Finite Group Databasebeta

← #44558071

Subgroups — #44558071

Order 32128.

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

normal subgroups only

This group has 428 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#30631607C211yesmaximal, characteristic, complemented
2#30631605C212yesmaximal
3#30631604C211yesmaximal, characteristic
4#30631600C212yesmaximal, complemented
5#30631578C212yesmaximal, complemented
6#30631577C212yesmaximal, complemented
7#30631576C212yesmaximal, complemented
8#30631551C212yesmaximal, complemented
9#382×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×(C2×D251)⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×C4×Dic251C2212yes—
21C251⋊#95C2211yescharacteristic
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2211yescharacteristic
25C251⋊#95C2212yes—
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2212yes—
31C23×D251C2312yes—
32C2×Dic502—41no—
33C2×Dic502—42no—
34C2×D1004—41no—
35C2×D1004—42no—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
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—24no—
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2312yes—
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—22no—
63Dic502—44no—
64D1004—44no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251#4114yes—
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—22no—
82C2×C4×C251—21no—
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—42no—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C2×D251—42no—
100C4×C251—22no—
101C4×C251—42no—
102C4×C251—42no—
103C4×C251—41no—
104C4×C251—41no—
105Dic251—24no—
106Dic251—42no—
107Dic251—42no—
108C2×C251#53911yescharacteristic
109C2×C251#53912yes—
110C2×C251#55511yescharacteristic
111C2×C251#55512yes—
112C2×C251H(3,𝔽4)11yescharacteristic
113D251—82no—
114C251#278511yescharacteristic, complemented, Sylow
115#2785—2511nomaximal, Sylow
116#395—2511no—
117#394—2511no—
118#394—2512no—
119#393—2512no—
120#388—2512no—
121#388—2512no—
122#388—2512no—
123#382—2512no—
124#382D25111yescharacteristic, complemented
125C22×Q8—2511no—
126C22×D4—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×C42C2×D25111yescharacteristic
139#95—2511no—
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2511no—
144#95C2×D25112yes—
145#95C2×D25112yes—
146C24—2512no—
147C2×Q8—10041no—
148C2×Q8—10042no—
149C2×D4—10041no—
150C2×D4—10042no—
151C22×C4—2512no—
152C22×C4—2512no—
153C22×C4—2512no—
154C22×C4C22×D25112yes—
155C22×C4C22×D25112yes—
156C22×C4C22×D25111yescharacteristic
157C22×C4C22×D25111yescharacteristic
158C22×C4C22×D25111yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—10042no—
162C4⋊C4—10042no—
163C22⋊C4—10042no—
164C22⋊C4—10042no—
165C22⋊C4—10042no—
166C22⋊C4—5024no—
167C22⋊C4—5024no—
168C22⋊C4—5024no—
169C42—22no—
170C42—5024no—
171C23—5022no—
172C23—5022no—
173C23—5022no—
174C23—5022no—
175C23—5022no—
176C23—5022no—
177C23—5022no—
178C23C23×D25111yescharacteristic
179Q8—10044no—
180D4—10044no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
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—2514no—
202C2×C4#1686081012yes—
203C22—10042no—
204C22—10042no—
205C22—10042no—
206C22—10042no—
207C22—10042no—
208C22—10042no—
209C22—10042no—
210C22#41×D25112yes—
211C22C2×#1686081011yescharacteristic
212C22#2254876011yescharacteristic
213C22#2254876112yes—
214C22#2254876311yescharacteristic
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—10042no—
221C4—10042no—
222C4—5024no—
223C2—20082no—
224C2#3063228411yescharacteristic
225C2#3063228512yes—
226C2#3063239911yescharacteristic
227C2#3063240012yes—
228C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#382×C251C2
prime radicalC22×D1004C22
hypercenter#382D251
hypocenterC251#2785
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #30631607C2semidirect
4. #30631600C2semidirect
5. #30631578C2semidirect
6. #30631577C2semidirect
7. #30631576C2semidirect
8. #30631551C2semidirect
9. #382×C251C2semidirect
114. C251#2785semidirect
124. #382D251semidirect

Maximal subgroups

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

subgroupindex
#306316072
#306316052
#306316042
#306316002
#306315782
#306315772
#306315762
#306315512
#382×C2512
#2785251