Finite Group Databasebeta

← #46835327

Subgroups — #46835327

Order 34688.

244 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 53 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#31774052C211yesmaximal, characteristic, complemented
2#31774035C211yesmaximal, characteristic
3#31773817C211yesmaximal, characteristic, complemented
4#31773634C211yesmaximal, characteristic, complemented
5#31773631C211yesmaximal, characteristic
6#31773623C211yesmaximal, characteristic, complemented
7#539×C271C211yesmaximal, characteristic
8C22×Dic542C2211yescharacteristic
9C2×#16864644—21no—
10C2×#16864642—21no—
11C2×Dic271⋊C4—21no—
12#22576083—21no—
13#22576083—21no—
14C22⋊(C271⋊C8)C2211yescharacteristic
15C22⋊(C271⋊C8)C2211yescharacteristic
16C271⋊#95—21no—
17C271⋊#95C2211yescharacteristic
18#126×C271—21no—
19#125×C271—21no—
20#123×C271—21no—
21#123×C271—21no—
22C22⋊Q8×C271—21no—
23C22⋊Q8×C271C2211yescharacteristic
24#121×C271C2211yescharacteristic
25C4×D4×C271—21no—
26C2×C4⋊C4×C271C2211yescharacteristic
27C2×Dic542—21no—
28C2×Dic542—21no—
29C2×Dic542—42no—
30C22×Dic271—21no—
31C22×Dic271—21no—
32#16864644—42no—
33#16864642—42no—
34C2×C271⋊C8—21no—
35C2×C271⋊C8—21no—
36Dic271⋊C4—42no—
37C2×Q8×C271D411yescharacteristic
38C2×D4×C271—21no—
39C2×D4×C271D411yescharacteristic
40C22×C4×C271—21no—
41C22×C4×C271D411yescharacteristic
42C22×C4×C271D411yescharacteristic
43C22×C4×C271C2311yescharacteristic
44C4⋊C4×C271—21no—
45C4⋊C4×C271—21no—
46C4⋊C4×C271—21no—
47C4⋊C4×C271—21no—
48C4⋊C4×C271—21no—
49C4⋊C4×C271—21no—
50C4⋊C4×C271D411yescharacteristic
51C4⋊C4×C271D411yescharacteristic
52C22⋊C4×C271—21no—
53C22⋊C4×C271—21no—
54C22⋊C4×C271—21no—
55C22⋊C4×C271—21no—
56C22⋊C4×C271—21no—
57C42×C271—21no—
58Dic542—22no—
59Dic542—22no—
60Dic542—41no—
61Dic542—41no—
62C2×Dic271—22no—
63C2×Dic271—22no—
64C2×Dic271—41no—
65C2×Dic271—41no—
66C2×Dic271—41no—
67C2×Dic271—41no—
68C271⋊C8—41no—
69C271⋊C8—41no—
70C23×C271—21no—
71C23×C271C2×D411yescharacteristic
72Q8×C271—41no—
73D4×C271—41no—
74D4×C271—41no—
75D4×C271—41no—
76C2×C4×C271—21no—
77C2×C4×C271—21no—
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—21no—
81C2×C4×C271—21no—
82C2×C4×C271—21no—
83C2×C4×C271—41no—
84C2×C4×C271—41no—
85C2×C4×C271—81no—
86C2×C4×C271C2×D411yescharacteristic
87C2×C4×C271C2×D411yescharacteristic
88C22×C271—21no—
89C22×C271—21no—
90C22×C271—41no—
91C22×C271—41no—
92C22×C271—41no—
93C22×C271#12011yescharacteristic
94C22×C271#13511yescharacteristic
95C22×C271#13711yescharacteristic
96C4×C271—21no—
97C4×C271—21no—
98C4×C271—41no—
99C4×C271—41no—
100C4×C271—41no—
101C4×C271—41no—
102C4×C271—41no—
103Dic271—42no—
104Dic271—42no—
105C2×C271—21no—
106C2×C271—21no—
107C2×C271—81no—
108C2×C271#45211yescharacteristic
109C2×C271#45611yescharacteristic
110C2×C271B(4,𝔽2)11yescharacteristic
111C271#160311yescharacteristic, complemented, Sylow
112#1603—2711nomaximal, Sylow
113#539D27111yescharacteristic
114#451—2711no—
115#448—2711no—
116#395—2711no—
117#328—2711no—
118#327—2711no—
119#324—2711no—
120C22×Q8—2711no—
121C2×Q16—5421no—
122C2×SD16—5421no—
123#126—21no—
124#125—21no—
125#123—21no—
126#123—21no—
127C22⋊Q8—21no—
128C22⋊Q8C2×D27111yescharacteristic
129#121C2×D27111yescharacteristic
130C4×D4—21no—
131C2×C4⋊C4—5421no—
132C2×C4⋊C4C2×D27111yescharacteristic
133#103—5421no—
134#103—5421no—
135C22⋊C8—2711no—
136C22⋊C8—2711no—
137#95—2711no—
138#95—5421no—
139C2×Q8—5421no—
140C2×Q8—5421no—
141C2×Q8—10842no—
142C2×Q8C22⋊D27111yescharacteristic
143C2×D4—21no—
144C2×D4C22⋊D27111yescharacteristic
145C22×C4—21no—
146C22×C4—5421no—
147C22×C4—5421no—
148C22×C4C22⋊D27111yescharacteristic
149C22×C4C22⋊D27111yescharacteristic
150C22×C4C22×D27111yescharacteristic
151Q16—10842no—
152SD16—10842no—
153C2×C8—5421no—
154C2×C8—5421no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—10842no—
162C4⋊C4C22⋊D27111yescharacteristic
163C4⋊C4C22⋊D27111yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C42—21no—
170C23—21no—
171C23C2×C22⋊D27111yescharacteristic
172Q8—41no—
173Q8—5422no—
174Q8—5422no—
175Q8—10841no—
176Q8—10841no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—5422no—
191C2×C4—5422no—
192C2×C4—10841no—
193C2×C4—10841no—
194C2×C4—10841no—
195C2×C4—10841no—
196C2×C4C2×C22⋊D27111yescharacteristic
197C2×C4C2×C22⋊D27111yescharacteristic
198C8—10841no—
199C8—10841no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2257612911yescharacteristic
206C22#2257618011yescharacteristic
207C22#2257619011yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—10842no—
216C4—10842no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#3177405911yescharacteristic
221C2#3177408011yescharacteristic
222C2#3177408411yescharacteristic
subgroupG/H
centerC22#22576129
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#539×C271C2
socleC22×C271#120
prime radical#121×C271C22
hypercenter#539D271
hypocenterC271#1603
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #31774052C2semidirect
3. #31773817C2semidirect
4. #31773634C2semidirect
6. #31773623C2semidirect
111. C271#1603semidirect

Maximal subgroups

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

subgroupindex
#317740522
#317740352
#317738172
#317736342
#317736312
#317736232
#539×C2712
#1603271