Q8—10041no—
146C2×Q8—2514no—
147C2×Q8C4×D25114yes—
148C22×C4—2512no—
149C22×C4—2512no—
150C22×C4—2512no—
151C22×C4—2512no—
152C22×C4C22×D25112yes—
153C22×C4C22×D25111yescharacteristic
154C22×C4C22×D25111yescharacteristic
155C22×C4C22×D25111yescharacteristic
156C22×C4C22×D25112yes—
157C4⋊C4—41no—
158C4⋊C4—10042no—
159C4⋊C4—10041no—
160C4⋊C4—5024no—
161C4⋊C4—2518no—
162C4⋊C4—2518no—
163C4⋊C4—2518no—
164C42—41no—
165C42—10042no—
166C42—10041no—
167C42—5024no—
168C42—5024no—
169C42—2514no—
170C23C23×D25111yescharacteristic
171Q8—44no—
172Q8—10044no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C2×C4—2514no—
192C2×C4—2514no—
193C2×C4—2514no—
194C2×C4—2514no—
195C2×C4—2514no—
196C2×C4C2×C4×D25112yes—
197C2×C4C2×C4×D25112yes—
198C2×C4C2×C4×D25112yes—
199C2×C4Q8×D25114yes—
200C22C22×C4×D25111yescharacteristic
201C22C2×Q8×D25112yes—
202C22#41×D25112yes—
203C22#2254876011yescharacteristic
204C22#2254876011yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—10042no—
211C4—5024no—
212C4—5024no—
213C4—2518no—
214C2C4×Q8×D25112yes—
215C2#3063216011yescharacteristic
216C2#3063239112yes—
217C2#3063239911yescharacteristic
218C2#3063241711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#391×C251C2
socle, prime radical, cosocleC23×C251C24
hypercenter#391D251
hypocenterC251#2369
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #22548713C4semidirect
44. Dic251⋊C4Q8semidirect
109. C251#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#225487132
#306315942
#306315912
#306315842
#306315822
C4×Dic251⋊C42
#306315382
#391×C2512
#2369251