Finite Group Databasebeta

← #44558421

Subgroups — #44558421

Order 32128.

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

normal subgroups only

This group has 464 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×#22548706C211yesmaximal, characteristic, complemented
2C2×#22548701C211yesmaximal, characteristic
3#30631624C211yesmaximal, characteristic, complemented
4#30631623C211yesmaximal, characteristic, complemented
5#30631613C211yesmaximal, characteristic, complemented
6#30631611C212yesmaximal, complemented
7#30631610C212yesmaximal, complemented
8#30631608C212yesmaximal, complemented
9#30631603C212yesmaximal
10#30631551C211yesmaximal, characteristic, complemented
11#397×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×C22⋊D251C2211yescharacteristic
14#22548706—24no—
15#22548701—24no—
16C2×C4⋊Dic251C2212yes—
17C2×C4⋊Dic251C2211yescharacteristic
18C2×C4⋊Dic251C2212yes—
19C2×Dic251⋊C4C2212yes—
20C2×Dic251⋊C4C2212yes—
21C2×Dic251⋊C4C2212yes—
22C2×C22⋊Dic251C2211yescharacteristic
23C2×C22⋊Dic251C2212yes—
24C2×(C2×D251)⋊C4C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C2×C4×Dic251C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2211yescharacteristic
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C22⋊C4×C251C2211yescharacteristic
34C2×C22⋊C4×C251C2212yes—
35#95×C251C2211yescharacteristic
36#95×C251C2212yes—
37C23×D251C2311yescharacteristic
38C2×Dic502—22no—
39C2×Dic502—42no—
40C2×C22⋊D251—22no—
41C2×C22⋊D251—42no—
42C22×Dic251C2311yescharacteristic
43C22×Dic251C2312yes—
44C22×Dic251C2312yes—
45C22×Dic251C2311yescharacteristic
46C22×Dic251C2311yescharacteristic
47C4⋊Dic251—24no—
48C4⋊Dic251—22no—
49C4⋊Dic251—42no—
50Dic251⋊C4—24no—
51Dic251⋊C4—42no—
52Dic251⋊C4—42no—
53C22⋊Dic251—22no—
54C22⋊Dic251—42no—
55(C2×D251)⋊C4—22no—
56(C2×D251)⋊C4—22no—
57(C2×D251)⋊C4—42no—
58(C2×D251)⋊C4—42no—
59C4×Dic251D414yes—
60C24×C251C2311yescharacteristic
61C22×C4×C251C2311yescharacteristic
62C22×C4×C251C2312yes—
63C22×C4×C251C2312yes—
64C22×C4×C251C2311yescharacteristic
65C4⋊C4×C251—22no—
66C22⋊C4×C251—22no—
67C22⋊C4×C251—42no—
68C22×D251—21no—
69C22×D251—21no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—21no—
73Dic502—44no—
74C22⋊D251—44no—
75C2×Dic251—21no—
76C2×Dic251—22no—
77C2×Dic251—21no—
78C2×Dic251—21no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251C2×D412yes—
87C2×Dic251C2×D412yes—
88C23×C251—21no—
89C23×C251—21no—
90C23×C251—22no—
91C23×C251—22no—
92C23×C251—21no—
93C23×C251C2411yescharacteristic
94C2×C4×C251—22no—
95C2×C4×C251—21no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D412yes—
104C22×C251—41no—
105C22×C251—41no—
106C22×C251—41no—
107C22×C251—42no—
108C22×C251—42no—
109C22×C251C22×D411yescharacteristic
110C22×C251H(5,𝔽2)11yescharacteristic
111C22×C251H(5,𝔽2)12yes—
112C22×C2512−1+412yes—
113C22×C2512−1+411yescharacteristic
114C2×D251—41no—
115C2×D251—41no—
116C2×D251—41no—
117C2×D251—42no—
118C2×D251—42no—
119C4×C251—22no—
120C4×C251—41no—
121C4×C251—42no—
122C4×C251—42no—
123Dic251—22no—
124Dic251—22no—
125Dic251—42no—
126Dic251—41no—
127Dic251—42no—
128C2×C251—81no—
129C2×C251#53511yescharacteristic
130C2×C251#53611yescharacteristic
131C2×C251#53711yescharacteristic
132C2×C251#56212yes—
