Finite Group Databasebeta

← #44558420

Subgroups — #44558420

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×#22548705C211yesmaximal, characteristic, complemented
2C2×#22548704C211yesmaximal, characteristic
3#30631622C212yesmaximal, complemented
4#30631616C212yesmaximal, complemented
5#30631611C212yesmaximal, complemented
6#30631609C212yesmaximal, complemented
7#30631606C211yesmaximal, characteristic, complemented
8#30631605C211yesmaximal, characteristic
9#30631604C211yesmaximal, characteristic
10#30631551C211yesmaximal, characteristic, complemented
11#395×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13#22548705—24no—
14#22548704—24no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×Dic251⋊C4C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2212yes—
21C2×Dic251⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×(C2×D251)⋊C4C2212yes—
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2212yes—
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C2×C4×Dic251C2211yescharacteristic
29C251⋊#95C2211yescharacteristic
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×Q8×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C4⋊C4×C251C2212yes—
35#95×C251C2211yescharacteristic
36#95×C251C2212yes—
37C23×D251C2312yes—
38C2×D1004—22no—
39C2×D1004—42no—
40C22×Dic251C2312yes—
41C22×Dic251C2311yescharacteristic
42C22×Dic251C2311yescharacteristic
43C22×Dic251C2311yescharacteristic
44C22×Dic251C2311yescharacteristic
45C4⋊Dic251—22no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48Dic251⋊C4—22no—
49Dic251⋊C4—22no—
50Dic251⋊C4—42no—
51Dic251⋊C4—42no—
52(C2×D251)⋊C4—24no—
53(C2×D251)⋊C4—24no—
54(C2×D251)⋊C4—42no—
55(C2×D251)⋊C4—42no—
56(C2×D251)⋊C4—42no—
57(C2×D251)⋊C4—42no—
58C4×Dic251D414yes—
59C2×Q8×C251—22no—
60C2×Q8×C251—42no—
61C22×C4×C251C2312yes—
62C22×C4×C251C2311yescharacteristic
63C22×C4×C251C2312yes—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—22no—
67C4⋊C4×C251—42no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—22no—
75D1004—44no—
76C2×Dic251—21no—
77C2×Dic251—22no—
78C2×Dic251—21no—
79C2×Dic251—21no—
80C2×Dic251—21no—
81C2×Dic251—21no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—21no—
86C2×Dic251—22no—
87C2×Dic251C2×D412yes—
88C2×Dic251C2×D412yes—
89C23×C251C2411yescharacteristic
90Q8×C251—44no—
91C2×C4×C251—22no—
92C2×C4×C251—21no—
93C2×C4×C251—21no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—21no—
100C2×C4×C251—22no—
101C2×C4×C251—21no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D412yes—
104C22×C251C22×D411yescharacteristic
105C22×C251H(5,𝔽2)12yes—
106C22×C251H(5,𝔽2)11yescharacteristic
107C22×C2512−1+412yes—
108C22×C2512−1+411yescharacteristic
109C2×D251—42no—
110C2×D251—42no—
111C2×D251—42no—
112C2×D251—42no—
113C2×D251—42no—
114C2×D251—42no—
115C2×D251—42no—
116C4×C251—22no—
117C4×C251—41no—
118C4×C251—42no—
119C4×C251—41no—
120C4×C251—42no—
121Dic251—22no—
122Dic251—22no—
123Dic251—41no—
124Dic251—41no—
125Dic251—42no—
126C2×C251#53511yescharacteristic
127C2×C251#53611yescharacteristic
128C2×C251#53711yescharacteristic
129C2×C251#56212yes—
