Finite Group Databasebeta

← #44558458

Subgroups — #44558458

Order 32128.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#30631623C211yesmaximal, characteristic, complemented
3#30631613C212yesmaximal, complemented
4#30631609C212yesmaximal, complemented
5#30631605C212yesmaximal, complemented
6#30631602C212yesmaximal, complemented
7#30631596C211yesmaximal, characteristic
8#30631566C212yesmaximal, complemented
9#385×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C23×C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2212yes—
28C2×C22⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2212yes—
31C23×D251C2312yes—
32C2×D1004—41no—
33C2×D1004—42no—
34C2×C22⋊D251—24no—
35C2×C22⋊D251—42no—
36C2×C22⋊D251—42no—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—41no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43C22⋊Dic251—42no—
44(C2×D251)⋊C4—42no—
45(C2×D251)⋊C4—42no—
46(C2×D251)⋊C4—42no—
47(C2×D251)⋊C4—42no—
48(C2×D251)⋊C4—42no—
49C24×C251C2311yescharacteristic
50C22×C4×C251—42no—
51C22×C4×C251—41no—
52C22×C4×C251C2311yescharacteristic
53C22×C4×C251C2312yes—
54C22×C4×C251C2312yes—
55C22×C4×C251C2311yescharacteristic
56C4⋊C4×C251—42no—
57C22⋊C4×C251—22no—
58C22⋊C4×C251—41no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—22no—
64C22×D251—22no—
65C22×D251—22no—
66D1004—44no—
67C22⋊D251—48no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C23×C251—21no—
78C23×C251—22no—
79C23×C251—22no—
80C23×C251—21no—
81C23×C251#4112yes—
82C23×C251C2411yescharacteristic
83C2×C4×C251—22no—
84C2×C4×C251—21no—
85C2×C4×C251—21no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—44no—
94C22×C251—24no—
95C22×C251—22no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251C2×#4111yescharacteristic
100C22×C251H(5,𝔽2)12yes—
101C22×C251H(5,𝔽2)11yescharacteristic
102C22×C251H(5,𝔽2)12yes—
103C22×C2512−1+411yescharacteristic
104C2×D251—42no—
105C2×D251—42no—
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—42no—
109C2×D251—42no—
110C2×D251—42no—
111C4×C251—42no—
112C4×C251—41no—
113C4×C251—42no—
114C4×C251—41no—
115Dic251—42no—
116Dic251—42no—
117Dic251—42no—
118C2×C251—42no—
119C2×C251#53812yes—
120C2×C251#53911yescharacteristic
121C2×C251#56011yescharacteristic
122C2×C251#56011yescharacteristic
123C2×C251#56212yes—
124D251—82no—
125C251#282611yescharacteristic, complemented, Sylow
126#2826—2511nomaximal, Sylow
127#400—2512no—
128#399—2511no—
129#397—2512no—
130#396—2512no—
131#394—2512no—
132#394—2512no—
133#392—2511no—
134#386—2512no—
135#385D25111yescharacteristic, complemented
136C22×D4—2512no—
137C22×D4—2511no—
138C23×C4C2×D25111yescharacteristic
139C2×C4⋊C4—2512no—
140C2×C4⋊C4—2512no—
141C2×C4⋊C4—2511no—
142C2×C4⋊C4C2×D25112yes—
143C2×C22⋊C4—2512no—
144C2×C22⋊C4—2512no—
145C2×C22⋊C4—2512no—
146C2×C22⋊C4—2512no—
147C2×C22⋊C4—2512no—
148C2×C22⋊C4—2512no—
149C2×C22⋊C4C2×D25111yescharacteristic
150C2×C22⋊C4C2×D25111yescharacteristic
151#95—2512no—
152#95—2512no—
153#95—2511no—
154#95—2512no—
155#95—2511no—
156#95C2×D25112yes—
157C24—2512no—
158C24C22×D25111yescharacteristic
159C2×D4—10041no—
160C2×D4—10042no—
161C2×D4—10042no—
162C2×D4—10042no—
163C2×D4—5024no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—2512no—
167C22×C4—2512no—
168C22×C4—2512no—
169C22×C4C22×D25112yes—
170C22×C4C22×D25111yescharacteristic
171C22×C4C22×D25112yes—
172C22×C4C22×D25111yescharacteristic
173C4⋊C4—42no—
174C4⋊C4—10042no—
175C4⋊C4—10042no—
176C4⋊C4—10041no—
177C22⋊C4—22no—
178C22⋊C4—41no—
179C22⋊C4—10042no—
180C22⋊C4—10042no—
181C22⋊C4—10042no—
182C22⋊C4—10042no—
183C22⋊C4—10042no—
184C22⋊C4—10042no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—5022no—
190C23—5022no—
191C23—5022no—
192C23—5022no—
193C23—5022no—
194C23—5022no—
195C23—5022no—
196C23#1686081012yes—
197C23C23×D25111yescharacteristic
198D4—10044no—
199D4—10048no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5022no—
219C2×C4—5022no—
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—10042no—
226C22—10042no—
227C22—10042no—
228C22—10042no—
229C22—10042no—
230C22—10042no—
231C22—10042no—
232C22C2×#1686081011yescharacteristic
233C22#2254876012yes—
234C22#2254876011yescharacteristic
235C22#2254876112yes—
236C22#2254876311yescharacteristic
237C4—42no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—10042no—
242C4—10042no—
243C4—10042no—
244C2—42no—
245C2—20082no—
246C2#3063227012yes—
247C2#3063228411yescharacteristic
248C2#3063242611yescharacteristic
249C2#3063242711yescharacteristic
250C2#3063243312yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#385×C251C2
prime radical#30631596C2
hypercenter#385D251
hypocenterC251#2826
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #30631628C2semidirect
2. #30631623C2semidirect
3. #30631613C2semidirect
4. #30631609C2semidirect
5. #30631605C2semidirect
6. #30631602C2semidirect
8. #30631566C2semidirect
9. #385×C251C2semidirect
125. C251#2826semidirect
135. #385D251semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316232
#306316132
#306316092
#306316052
#306316022
#306315962
#306315662
#385×C2512
#2826251