Finite Group Databasebeta

← #46838351

Subgroups — #46838351

Order 34688.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#31774575C211yesmaximal, characteristic, complemented
2#31774084C211yesmaximal, characteristic
3#31774075C211yesmaximal, characteristic, complemented
4#31773691C211yesmaximal, characteristic, complemented
5#31773687C211yesmaximal, characteristic, complemented
6#31773683C211yesmaximal, characteristic, complemented
7#458×C271C211yesmaximal, characteristic, complemented
8#22576190—21no—
9#22576153—21no—
10#22576147C2211yescharacteristic
11#22576142—21no—
12#22576138—21no—
13#22576128—21no—
14#22576128C2211yescharacteristic
15C4×C22⋊D271—21no—
16#22576118—21no—
17C22⋊C4×D271—21no—
18#22576087—21no—
19#22576077C2211yescharacteristic
20#22576075—21no—
21#22576075C2211yescharacteristic
22H(5,𝔽2)×C271C2211yescharacteristic
23#123×C271—21no—
24#123×C271C2211yescharacteristic
25#99×C271—21no—
26#99×C271C2211yescharacteristic
27C23×D271D411yescharacteristic
28C2×Dic542D411yescharacteristic, complemented
29C2×C22⋊D271—21no—
30C2×C22⋊D271—41no—
31C2×C4×D271—21no—
32C2×C4×D271—21no—
33#16864644—41no—
34#16864642—41no—
35#16864638—21no—
36C4⋊Dic271—21no—
37Dic271⋊C4—21no—
38Dic271⋊C4—21no—
39C22⋊Dic271—21no—
40C22⋊Dic271—21no—
41C22⋊Dic271—21no—
42C22⋊Dic271D411yescharacteristic
43(C2×D271)⋊C4—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46(C2×D271)⋊C4—41no—
47C4×Dic271—21no—
48C4×Dic271D411yescharacteristic, complemented
49#41×C271—21no—
50#41×C271—41no—
51C2×D4×C271—21no—
52C2×D4×C271—21no—
53C2×D4×C271—41no—
54C2×D4×C271C2311yescharacteristic
55C22×C4×C271D411yescharacteristic, complemented
56C4⋊C4×C271—21no—
57C4⋊C4×C271—41no—
58C22⋊C4×C271—21no—
59C22⋊C4×C271—21no—
60C22⋊C4×C271—41no—
61C22⋊C4×C271D411yescharacteristic, complemented
62C22×D271—21no—
63C22×D271—42no—
64C22×D271—41no—
65Dic542—41no—
66C22⋊D271—42no—
67C22⋊D271—82no—
68C2×Dic271—21no—
69C2×Dic271—21no—
70C2×Dic271—21no—
71C2×Dic271—41no—
72C4×D271—42no—
73C4×D271—81no—
74C271⋊C8—41no—
75C23×C271—21no—
76C23×C271—21no—
77C23×C271C2×D411yescharacteristic
78C23×C271C2×D411yescharacteristic
79Q8×C271—21no—
80D4×C271—21no—
81D4×C271—41no—
82D4×C271—41no—
83D4×C271—41no—
84D4×C271—41no—
85C2×C4×C271—21no—
86C2×C4×C271—21no—
87C2×C4×C271—21no—
88C2×C4×C271—21no—
89C2×C4×C271—41no—
90C2×C4×C271—41no—
91C2×C4×C271—41no—
92C2×C4×C271C2×D411yescharacteristic
93C22×C271—21no—
94C22×C271—21no—
95C22×C271—41no—
96C22×C271—41no—
97C22×C271—41no—
98C22×C271—81no—
99C22×C271—81no—
100C22×C271#12011yescharacteristic
101C2×D271—22no—
102C2×D271—41no—
103C2×D271—41no—
104C2×D271—42no—
105C2×D271—81no—
106C4×C271—21no—
107C4×C271—41no—
108C4×C271—41no—
109C4×C271—41no—
110C4×C271—81no—
111Dic271—41no—
112Dic271—41no—
113Dic271—41no—
114Dic271—81no—
115C2×C271—21no—
116C2×C271—41no—
117C2×C271—41no—
118C2×C271—81no—
119C2×C271B(4,𝔽2)11yescharacteristic
120D271—42no—