130C2×C251#56312yes—
131D251—82no—
132C251#282011yescharacteristic, complemented, Sylow
133#2820—2511nomaximal, Sylow
134C2×#125—2511no—
135C2×#124—2511no—
136#399—2512no—
137#397—2512no—
138#396—2512no—
139#396—2512no—
140#395—2511no—
141#395D25111yescharacteristic, complemented
142#394—2511no—
143#394—2511no—
144#382—2511no—
145C22×Q8C2×D25111yescharacteristic
146C22×D4—2511no—
147#125—5024no—
148#124—5024no—
149C2×C4⋊C4—2512no—
150C2×C4⋊C4—2512no—
151C2×C4⋊C4—2511no—
152C2×C4⋊C4—2511no—
153C2×C4⋊C4—2511no—
154C2×C4⋊C4—2511no—
155C2×C4⋊C4—2511no—
156C2×C4⋊C4C2×D25111yescharacteristic
157C2×C4⋊C4C2×D25112yes—
158C2×C22⋊C4—2512no—
159C2×C22⋊C4—2512no—
160C2×C22⋊C4—2512no—
161C2×C22⋊C4—2512no—
162C2×C22⋊C4—2512no—
163C2×C22⋊C4—2512no—
164C2×C42—2511no—
165#95—2511no—
166#95—2512no—
167#95—2512no—
168#95C2×D25111yescharacteristic
169#95C2×D25112yes—
170C24—2512no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×D4—10042no—
174C2×D4—5022no—
175C22×C4—2512no—
176C22×C4—2511no—
177C22×C4—2511no—
178C22×C4—2511no—
179C22×C4—2511no—
180C22×C4C22×D25112yes—
181C22×C4C22×D25112yes—
182C22×C4C22×D25111yescharacteristic
183C22×C4C22×D25111yescharacteristic
184C22×C4C22×D25111yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—10042no—
188C4⋊C4—10042no—
189C4⋊C4—5022no—
190C4⋊C4—5022no—
191C4⋊C4—5022no—
192C4⋊C4—5022no—
193C4⋊C4—5022no—
194C22⋊C4—10042no—
195C22⋊C4—10042no—
196C22⋊C4—10042no—
197C22⋊C4—10042no—
198C22⋊C4—5024no—
199C22⋊C4—5024no—
200C42—2514no—
201C23—5022no—
202C23—5022no—
203C23—5022no—
204C23—5022no—
205C23—5022no—
206C23—5022no—
207C23—5022no—
208C23C23×D25111yescharacteristic
209Q8—44no—
210D4—10044no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—5021no—
224C2×C4—5021no—
225C2×C4—5022no—
226C2×C4—5022no—
227C2×C4—5021no—
228C2×C4—5022no—
229C2×C4—5021no—
230C2×C4—5022no—
231C2×C4—5021no—
232C2×C4—5022no—
233C2×C4—5021no—
234C2×C4—2512no—
235C2×C4—2512no—
236C2×C4D4×D25112yes—
237C22—10042no—
238C22—10042no—
239C22—10042no—
240C22—10042no—
241C22—10042no—
242C22—10042no—
243C22—10042no—
244C22C2×D4×D25111yescharacteristic
245C22#2254876012yes—
246C22#2254876111yescharacteristic
247C22#2254876312yes—
248C22#2254876311yescharacteristic
249C4—22no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—42no—
254C4—10041no—
255C4—10041no—
256C4—10042no—
257C4—5022no—
258C4—5022no—
259C2—20082no—
260C2#3063225211yescharacteristic
261C2#3063226211yescharacteristic
262C2#3063226711yescharacteristic
263C2#3063243312yes—
264C2#3063243612yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#395×C251C2
prime radicalC22×D1004C22
hypercenter#395D251
hypocenterC251#2820
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#22548705C2semidirect
3. #30631622C2semidirect
4. #30631616C2semidirect
5. #30631611C2semidirect
6. #30631609C2semidirect
7. #30631606C2semidirect
10. #30631551C2semidirect
11. #395×C251C2semidirect
132. C251#2820semidirect
141. #395D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487052
C2×#225487042
#306316222
#306316162
#306316112
#306316092
#306316062
#306316052
#306316042
#306315512
#395×C2512
#2820251