Finite Group Databasebeta

← #44558199

Subgroups — #44558199

Order 32128.

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

normal subgroups only

This group has 420 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#30631609C212yesmaximal, complemented
2#30631605C212yesmaximal
3#30631604C211yesmaximal, characteristic
4#30631579C212yesmaximal, complemented
5#30631579C212yesmaximal, complemented
6#30631576C212yesmaximal, complemented
7#30631570C212yesmaximal, complemented
8#30631556C211yesmaximal, characteristic, complemented
9#400×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C2×C4⋊Dic251C2212yes—
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—
20C2×(C2×D251)⋊C4C2212yes—
21C2×C4×Dic251C2211yescharacteristic
22C2×C4×Dic251C2212yes—
23C251⋊#95C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#95×C251C2212yes—
29#95×C251C2211yescharacteristic
30#95×C251C2212yes—
31C23×D251C2312yes—
32C2×D1004—41no—
33C2×D1004—42no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—24no—
38C4⋊Dic251—41no—
39Dic251⋊C4—42no—
40Dic251⋊C4—42no—
41(C2×D251)⋊C4—24no—
42(C2×D251)⋊C4—24no—
43(C2×D251)⋊C4—24no—
44(C2×D251)⋊C4—42no—
45(C2×D251)⋊C4—42no—
46(C2×D251)⋊C4—42no—
47C4×Dic251—22no—
48C4×Dic251—42no—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2312yes—
52C22×C4×C251C2311yescharacteristic
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251—41no—
55C4⋊C4×C251—41no—
56C22×D251—22no—
57C22×D251—22no—
58C22×D251—22no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—22no—
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#4114yes—
73C23×C251C2411yescharacteristic
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—21no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251#4112yes—
87C22×C251C2×#4112yes—
88C22×C251C2×#4111yescharacteristic
89C22×C251H(5,𝔽2)11yescharacteristic
90C22×C251H(5,𝔽2)12yes—
91C22×C2512−1+411yescharacteristic
92C2×D251—42no—
93C2×D251—42no—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C4×C251—22no—
100C4×C251—42no—
101C4×C251—42no—
102C4×C251—41no—
103C4×C251—41no—
104Dic251—24no—
105Dic251—42no—
106Dic251—42no—
107C2×C251#53911yescharacteristic
108C2×C251#54212yes—
109C2×C251#55312yes—
110C2×C251#55511yescharacteristic
111C2×C251#56011yescharacteristic
112D251—82no—
113C251#279511yescharacteristic, complemented, Sylow
114#2795—2511nomaximal, Sylow
115#400D25111yescharacteristic, complemented
116#396—2512no—
117#394—2511no—
118#394—2512no—
119#388—2512no—
120#388—2512no—
121#388—2512no—
122#387—2512no—
123#384—2511no—
124C22×D4—2511no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2511no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4C2×D25111yescharacteristic
130C2×C4⋊C4C2×D25111yescharacteristic
131C2×C22⋊C4—2512no—
132C2×C22⋊C4—2512no—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C42—2512no—
138C2×C42—2511no—
139#95—2511no—
140#95—2512no—
141#95—2512no—
142#95C2×D25112yes—
143#95C2×D25111yescharacteristic
144#95C2×D25112yes—
145C24—2512no—
146C2×D4—10041no—
147C2×D4—10042no—
148C22×C4—2512no—
149C22×C4—2512no—
150C22×C4—2512no—
151C22×C4C22×D25112yes—
152C22×C4C22×D25112yes—
153C22×C4C22×D25111yescharacteristic
154C22×C4C22×D25111yescharacteristic
155C22×C4C22×D25111yescharacteristic
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—10042no—
159C4⋊C4—10042no—
160C4⋊C4—10041no—
161C4⋊C4—5024no—
162C22⋊C4—10042no—
163C22⋊C4—10042no—
164C22⋊C4—10042no—
165C22⋊C4—5024no—
166C22⋊C4—5024no—
167C22⋊C4—5024no—
168C42—10042no—
169C42—5022no—
170C23—5022no—
171C23—5022no—
172C23—5022no—
173C23—5022no—
174C23—5022no—
175C23—5022no—
176C23—5022no—
177C23C23×D25111yescharacteristic
178D4—10044no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
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—2514no—
200C2×C4#1686081112yes—
201C22—10042no—
202C22—10042no—
203C22—10042no—
204C22—10042no—
205C22—10042no—
206C22—10042no—
207C22—10042no—
208C22#41×D25112yes—
209C22C2×#1686081111yescharacteristic
210C22#2254876011yescharacteristic
211C22#2254876112yes—
212C22#2254876211yescharacteristic
213C4—22no—
214C4—42no—
215C4—42no—
216C4—41no—
217C4—41no—
218C4—10042no—
219C4—10042no—
220C4—5024no—
221C2—20082no—
222C2#3063228611yescharacteristic
223C2#3063230312yes—
224C2#3063239012yes—
225C2#3063239911yescharacteristic
226C2#3063242711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#400×C251C2
prime radicalC22×D1004C22
hypercenter#400D251
hypocenterC251#2795
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #30631609C2semidirect
4. #30631579C2semidirect
5. #30631579C2semidirect
6. #30631576C2semidirect
7. #30631570C2semidirect
8. #30631556C2semidirect
9. #400×C251C2semidirect
113. C251#2795semidirect
115. #400D251semidirect

Maximal subgroups

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

subgroupindex
#306316092
#306316052
#306316042
#306315792
#306315762
#306315702
#306315562
#400×C2512
#2795251