Finite Group Databasebeta

← #44558043

Subgroups — #44558043

Order 32128.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#30631627C212yesmaximal, complemented
2#30631626C211yesmaximal, characteristic, complemented
3#30631615C212yesmaximal, complemented
4#30631575C212yesmaximal, complemented
5#30631567C212yesmaximal
6#30631562C212yesmaximal, complemented
7#30631562C212yesmaximal, complemented
8#30631552C211yesmaximal, characteristic, complemented
9#394×C251C211yesmaximal, characteristic
10C23×Dic251C2212yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C4×Dic251C2211yescharacteristic
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×D4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2212yes—
30#95×C251C2211yescharacteristic
31C22×Dic251—42no—
32C22×Dic251—42no—
33C22×Dic251—42no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C4⋊Dic251—41no—
39C4⋊Dic251—41no—
40C4⋊Dic251—42no—
41Dic251⋊C4—42no—
42C22⋊Dic251—24no—
43C22⋊Dic251—42no—
44C22⋊Dic251—42no—
45C4×Dic251—22no—
46C24×C251C2312yes—
47C2×D4×C251—24no—
48C2×D4×C251—41no—
49C22×C4×C251C2312yes—
50C22×C4×C251C2311yescharacteristic
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2311yescharacteristic
53C4⋊C4×C251—41no—
54C22⋊C4×C251—24no—
55C22⋊C4×C251—24no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—48no—
69C23×C251—22no—
70C23×C251—22no—
71C23×C251—22no—
72C23×C251—22no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251#4114yes—
76C23×C251C2411yescharacteristic
77D4×C251—44no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—22no—
87C2×C4×C251#4112yes—
88C22×C251—24no—
89C22×C251—24no—
90C22×C251—24no—
91C22×C251—42no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251C2×#4112yes—
96C22×C251C2×#4111yescharacteristic
97C22×C251H(5,𝔽2)11yescharacteristic
98C22×C251H(5,𝔽2)12yes—
99C22×C2512−1+411yescharacteristic
100C4×C251—22no—
101C4×C251—41no—
102C4×C251—41no—
103C4×C251—42no—
104Dic251—42no—
105Dic251—42no—
106Dic251—42no—
107Dic251—42no—
108C2×C251—44no—
109C2×C251#53912yes—
110C2×C251#53911yescharacteristic
111C2×C251#55111yescharacteristic
112C2×C251#55312yes—
113C2×C251H(3,𝔽4)11yescharacteristic
114C251#278211yescharacteristic, complemented, Sylow
115#2782—2511nomaximal, Sylow
116#400—2511no—
117#400—2512no—
118#397—2512no—
119#394D25111yescharacteristic
120#388—2512no—
121#386—2512no—
122#385—2512no—
123#385—2512no—
124#382—2511no—
125C22×D4C2×D25111yescharacteristic
126C23×C4—2512no—
127C2×C4⋊C4—2511no—
128C2×C4⋊C4—2511no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4—2512no—
131C2×C4⋊C4C2×D25111yescharacteristic
132C2×C22⋊C4—2512no—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4C2×D25112yes—
136C2×C22⋊C4C2×D25112yes—
137C2×C42—2511no—
138#95—2512no—
139#95—2512no—
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2512no—
144#95—2511no—
145#95C2×D25111yescharacteristic
146C24C22×D25112yes—
147C2×D4—24no—
148C2×D4—41no—
149C22×C4—10042no—
150C22×C4—10042no—
151C22×C4—10042no—
152C22×C4—2512no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4—2512no—
156C22×C4C22×D25111yescharacteristic
157C22×C4C22×D25111yescharacteristic
158C22×C4C22×D25112yes—
159C22×C4C22×D25111yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—10042no—
162C4⋊C4—10041no—
163C4⋊C4—10041no—
164C4⋊C4—10042no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—10042no—
168C22⋊C4—10042no—
169C22⋊C4—5024no—
170C42—5022no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23#1686080914yes—
178C23C23×D25111yescharacteristic
179D4—44no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—10048no—
190C2×C4—5022no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4#1686080912yes—
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#1686080912yes—
211C22C2×#1686080911yescharacteristic
212C22#2254876011yescharacteristic
213C22#2254876112yes—
214C22#2254876211yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—10042no—
220C4—10042no—
221C4—10042no—
222C4—10042no—
223C2—44no—
224C2#3063228112yes—
225C2#3063228211yescharacteristic
226C2#3063236811yescharacteristic
227C2#3063238912yes—
228C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#2782
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #30631627C2semidirect
2. #30631626C2semidirect
3. #30631615C2semidirect
4. #30631575C2semidirect
6. #30631562C2semidirect
7. #30631562C2semidirect
8. #30631552C2semidirect
114. C251#2782semidirect

Maximal subgroups

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

subgroupindex
#306316272
#306316262
#306316152
#306315752
#306315672
#306315622
#306315522
#394×C2512
#2782251