Finite Group Databasebeta

← #44558430

Subgroups — #44558430

Order 32128.

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

normal subgroups only

This group has 430 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
1C2×#22548714C211yesmaximal, characteristic, complemented
2C2×#22548710C211yesmaximal, characteristic
3#30631626C211yesmaximal, characteristic, complemented
4#30631624C211yesmaximal, characteristic
5#30631622C211yesmaximal, characteristic
6#30631621C212yesmaximal, complemented
7#30631616C212yesmaximal
8#30631608C212yesmaximal, complemented
9#30631600C212yesmaximal
10#30631554C211yesmaximal, characteristic, complemented
11#400×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2212yes—
13#22548714—24no—
14#22548710—24no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2212yes—
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2212yes—
20C2×Dic251⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2211yescharacteristic
22C2×(C2×D251)⋊C4C2211yescharacteristic
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2212yes—
26C2×C4×Dic251C2211yescharacteristic
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2211yescharacteristic
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34#95×C251C2211yescharacteristic
35#95×C251C2212yes—
36#95×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×Dic502—24no—
39C2×Dic502—42no—
40C2×Dic502—42no—
41C22×Dic251C2312yes—
42C22×Dic251C2312yes—
43C22×Dic251C2311yescharacteristic
44C22×Dic251C2311yescharacteristic
45C22×Dic251C2311yescharacteristic
46C4⋊Dic251—22no—
47C4⋊Dic251—22no—
48C4⋊Dic251—42no—
49Dic251⋊C4—22no—
50Dic251⋊C4—24no—
51Dic251⋊C4—42no—
52(C2×D251)⋊C4—22no—
53(C2×D251)⋊C4—22no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—42no—
56(C2×D251)⋊C4—42no—
57C4×Dic251D414yes—
58C22×C4×C251C2312yes—
59C22×C4×C251C2312yes—
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2311yescharacteristic
62C22×C4×C251C2311yescharacteristic
63C4⋊C4×C251—22no—
64C4⋊C4×C251—22no—
65C22×D251—21no—
66C22×D251—22no—
67C22×D251—22no—
68C22×D251—21no—
69C22×D251—21no—
70Dic502—48no—
71C2×Dic251—22no—
72C2×Dic251—21no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251C2×D412yes—
83C2×Dic251C2×D412yes—
84C23×C251C2411yescharacteristic
85C2×C4×C251—22no—
86C2×C4×C251—21no—
87C2×C4×C251—21no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—21no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251—21no—
97C2×C4×C251C2×D412yes—
98C22×C251C22×D411yescharacteristic
99C22×C251H(5,𝔽2)12yes—
100C22×C2512−1+412yes—
101C22×C2512−1+411yescharacteristic
102C22×C2512−1+411yescharacteristic
103C2×D251—41no—
104C2×D251—41no—
105C2×D251—41no—
106C2×D251—42no—
107C2×D251—42no—
108C4×C251—22no—
109C4×C251—42no—
110C4×C251—41no—
111C4×C251—41no—
112C4×C251—42no—
113Dic251—22no—
114Dic251—22no—
115Dic251—41no—
116Dic251—42no—
117Dic251—42no—
118C2×C251#53511yescharacteristic
119C2×C251#53611yescharacteristic
120C2×C251#53611yescharacteristic
121C2×C251#56312yes—
122C2×C251#56312yes—
123D251—81no—
124C251#282111yescharacteristic, complemented, Sylow
125#2821—2511nomaximal, Sylow
126C2×C4⋊Q8—2511no—
127C2×#126—2511no—
128#400—2511no—
129#400D25111yescharacteristic, complemented
130#399—2511no—
131#399—2511no—
132#398—2512no—
133#397—2512no—
134#395—2512no—
135#393—2512no—
136#383—2511no—
137C22×Q8—2512no—
138C4⋊Q8—5024no—
139#126—5024no—
140C2×C4⋊C4—2512no—
141C2×C4⋊C4—2512no—
142C2×C4⋊C4—2511no—
143C2×C4⋊C4—2511no—
144C2×C4⋊C4—2512no—
145C2×C4⋊C4—2511no—
146C2×C4⋊C4C2×D25111yescharacteristic
147C2×C4⋊C4C2×D25111yescharacteristic
148C2×C22⋊C4—2512no—
149C2×C22⋊C4—2512no—
150C2×C22⋊C4—2511no—
151C2×C22⋊C4—2511no—
152C2×C22⋊C4—2511no—
153C2×C42—2511no—
154#95—2512no—
155#95—2512no—
156#95—2511no—
157#95—2512no—
158#95—2512no—
159#95C2×D25111yescharacteristic
160#95C2×D25112yes—
161#95C2×D25112yes—
162C24—2511no—
163C2×Q8—10042no—
164C2×Q8—10042no—
165C2×Q8—5024no—
166C22×C4—2512no—
167C22×C4—2512no—
168C22×C4—2511no—
169C22×C4—2511no—
170C22×C4—2511no—
171C22×C4C22×D25112yes—
172C22×C4C22×D25111yescharacteristic
173C22×C4C22×D25111yescharacteristic
174C22×C4C22×D25112yes—
175C22×C4C22×D25111yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—10042no—
179C4⋊C4—10042no—
180C4⋊C4—5022no—
181C4⋊C4—5024no—
182C4⋊C4—5022no—
183C4⋊C4—5022no—
184C22⋊C4—10042no—
185C22⋊C4—10042no—
186C22⋊C4—5022no—
187C22⋊C4—5022no—
188C22⋊C4—5022no—
189C42—2514no—
190C23—5021no—
191C23—5022no—
192C23—5022no—
193C23—5021no—
194C23—5021no—
195C23C23×D25111yescharacteristic
196Q8—10048no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—5021no—
210C2×C4—5022no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5021no—
219C2×C4—5021no—
220C2×C4—2512no—
221C2×C4—2512no—
222C2×C4D4×D25112yes—
223C22—10041no—
224C22—10041no—
225C22—10041no—
226C22—10042no—
227C22—10042no—
228C22C2×D4×D25111yescharacteristic
229C22#2254876112yes—
230C22#2254876212yes—
231C22#2254876211yescharacteristic
232C22#2254876311yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—10042no—
239C4—10041no—
240C4—10042no—
241C4—5022no—
242C4—5022no—
243C2—20081no—
244C2#3063225511yescharacteristic
245C2#3063225911yescharacteristic
246C2#3063226011yescharacteristic
247C2#3063243912yes—
248C2#3063244112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#400×C251C2
prime radicalC23×D251C23
hypercenter#400D251
hypocenterC251#2821
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#22548714C2semidirect
3. #30631626C2semidirect
6. #30631621C2semidirect
8. #30631608C2semidirect
10. #30631554C2semidirect
11. #400×C251C2semidirect
124. C251#2821semidirect
129. #400D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487142
C2×#225487102
#306316262
#306316242
#306316222
#306316212
#306316162
#306316082
#306316002
#306315542
#400×C2512
#2821251