133C2×C251#56312yes—
134D251—81no—
135C251#282011yescharacteristic, complemented, Sylow
136#2820—2511nomaximal, Sylow
137C2×#125—2511no—
138C2×#124—2511no—
139#399—2511no—
140#399—2511no—
141#397—2511no—
142#397D25111yescharacteristic, complemented
143#396—2512no—
144#396—2512no—
145#395—2512no—
146#394—2512no—
147#382—2511no—
148C22×Q8—2511no—
149C22×D4—2511no—
150#125—5024no—
151#124—5024no—
152C2×C4⋊C4—2512no—
153C2×C4⋊C4—2512no—
154C2×C4⋊C4—2512no—
155C2×C4⋊C4—2512no—
156C2×C4⋊C4—2512no—
157C2×C4⋊C4—2511no—
158C2×C4⋊C4C2×D25111yescharacteristic
159C2×C22⋊C4—2512no—
160C2×C22⋊C4—2512no—
161C2×C22⋊C4—2512no—
162C2×C22⋊C4—2511no—
163C2×C22⋊C4—2511no—
164C2×C22⋊C4—2511no—
165C2×C22⋊C4C2×D25111yescharacteristic
166C2×C22⋊C4C2×D25112yes—
167C2×C42—2511no—
168#95—2512no—
169#95—2511no—
170#95—2512no—
171#95C2×D25111yescharacteristic
172#95C2×D25112yes—
173C24—2511no—
174C24C22×D25111yescharacteristic
175C2×Q8—10042no—
176C2×Q8—5022no—
177C2×D4—10042no—
178C2×D4—5022no—
179C22×C4—2512no—
180C22×C4—2511no—
181C22×C4—2512no—
182C22×C4—2511no—
183C22×C4—2511no—
184C22×C4C22×D25112yes—
185C22×C4C22×D25112yes—
186C22×C4C22×D25111yescharacteristic
187C22×C4C22×D25111yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—10042no—
190C4⋊C4—10042no—
191C4⋊C4—10042no—
192C4⋊C4—5024no—
193C4⋊C4—5024no—
194C4⋊C4—5022no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—10042no—
198C22⋊C4—10042no—
199C22⋊C4—10042no—
200C22⋊C4—5022no—
201C22⋊C4—5022no—
202C22⋊C4—5022no—
203C42—2514no—
204C23—21no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—21no—
209C23—5021no—
210C23—5021no—
211C23—5022no—
212C23—5022no—
213C23—5021no—
214C23C23×D25111yescharacteristic
215Q8—10044no—
216D4—10044no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—5021no—
227C2×C4—5022no—
228C2×C4—5022no—
229C2×C4—5022no—
230C2×C4—5022no—
231C2×C4—5022no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5021no—
235C2×C4—5022no—
236C2×C4—5021no—
237C2×C4—2512no—
238C2×C4—2512no—
239C2×C4D4×D25112yes—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—10041no—
246C22—10041no—
247C22—10041no—
248C22—10042no—
249C22—10042no—
250C22C2×D4×D25111yescharacteristic
251C22#2254876011yescharacteristic
252C22#2254876112yes—
253C22#2254876212yes—
254C22#2254876211yescharacteristic
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—10041no—
260C4—10042no—
261C4—10042no—
262C4—5022no—
263C4—5022no—
264C2—81no—
265C2—20081no—
266C2#3063225511yescharacteristic
267C2#3063225911yescharacteristic
268C2#3063226611yescharacteristic
269C2#3063243412yes—
270C2#3063243912yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#397×C251C2
prime radicalC22×C22⋊D251C22
hypercenter#397D251
hypocenterC251#2820
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#22548706C2semidirect
3. #30631624C2semidirect
4. #30631623C2semidirect
5. #30631613C2semidirect
6. #30631611C2semidirect
7. #30631610C2semidirect
8. #30631608C2semidirect
10. #30631551C2semidirect
11. #397×C251C2semidirect
135. C251#2820semidirect
142. #397D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487062
C2×#225487012
#306316242
#306316232
#306316132
#306316112
#306316102
#306316082
#306316032
#306315512
#397×C2512
#2820251