Finite Group Databasebeta

← #46839653

Subgroups — #46839653

Order 34688.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22576125C211yesmaximal, characteristic, complemented
2C2×#22576122C211yesmaximal, characteristic
3#31773835C211yesmaximal, characteristic, complemented
4#31773826C212yesmaximal, complemented
5#31773815C211yesmaximal, characteristic
6#31773814C211yesmaximal, characteristic
7#31773810C212yesmaximal, complemented
8#31773744C212yesmaximal, complemented
9#31773744C212yesmaximal, complemented
10#31773742C211yesmaximal, characteristic, complemented
11#400×C271C211yesmaximal, characteristic, complemented
12C22×Dic542C2211yescharacteristic
13C22×D1084C2211yescharacteristic
14C22×C4×D271C2212yes—
15#22576125—24no—
16#22576122—24no—
17C2×Dic271⋊C4C2211yescharacteristic
18C2×Dic271⋊C4C2211yescharacteristic
19C2×(C2×D271)⋊C4C2212yes—
20C2×(C2×D271)⋊C4C2212yes—
21C2×(C2×D271)⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2212yes—
24C2×C4×Dic271C2212yes—
25C2×C4×Dic271C2211yescharacteristic
26C2×C4×Dic271C2211yescharacteristic
27C2×C4×Dic271C2211yescharacteristic
28C271⋊#95C2212yes—
29C271⋊#95C2211yescharacteristic
30C271⋊#95C2212yes—
31C271⋊#95C2212yes—
32C2×C4⋊C4×C271C2211yescharacteristic
33C2×C4⋊C4×C271C2211yescharacteristic
34#95×C271C2211yescharacteristic
35#95×C271C2212yes—
36#95×C271C2212yes—
37C23×D271C2312yes—
38C2×Dic542—22no—
39C2×Dic542—22no—
40C2×Dic542—22no—
41C2×D1084—24no—
42C2×D1084—22no—
43C22×Dic271C2312yes—
44C22×Dic271C2311yescharacteristic
45C22×Dic271C2311yescharacteristic
46C22×Dic271C2311yescharacteristic
47C22×Dic271C2311yescharacteristic
48C2×C4×D271—24no—
49C2×C4×D271—24no—
50C2×C4×D271—24no—
51Dic271⋊C4—22no—
52Dic271⋊C4—22no—
53(C2×D271)⋊C4—24no—
54(C2×D271)⋊C4—24no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57(C2×D271)⋊C4—24no—
58C4×Dic271—22no—
59C4×Dic271—22no—
60C4×Dic271—24no—
61C4×Dic271D414yes—
62C22×C4×C271C2312yes—
63C22×C4×C271C2312yes—
64C22×C4×C271C2311yescharacteristic
65C22×C4×C271C2311yescharacteristic
66C22×C4×C271C2311yescharacteristic
67C4⋊C4×C271—22no—
68C4⋊C4×C271—22no—
69C22×D271—22no—
70C22×D271—22no—
71C22×D271—22no—
72C22×D271—22no—
73C22×D271—22no—
74C22×D271—22no—
75C22×D271#4114yes—
76Dic542—28no—
77D1084—28no—
78C2×Dic271—21no—
79C2×Dic271—21no—
80C2×Dic271—21no—
81C2×Dic271—21no—
82C2×Dic271—21no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271C2×D412yes—
88C2×Dic271C2×D412yes—
89C2×Dic271#4112yes—
90C2×Dic271#4114yes—
91C4×D271—48no—
92C23×C271C2411yescharacteristic
93C2×C4×C271—21no—
94C2×C4×C271—22no—
95C2×C4×C271—21no—
96C2×C4×C271—21no—
97C2×C4×C271—22no—
98C2×C4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271C2×D412yes—
105C2×C4×C271#4112yes—
106C22×C271C22×D411yescharacteristic
107C22×C271C2×#4112yes—
108C22×C271C2×#4111yescharacteristic
109C22×C271C2×#4112yes—
110C22×C271C2×#4111yescharacteristic
111C2×D271—24no—
112C2×D271—24no—
113C2×D271—24no—
114C2×D271—42no—
115C2×D271—42no—
116C2×D271—42no—
117C2×D271—42no—
118C4×C271—22no—
119C4×C271—22no—
120C4×C271—41no—
121C4×C271—42no—
122C4×C271—42no—
123Dic271—22no—
124Dic271—22no—
125Dic271—41no—
126Dic271—42no—
127Dic271#12414yes—
128C2×C271#52511yescharacteristic
