Finite Group Databasebeta

← #46839649

Subgroups — #46839649

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×(C22⋊D271)⋊C4C211yesmaximal, characteristic
3#31773838C212yesmaximal, complemented
4#31773823C211yesmaximal, characteristic, complemented
5#31773812C212yesmaximal
6#31773810C211yesmaximal, characteristic, complemented
7#31773809C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic271C212yesmaximal, complemented
9#31773744C212yesmaximal, complemented
10#31773742C211yesmaximal, characteristic, complemented
11#397×C271C211yesmaximal, characteristic, complemented
12C22×Dic542C2211yescharacteristic
13C22×C22⋊D271C2211yescharacteristic
14C23×Dic271C2211yescharacteristic
15C22×C4×D271C2211yescharacteristic
16#22576125—24no—
17(C22⋊D271)⋊C4—24no—
18C2×Dic271⋊C4C2212yes—
19C2×Dic271⋊C4C2211yescharacteristic
20C2×C22⋊Dic271C2212yes—
21C2×(C2×D271)⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2211yescharacteristic
24C2×C4×Dic271C2212yes—
25C2×C4×Dic271C2212yes—
26C2×C4×Dic271C2211yescharacteristic
27C271⋊#95C2211yescharacteristic
28C271⋊#95C2212yes—
29C271⋊#95C2212yes—
30C271⋊#95C2212yes—
31C271⋊#95C2212yes—
32C2×C4⋊C4×C271C2211yescharacteristic
33C2×C22⋊C4×C271C2212yes—
34C2×C22⋊C4×C271C2211yescharacteristic
35#95×C271C2211yescharacteristic
36#95×C271C2212yes—
37C23×D271C2311yescharacteristic
38C2×Dic542—24no—
39C2×Dic542—22no—
40C2×C22⋊D271—22no—
41C2×C22⋊D271—22no—
42C2×C22⋊D271—22no—
43C22×Dic271—22no—
44C22×Dic271—24no—
45C22×Dic271C2311yescharacteristic
46C22×Dic271C2312yes—
47C22×Dic271C2311yescharacteristic
48C22×Dic271C2312yes—
49C22×Dic271C2311yescharacteristic
50C2×C4×D271—22no—
51C2×C4×D271—24no—
52Dic271⋊C4—22no—
53Dic271⋊C4—24no—
54C22⋊Dic271—24no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57(C2×D271)⋊C4—22no—
58C4×Dic271—24no—
59C4×Dic271—24no—
60C4×Dic271D414yes—
61C24×C271C2311yescharacteristic
62C22×C4×C271C2311yescharacteristic
63C22×C4×C271C2312yes—
64C22×C4×C271C2312yes—
65C22×C4×C271C2311yescharacteristic
66C4⋊C4×C271—22no—
67C22⋊C4×C271—24no—
68C22⋊C4×C271—22no—
69C22×D271—21no—
70C22×D271—22no—
71C22×D271—22no—
72C22×D271—21no—
73C22×D271#4112yes—
74Dic542—28no—
75C22⋊D271—28no—
76C2×Dic271—22no—
77C2×Dic271—21no—
78C2×Dic271—21no—
79C2×Dic271—22no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—44no—
86C2×Dic271C2×D412yes—
87C2×Dic271C2×D412yes—
88C2×Dic271#4114yes—
89C2×Dic271#4114yes—
90C4×D271—44no—
91C23×C271—21no—
92C23×C271—22no—
93C23×C271—22no—
94C23×C271—21no—
95C23×C271#4112yes—
96C23×C271C2411yescharacteristic
97C2×C4×C271—22no—
98C2×C4×C271—21no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271C2×D412yes—
107C22×C271—22no—
108C22×C271—24no—
109C22×C271—41no—
110C22×C271—41no—
111C22×C271—42no—
112C22×C271C22×D411yescharacteristic
113C22×C271C2×#4111yescharacteristic
114C22×C271C2×#4112yes—
115C22×C271C2×#4112yes—
116C22×C271C2×#4111yescharacteristic
117C2×D271—22no—
118C2×D271—24no—
119C2×D271—41no—
120C2×D271—41no—
121C2×D271—42no—
122C4×C271—22no—
123C4×C271—42no—
124C4×C271—41no—
125C4×C271—42no—
126Dic271—22no—
127Dic271—24no—
128Dic271—41no—
129Dic271—42no—
130Dic271#12414yes—
131C2×C271—42no—
132C2×C271#52511yescharacteristic
133C2×C271#52511yescharacteristic
134C2×C271C2×#12411yescharacteristic
135C2×C271#52912yes—
