Finite Group Databasebeta

← #44558039

Subgroups — #44558039

Order 32128.

448 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 121 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#30631628C212yesmaximal, complemented
2#30631626C211yesmaximal, characteristic, complemented
3#30631616C212yesmaximal, complemented
4#30631604C211yesmaximal, characteristic
5#30631578C212yesmaximal, complemented
6#30631565C212yesmaximal
7#30631561C212yesmaximal, complemented
8#30631558C212yesmaximal, complemented
9#382×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C4×D251C2212yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2211yescharacteristic
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2212yes—
31C23×D251C2312yes—
32C2×D1004—24no—
33C2×D1004—41no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C2×C4×D251—42no—
38C2×C4×D251—42no—
39C2×C4×D251—42no—
40C4⋊Dic251—41no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43(C2×D251)⋊C4—24no—
44(C2×D251)⋊C4—24no—
45(C2×D251)⋊C4—24no—
46(C2×D251)⋊C4—42no—
47(C2×D251)⋊C4—42no—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2311yescharacteristic
53C4⋊C4×C251—41no—
54C4⋊C4×C251—41no—
55C42×C251—22no—
56C22×D251—22no—
57C22×D251—22no—
58C22×D251—22no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251#4114yes—
63D1004—44no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C4×D251—48no—
74C23×C251C2411yescharacteristic
75C2×C4×C251—21no—
76C2×C4×C251—21no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—21no—
82C2×C4×C251—22no—
83C2×C4×C251—21no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251#4112yes—
88C22×C251C2×#4112yes—
89C22×C251C2×#4111yescharacteristic
90C22×C251H(5,𝔽2)11yescharacteristic
91C22×C251H(5,𝔽2)12yes—
92C22×C2512−1+411yescharacteristic
93C2×D251—24no—
94C2×D251—24no—
95C2×D251—24no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C2×D251—42no—
100C4×C251—22no—
101C4×C251—41no—
102C4×C251—41no—
103C4×C251—42no—
104C4×C251—42no—
105Dic251—42no—
106Dic251—42no—
107Dic251—42no—
108C2×C251#53911yescharacteristic
109C2×C251#53912yes—
110C2×C251#55111yescharacteristic
111C2×C251#55312yes—
112C2×C251H(3,𝔽4)11yescharacteristic
113D251—44no—
114C251#278211yescharacteristic, complemented, Sylow
115#2782—2511nomaximal, Sylow
116#400—2511no—
117#400—2512no—
118#397—2512no—
119#394—2511no—
120#388—2512no—
121#386—2512no—
122#385—2512no—
123#385—2512no—
124#382D25111yescharacteristic, complemented
125C22×D4—2511no—
126C23×C4—2512no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2511no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4C2×D25111yescharacteristic
131C2×C4⋊C4C2×D25111yescharacteristic
132C2×C22⋊C4—2512no—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C42C2×D25111yescharacteristic
138#95—2512no—
139#95—2512no—
140#95—2512no—
141#95—2512no—
142#95—2511no—
143#95—2511no—
144#95C2×D25112yes—
145#95C2×D25112yes—
146C24—2512no—
147C2×D4—10041no—
148C2×D4—5024no—
149C22×C4—10042no—
150C22×C4—10042no—
151C22×C4—10042no—
152C22×C4—2512no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4C22×D25111yescharacteristic
156C22×C4C22×D25112yes—
157C22×C4C22×D25111yescharacteristic
158C22×C4C22×D25112yes—
159C22×C4C22×D25111yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—10042no—
163C4⋊C4—10042no—
164C4⋊C4—10041no—
165C22⋊C4—10042no—
166C22⋊C4—10042no—
167C22⋊C4—5024no—
168C22⋊C4—5024no—
169C22⋊C4—5024no—
170C42—22no—
171C23—5022no—
172C23—5022no—
173C23—5022no—
174C23—5022no—
175C23—5022no—
176C23—5022no—
177C23—2514no—
178C23C23×D25111yescharacteristic
179D4—10044no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—10048no—
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#1686081012yes—
203C22—10042no—
204C22—10042no—
205C22—10042no—
206C22—10042no—
207C22—5024no—
208C22—5024no—
209C22—5024no—
210C22#41×D25112yes—
211C22C2×#1686081011yescharacteristic
212C22#2254876011yescharacteristic
213C22#2254876112yes—
214C22#2254876311yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—42no—
220C4—10042no—
221C4—10042no—
222C4—10042no—
223C2—10044no—
224C2#3063228411yescharacteristic
225C2#3063228512yes—
226C2#3063236711yescharacteristic
227C2#3063238112yes—
228C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#382×C251C2
prime radicalC22×D1004C22
hypercenter#382D251
hypocenterC251#2782
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #30631628C2semidirect
2. #30631626C2semidirect
3. #30631616C2semidirect
5. #30631578C2semidirect
7. #30631561C2semidirect
8. #30631558C2semidirect
9. #382×C251C2semidirect
114. C251#2782semidirect
124. #382D251semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316262
#306316162
#306316042
#306315782
#306315652
#306315612
#306315582
#382×C2512
#2782251