129C2×C271#52511yescharacteristic
130C2×C271C2×#12411yescharacteristic
131C2×C271#52912yes—
132C2×C271#52912yes—
133D271—44no—
134C271#239811yescharacteristic, complemented, Sylow
135#2398—2711nomaximal, Sylow
136C2×C4×Q8—2711no—
137C2×C4×D4—2711no—
138#400—2711no—
139#400D27111yescharacteristic, complemented
140#397—2712no—
141#394—2711no—
142#394—2711no—
143#393—2712no—
144C4×C22⋊C4—2712no—
145C4×C22⋊C4—2712no—
146#376—2711no—
147C22×Q8—2711no—
148C22×D4—2711no—
149C23×C4—2712no—
150C4×Q8—5424no—
151C4×D4—5424no—
152C2×C4⋊C4—2711no—
153C2×C4⋊C4—2711no—
154C2×C4⋊C4C2×D27111yescharacteristic
155C2×C4⋊C4C2×D27111yescharacteristic
156C2×C22⋊C4—2712no—
157C2×C22⋊C4—2712no—
158C2×C22⋊C4—2712no—
159C2×C22⋊C4—2712no—
160C2×C22⋊C4—2712no—
161C2×C42—2711no—
162C2×C42—2711no—
163C2×C42—2711no—
164C2×C42—2712no—
165#95—2712no—
166#95—2711no—
167#95—2712no—
168#95—2712no—
169#95C2×D27111yescharacteristic
170#95C2×D27112yes—
171#95C2×D27112yes—
172C24—2712no—
173C2×Q8—5422no—
174C2×Q8—5422no—
175C2×Q8—5422no—
176C2×D4—5424no—
177C2×D4—5422no—
178C22×C4—2711no—
179C22×C4—2711no—
180C22×C4—2711no—
181C22×C4—2712no—
182C22×C4—2711no—
183C22×C4—5424no—
184C22×C4—5424no—
185C22×C4—5424no—
186C22×C4C22×D27112yes—
187C22×C4C22×D27112yes—
188C22×C4C22×D27111yescharacteristic
189C22×C4C22×D27111yescharacteristic
190C22×C4C22×D27111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—5422no—
194C4⋊C4—5422no—
195C22⋊C4—5424no—
196C22⋊C4—5424no—
197C22⋊C4—5424no—
198C22⋊C4—5424no—
199C22⋊C4—5424no—
200C42—2714no—
201C42—5424no—
202C42—5422no—
203C42—5422no—
204C23—2714no—
205C23—5422no—
206C23—5422no—
207C23—5422no—
208C23—5422no—
209C23—5422no—
210C23—5422no—
211C23C23×D27111yescharacteristic
212Q8—5428no—
213D4—5428no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—2712no—
226C2×C4—2714no—
227C2×C4—2712no—
228C2×C4—2712no—
229C2×C4—5421no—
230C2×C4—5421no—
231C2×C4—5421no—
232C2×C4—5421no—
233C2×C4—5421no—
234C2×C4—5422no—
235C2×C4—5422no—
236C2×C4—5422no—
237C2×C4—5422no—
238C2×C4—10848no—
239C2×C4D4×D27112yes—
240C2×C4#1686465512yes—
241C22—5424no—
242C22—5424no—
243C22—5424no—
244C22—10842no—
245C22—10842no—
246C22—10842no—
247C22—10842no—
248C22C2×D4×D27111yescharacteristic
249C22#41×D27112yes—
250C22#41×D27111yescharacteristic
251C22#41×D27112yes—
252C22C2×#1686465511yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—2714no—
259C4—5422no—
260C4—5422no—
261C4—10841no—
262C4—10842no—
263C2—10844no—
264C2#3177440511yescharacteristic
265C2#3177441011yescharacteristic
266C2#124×D27111yescharacteristic
267C2#3177443012yes—
268C2#3177443212yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#400×C271C2
prime radicalC22×D1084C22
hypercenter#400D271
hypocenterC271#2398
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#22576125C2semidirect
3. #31773835C2semidirect
4. #31773826C2semidirect
7. #31773810C2semidirect
8. #31773744C2semidirect
9. #31773744C2semidirect
10. #31773742C2semidirect
11. #400×C271C2semidirect
134. C271#2398semidirect
139. #400D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761252
C2×#225761222
#317738352
#317738262
#317738152
#317738142
#317738102
#317737442
#317737422
#400×C2712
#2398271