121C271#219711yescharacteristic, complemented, Sylow
122#2197—2711nomaximal, Sylow
123#551—2711no—
124#458D27111yescharacteristic, complemented
125B(4,𝔽2)—2711no—
126#455—2711no—
127#354—2711no—
128#352—2711no—
129AΣL(1,𝔽16)—2711no—
130H(5,𝔽2)C2×D27111yescharacteristic
131#137—5421no—
132#126—5421no—
133#124—2711no—
134#123—21no—
135#123—5421no—
136#123C2×D27111yescharacteristic
137C22⋊Q8—5421no—
138#120—2711no—
139#120—5421no—
140C4×D4—5421no—
141#117—5421no—
142C2×C22⋊C4—5421no—
143#104—5421no—
144#100—2711no—
145#99—21no—
146#99—2711no—
147#99—5421no—
148#99C2×D27111yescharacteristic
149C24—2711no—
150#41—21no—
151#41—41no—
152C2×Q8—2711no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—5421no—
157C2×D4—10841no—
158C2×D4C22×D27111yescharacteristic
159C22×C4—5421no—
160C22×C4—5421no—
161C22×C4C22⋊D27111yescharacteristic, complemented
162Q16—10841no—
163SD16—10841no—
164M16—5421no—
165C4⋊C4—21no—
166C4⋊C4—41no—
167C4⋊C4—5421no—
168C4⋊C4—5421no—
169C4⋊C4—5421no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—41no—
173C22⋊C4—2711no—
174C22⋊C4—5421no—
175C22⋊C4—5421no—
176C22⋊C4—5421no—
177C22⋊C4—5421no—
178C22⋊C4—5421no—
179C22⋊C4—5421no—
180C22⋊C4—10841no—
181C22⋊C4C22⋊D27111yescharacteristic, complemented
182C42—2711no—
183C42—5421no—
184C23—21no—
185C23—21no—
186C23—5421no—
187C23—10841no—
188C23—10842no—
189C23D4×D27111yescharacteristic
190C23C2×C22⋊D27111yescharacteristic
191Q8—21no—
192Q8—10841no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—41no—
198D4—10842no—
199D4—21682no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—5421no—
208C2×C4—5421no—
209C2×C4—5421no—
210C2×C4—10842no—
211C2×C4—10841no—
212C2×C4—21681no—
213C2×C4D4×D27111yescharacteristic
214C8—10841no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—81no—
221C22—81no—
222C22—5422no—
223C22—10841no—
224C22—10842no—
225C22—10841no—
226C22—21681no—
227C22#2257612811yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—10841no—
234C4—10841no—
235C4—10841no—
236C4—21681no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—81no—
241C2—10842no—
242C2#3177408311yescharacteristic
subgroupG/H
centerC2#31774083
commutator, cosocleC2×D4×C271C23
FrattiniC2×D4C22×D271
Fitting#458×C271C2
socleC2×C271B(4,𝔽2)
prime radical#31774084C2
hypercenter#458D271
hypocenterC271#2197
LCSG ▷ C2×D4×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #458

Complements

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

NH ≅ G/Ntype
1. #31774575C2semidirect
3. #31774075C2semidirect
4. #31773691C2semidirect
5. #31773687C2semidirect
6. #31773683C2semidirect
7. #458×C271C2semidirect
28. C2×Dic542D4semidirect
48. C4×Dic271D4semidirect
55. C22×C4×C271D4semidirect
61. C22⋊C4×C271D4semidirect
121. C271#2197semidirect
124. #458D271semidirect
161. C22×C4C22⋊D271semidirect
181. C22⋊C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317745752
#317740842
#317740752
#317736912
#317736872
#317736832
#458×C2712
#2197271