Finite Group Databasebeta

← #46837430

Subgroups — #46837430

Order 34688.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#31774695C211yesmaximal, characteristic, complemented
2#31774409C211yesmaximal, characteristic, complemented
3#31774404C211yesmaximal, characteristic, complemented
4#31773913C211yesmaximal, characteristic, complemented
5#31773860C211yesmaximal, characteristic, complemented
6#31773653C211yesmaximal, characteristic, complemented
7#421×C271C211yesmaximal, characteristic, complemented
8C2×#16864654C2211yescharacteristic
9C22×C4×D271C2211yescharacteristic
10#22576191—21no—
11#22576190—21no—
12#22576186—21no—
13#22576185—21no—
14#22576180—22no—
15C2×#16864638C2211yescharacteristic
16#22576144—21no—
17#22576141—21no—
18#22576140—21no—
19#22576136—21no—
20#22576135—21no—
21#22576131—21no—
22#22576130—21no—
23#22576128—21no—
24C4×D1084—22no—
25C4×C22⋊D271—22no—
26#22576119C2211yescharacteristic
27#22576086—22no—
28(C2×D271)⋊C8—22no—
29C2×#41×C271C2211yescharacteristic
30C2×M16×C271C2211yescharacteristic
31#117×C271C2211yescharacteristic
32#104×C271—22no—
33C23×D271D411yescharacteristic
34#16864654—21no—
35#16864654—21no—
36#16864654—21no—
37#16864654—21no—
38C2×Dic542D411yescharacteristic, complemented
39C2×D1084—22no—
40C2×D1084D411yescharacteristic, complemented
41C2×C22⋊D271—22no—
42C2×C22⋊D271—21no—
43C22×Dic271D411yescharacteristic
44C2×C4×D271—21no—
45C2×C4×D271—21no—
46C2×C4×D271—21no—
47C2×C4×D271—21no—
48C2×C4×D271—21no—
49C2×C4×D271—21no—
50C2×C4×D271—21no—
51#16864644—22no—
52#16864642—22no—
53#16864641—22no—
54#16864639—22no—
55#16864638—21no—
56#16864638—21no—
57C2×C271⋊C8D412yescomplemented
58C4⋊Dic271—22no—
59C4⋊Dic271—21no—
60Dic271⋊C4—22no—
61C22⋊Dic271—22no—
62C22⋊Dic271—21no—
63(C2×D271)⋊C4—22no—
64(C2×D271)⋊C4—22no—
65(C2×D271)⋊C4—22no—
66C4×Dic271D412yescomplemented
67#41×C271—21no—
68#41×C271—21no—
69#41×C271—21no—
70#41×C271—21no—
71C2×Q8×C271D411yescharacteristic, complemented
72C2×D4×C271—21no—
73C2×D4×C271D411yescharacteristic, complemented
74C22×C4×C271—21no—
75C22×C4×C271C2311yescharacteristic
76M16×C271—21no—
77M16×C271—21no—
78C2×C8×C271—21no—
79C4⋊C4×C271—21no—
80C22⋊C4×C271—21no—
81C42×C271D412yescomplemented
82C22×D271—21no—
83C22×D271—21no—
84C22×D271—21no—
85C22×D271—21no—
86C22×D271—42no—
87Dic542—21no—
88Dic542—21no—
89D1084—21no—
90D1084—21no—
91D1084—44no—
92C22⋊D271—44no—
93C22⋊D271—41no—
94C22⋊D271—41no—
95C2×Dic271—22no—
96C2×Dic271—21no—
97C2×Dic271—21no—
98C2×Dic271—21no—
99C2×Dic271—21no—
100C4×D271—41no—
101C4×D271—41no—
102C4×D271—41no—
103C4×D271—41no—
104C4×D271—41no—
105C4×D271—41no—
106C271⋊C8—22no—
107C23×C271—21no—
108C23×C271C2×D411yescharacteristic
109Q8×C271—21no—
110Q8×C271—21no—
111D4×C271—21no—
112D4×C271—21no—
113D4×C271—41no—
114D4×C271—41no—
115C2×C4×C271—22no—
116C2×C4×C271—21no—
117C2×C4×C271—41no—
118C2×C4×C271—41no—
119C2×C4×C271C2×D411yescharacteristic
120C2×C4×C271C2×D411yescharacteristic
121C2×C4×C271C2×D411yescharacteristic
122C2×C4×C271C2×D411yescharacteristic
123C2×C4×C271C2×D411yescharacteristic
124C2×C4×C271#4111yescharacteristic
125C8×C271—41no—
126C22×C271—41no—
127C22×C271—42no—
128C22×C271—41no—
129C22×C271#12011yescharacteristic
130C22×C271#12011yescharacteristic
131C22×C271#12111yescharacteristic
132C2×D271—22no—
133C2×D271—22no—
134C2×D271—22no—
135C2×D271—41no—
136C2×D271—41no—
137C2×D271—41no—
138C2×D271—41no—
139C2×D271—42no—
140C2×D271—41no—
141C4×C271—42no—
142C4×C271—41no—
143C4×C271#12011yescharacteristic
144C4×C271#12111yescharacteristic
145C4×C271#12111yescharacteristic
146C4×C271#12711yescharacteristic
147Dic271—42no—
148Dic271—41no—
149Dic271—41no—
150C2×C271—21no—
151C2×C271—21no—
152C2×C271—21no—
153C2×C271—81no—
154C2×C271#39211yescharacteristic
155D271—42no—
156D271—81no—
