Finite Group Databasebeta

← #46835329

Subgroups — #46835329

Order 34688.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774494C211yesmaximal, characteristic
2#31774055C211yesmaximal, characteristic, complemented
3#31774038C211yesmaximal, characteristic
4#31773817C211yesmaximal, characteristic, complemented
5#31773635C211yesmaximal, characteristic, complemented
6#31773632C211yesmaximal, characteristic
7#324×C271C211yesmaximal, characteristic, complemented
8C22×Dic542C2211yescharacteristic
9C2×Dic1084—21no—
10C2×#16864643—21no—
11#22576155—21no—
12#22576150—21no—
13#22576145—21no—
14#22576143—21no—
15#22576139C2211yescharacteristic
16#22576138—21no—
17#22576134C2211yescharacteristic
18C4×C22⋊D271—21no—
19C2×Dic271⋊C4—21no—
20#22576085—21no—
21#22576085—21no—
22C271⋊#95—21no—
23C271⋊#95C2211yescharacteristic
24C2×C4⋊C4×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26C22⋊C8×C271C2211yescharacteristic
27C2×Dic542—21no—
28C2×Dic542—21no—
29C2×Dic542—42no—
30C2×Dic542D411yescharacteristic
31C2×D1084D411yescharacteristic
32C2×C22⋊D271—21no—
33C22×Dic271—21no—
34C22×Dic271—21no—
35C2×C4×D271—21no—
36Dic1084—42no—
37#16864643—42no—
38C4⋊Dic271D411yescharacteristic
39C4⋊Dic271D411yescharacteristic
40Dic271⋊C4—21no—
41Dic271⋊C4—21no—
42Dic271⋊C4—21no—
43Dic271⋊C4—21no—
44Dic271⋊C4—42no—
45C22⋊Dic271—21no—
46(C2×D271)⋊C4—21no—
47(C2×D271)⋊C4—21no—
48(C2×D271)⋊C4—21no—
49(C2×D271)⋊C4—21no—
50C4×Dic271—21no—
51C22×C4×C271D411yescharacteristic
52C22×C4×C271D411yescharacteristic
53C22×C4×C271C2311yescharacteristic
54C2×C8×C271—21no—
55C2×C8×C271—21no—
56C4⋊C4×C271—21no—
57C4⋊C4×C271—21no—
58C22×D271—21no—
59Dic542—22no—
60Dic542—22no—
61Dic542—41no—
62Dic542—41no—
63Dic542—41no—
64D1084—41no—
65C22⋊D271—41no—
66C22⋊D271—41no—
67C2×Dic271—22no—
68C2×Dic271—22no—
69C2×Dic271—21no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—41no—
73C2×Dic271—41no—
74C2×Dic271—41no—
75C2×Dic271—41no—
76C4×D271—81no—
77C23×C271C2×D411yescharacteristic
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—21no—
81C2×C4×C271—21no—
82C2×C4×C271—41no—
83C2×C4×C271—41no—
84C2×C4×C271C2×D411yescharacteristic
85C2×C4×C271C2×D411yescharacteristic
86C8×C271—41no—
87C8×C271—41no—
88C22×C271—21no—
89C22×C271—21no—
90C22×C271#12011yescharacteristic
91C22×C271#13511yescharacteristic
92C22×C271#13711yescharacteristic
93C2×D271—41no—
94C2×D271—41no—
95C2×D271—41no—
96C4×C271—21no—
97C4×C271—21no—
98C4×C271—41no—
99C4×C271—41no—
100Dic271—42no—
101Dic271—41no—
102Dic271—41no—
103Dic271—41no—
104Dic271—42no—
105C2×C271—21no—
106C2×C271—21no—
107C2×C271#45211yescharacteristic
108C2×C271#45611yescharacteristic
109C2×C271B(4,𝔽2)11yescharacteristic
110D271—81no—
111C271#160311yescharacteristic, complemented, Sylow
112#1603—2711nomaximal, Sylow
113#539—2711no—
114#451—2711no—
115#448—2711no—
116#395—2711no—
117#328—2711no—
118#327—2711no—
119#324D27111yescharacteristic, complemented
120C22×Q8—2711no—
121C2×Q16—5421no—
122C2×SD16—5421no—
123#126—5421no—
124#125—5421no—
125#123—5421no—
126#123—5421no—
127C22⋊Q8—2711no—
128C22⋊Q8—5421no—
129#121—2711no—
130C4×D4—5421no—
131C2×C4⋊C4—5421no—
132C2×C4⋊C4C2×D27111yescharacteristic
133#103—5421no—
134#103—5421no—
135C22⋊C8C2×D27111yescharacteristic
136C22⋊C8C2×D27111yescharacteristic
137#95—2711no—
138#95—5421no—
139C2×Q8—2711no—
140C2×Q8—5421no—
141C2×Q8—5421no—
142C2×Q8—10842no—
143C2×D4—2711no—
144C2×D4—5421no—
145C22×C4—5421no—
146C22×C4—5421no—
147C22×C4—5421no—
148C22×C4D108411yescharacteristic
149C22×C4D108411yescharacteristic
150C22×C4C22×D27111yescharacteristic
151Q16—10842no—
152SD16—10842no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—2711no—
158C4⋊C4—2711no—
159C4⋊C4—5421no—
160C4⋊C4—5421no—
161C4⋊C4—5421no—
162C4⋊C4—5421no—
163C4⋊C4—10842no—
164C22⋊C4—5421no—
165C22⋊C4—5421no—
166C22⋊C4—5421no—
167C22⋊C4—5421no—
168C22⋊C4—5421no—
169C42—5421no—
170C23—5421no—
171C23C2×D108411yescharacteristic
172Q8—5422no—
173Q8—5422no—
174Q8—10841no—
175Q8—10841no—
176Q8—10841no—
177D4—10841no—
178D4—10841no—
179D4—10841no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—5422no—
187C2×C4—5422no—
188C2×C4—5421no—
189C2×C4—5421no—
190C2×C4—5421no—
191C2×C4—10841no—
192C2×C4—10841no—
193C2×C4—10841no—
194C2×C4—10841no—
195C2×C4—21681no—
196C2×C4D4×D27111yescharacteristic
197C2×C4D4×D27111yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—10841no—
203C22—10841no—
204C22—10841no—
205C22#2257613011yescharacteristic
206C22#2257618211yescharacteristic
207C22#2257619311yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—10842no—
213C4—10841no—
214C4—10841no—
215C4—10842no—
216C4—10841no—
217C2—21no—
218C2—21no—
219C2—21681no—
220C2#3177406111yescharacteristic
221C2#3177408211yescharacteristic
222C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radical#22576134C22
hypercenter#324D271
hypocenterC271#1603
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #31774055C2semidirect
4. #31773817C2semidirect
5. #31773635C2semidirect
7. #324×C271C2semidirect
111. C271#1603semidirect
119. #324D271semidirect

Maximal subgroups

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

subgroupindex
#317744942
#317740552
#317740382
#317738172
#317736352
#317736322
#324×C2712
#1603271