136C2×C271#52912yes—
137D271—42no—
138C271#239811yescharacteristic, complemented, Sylow
139#2398—2711nomaximal, Sylow
140C2×C4×Q8—2711no—
141C2×C4×D4—2711no—
142#400—2712no—
143#397—2711no—
144#397D27111yescharacteristic, complemented
145#394—2712no—
146#393—2711no—
147#393—2711no—
148C4×C22⋊C4—2712no—
149C4×C22⋊C4—2712no—
150#376—2711no—
151C22×Q8—2711no—
152C22×D4—2711no—
153C23×C4—2711no—
154C23×C4—2711no—
155C4×Q8—5424no—
156C4×D4—5424no—
157C2×C4⋊C4—2711no—
158C2×C4⋊C4—2712no—
159C2×C4⋊C4C2×D27111yescharacteristic
160C2×C22⋊C4—2712no—
161C2×C22⋊C4—2711no—
162C2×C22⋊C4—2712no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4C2×D27111yescharacteristic
165C2×C22⋊C4C2×D27112yes—
166C2×C42—2711no—
167C2×C42—2712no—
168C2×C42—2712no—
169#95—2711no—
170#95—2712no—
171#95—2712no—
172#95—2712no—
173#95—2712no—
174#95C2×D27111yescharacteristic
175#95C2×D27112yes—
176C24—2711no—
177C24C22×D27111yescharacteristic
178C2×Q8—5424no—
179C2×Q8—5422no—
180C2×D4—5422no—
181C2×D4—5422no—
182C2×D4—5422no—
183C22×C4—2711no—
184C22×C4—2712no—
185C22×C4—2711no—
186C22×C4—2712no—
187C22×C4—2711no—
188C22×C4—5422no—
189C22×C4—5424no—
190C22×C4—5422no—
191C22×C4—5424no—
192C22×C4C22×D27112yes—
193C22×C4C22×D27111yescharacteristic
194C22×C4C22×D27112yes—
195C22×C4C22×D27111yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—5422no—
198C4⋊C4—5424no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—5424no—
202C22⋊C4—5424no—
203C22⋊C4—5424no—
204C22⋊C4—5422no—
205C42—2714no—
206C42—5424no—
207C42—5424no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—2712no—
213C23—5421no—
214C23—5422no—
215C23—5422no—
216C23—5421no—
217C23#1686465312yes—
218C23C23×D27111yescharacteristic
219Q8—5428no—
220D4—5428no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—2712no—
231C2×C4—2714no—
232C2×C4—2712no—
233C2×C4—2714no—
234C2×C4—5422no—
235C2×C4—5421no—
236C2×C4—5422no—
237C2×C4—5421no—
238C2×C4—5421no—
239C2×C4—5422no—
240C2×C4—5422no—
241C2×C4—5422no—
242C2×C4—5422no—
243C2×C4—10844no—
244C2×C4—10844no—
245C2×C4D4×D27112yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—5422no—
252C22—5424no—
253C22—10841no—
254C22—10841no—
255C22—10842no—
256C22C2×D4×D27111yescharacteristic
257C22#41×D27112yes—
258C22#41×D27111yescharacteristic
259C22#41×D27112yes—
260C22C2×#1686465311yescharacteristic
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—2714no—
266C4—5424no—
267C4—5422no—
268C4—10841no—
269C4—10842no—
270C2—42no—
271C2—10842no—
272C2#3177440511yescharacteristic
273C2#3177440711yescharacteristic
274C2#124×D27111yescharacteristic
275C2#3177442412yes—
276C2#3177443012yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#397×C271C2
prime radicalC22×C22⋊D271C22
hypercenter#397D271
hypocenterC271#2398
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#22576125C2semidirect
3. #31773838C2semidirect
4. #31773823C2semidirect
6. #31773810C2semidirect
7. #31773809C2semidirect
8. C22⋊C4×Dic271C2semidirect
9. #31773744C2semidirect
10. #31773742C2semidirect
11. #397×C271C2semidirect
138. C271#2398semidirect
144. #397D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761252
C2×(C22⋊D271)⋊C42
#317738382
#317738232
#317738122
#317738102
#317738092
C22⋊C4×Dic2712
#317737442
#317737422
#397×C2712
#2398271