157C271#201011yescharacteristic, complemented, Sylow
158#2010—2711nomaximal, Sylow
159#575—2711no—
160#525—2711no—
161#525—2711no—
162#421—2711no—
163#421D27111yescharacteristic, complemented
164#407—2711no—
165H(3,ℤ/4)—2711no—
166C2×#41—2711no—
167C2×#41C2×D27111yescharacteristic
168C23×C4—2711no—
169#137—5421no—
170#137—5421no—
171AGL(1,ℤ/8)—5421no—
172AGL(1,ℤ/8)—5421no—
173#135—5422no—
174C2×M16—2711no—
175C2×M16C2×D27111yescharacteristic
176#123—5421no—
177#123—5421no—
178C22⋊Q8—5421no—
179C22⋊Q8—5421no—
180#121—5421no—
181#121—5421no—
182#120—5421no—
183#120—5421no—
184C4×D4—5422no—
185C4×D4—5422no—
186#117—2711no—
187#117C2×D27111yescharacteristic
188#104—22no—
189#104—5422no—
190C22⋊C8—5422no—
191C24—2711no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—5421no—
197#41—5421no—
198#41—5421no—
199#41—5421no—
200C2×Q8—2711no—
201C2×Q8D108411yescharacteristic, complemented
202C2×D4—21no—
203C2×D4—2711no—
204C2×D4—5422no—
205C2×D4—5422no—
206C2×D4—5421no—
207C2×D4D108411yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—2711no—
210C22×C4—5421no—
211C22×C4—5421no—
212C22×C4—5421no—
213C22×C4—5421no—
214C22×C4—5421no—
215C22×C4—5421no—
216C22×C4—5421no—
217C22×C4C22×D27111yescharacteristic
218Q16—5422no—
219SD16—5422no—
220SD16—5422no—
221D8—5422no—
222M16—21no—
223M16—21no—
224M16—5421no—
225M16—5421no—
226C2×C8—21no—
227C2×C8—2712no—
228C4⋊C4—21no—
229C4⋊C4—5422no—
230C4⋊C4—5422no—
231C4⋊C4—5421no—
232C22⋊C4—21no—
233C22⋊C4—5422no—
234C22⋊C4—5422no—
235C22⋊C4—5422no—
236C22⋊C4—5422no—
237C22⋊C4—5421no—
238C42—2712no—
239C42C22⋊D27112yescomplemented
240C23—21no—
241C23—5421no—
242C23—5421no—
243C23—5421no—
244C23—5421no—
245C23—10842no—
246C23D4×D27111yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—5421no—
250Q8—5421no—
251D4—21no—
252D4—21no—
253D4—41no—
254D4—41no—
255D4—5421no—
256D4—5421no—
257D4—10844no—
258D4—10844no—
259D4—10841no—
260D4—10841no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—5421no—
266C2×C4—5421no—
267C2×C4—5422no—
268C2×C4—5421no—
269C2×C4—5421no—
270C2×C4—10841no—
271C2×C4—10841no—
272C2×C4—10841no—
273C2×C4—10841no—
274C2×C4—10841no—
275C2×C4—10841no—
276C2×C4D4×D27111yescharacteristic
277C2×C4D4×D27111yescharacteristic
278C2×C4D4×D27111yescharacteristic
279C2×C4C2×C22⋊D27111yescharacteristic
280C2×C4C2×D108411yescharacteristic
281C2×C4#1686465411yescharacteristic
282C8—41no—
283C8—5422no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—5422no—
288C22—5422no—
289C22—5422no—
290C22—10841no—
291C22—10841no—
292C22—10841no—
293C22—10842no—
294C22—10841no—
295C22—10841no—
296C22#2257612811yescharacteristic
297C22#2257613011yescharacteristic
298C22#2257613311yescharacteristic
299C4—42no—
300C4—41no—
301C4—10842no—
302C4—10841no—
303C4—10841no—
304C4#2257613011yescharacteristic
305C4#2257613311yescharacteristic
306C4#2257613411yescharacteristic
307C4#2257615611yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2—10842no—
313C2—21681no—
314C2#3177380611yescharacteristic
subgroupG/H
centerC4#22576156
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#421×C271C2
socleC2×C271#392
hypercenter#421D271
hypocenterC271#2010
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #31774695C2semidirect
2. #31774409C2semidirect
3. #31774404C2semidirect
4. #31773913C2semidirect
5. #31773860C2semidirect
6. #31773653C2semidirect
7. #421×C271C2semidirect
38. C2×Dic542D4semidirect
40. C2×D1084D4semidirect
57. C2×C271⋊C8D4semidirect
66. C4×Dic271D4semidirect
71. C2×Q8×C271D4semidirect
73. C2×D4×C271D4semidirect
81. C42×C271D4semidirect
157. C271#2010semidirect
163. #421D271semidirect
201. C2×Q8D1084semidirect
207. C2×D4D1084semidirect
239. C42C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317746952
#317744092
#317744042
#317739132
#317738602
#317736532
#421×C2712
#2010271