Finite Group Databasebeta

← #46836330

Subgroups — #46836330

Order 34688.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#31774733C211yesmaximal, characteristic
2#31773915C212yesmaximal, complemented
3#31773891C212yesmaximal, complemented
4#31773864C211yesmaximal, characteristic, complemented
5#410×C271C211yesmaximal, characteristic, complemented
6#22576208C412yescomplemented
7#22576207—22no—
8#22576206—22no—
9#22576205—22no—
10#22576205C412yescomplemented
11C2×#16864654—22no—
12C2×#16864654C2212yes—
13#41×D271—24no—
14C2×#16864638C2211yescharacteristic, complemented
15C22×C271⋊C8C2211yescharacteristic, complemented
16#22576088—24no—
17#22576083—22no—
18#22576080—22no—
19C22⋊(C271⋊C8)—22no—
20C2×#41×C271C2211yescharacteristic
21#117×C271C2212yescomplemented
22#99×C271—22no—
23#16864655—24no—
24#16864654—24no—
25#16864654—24no—
26#16864654—24no—
27#16864654—22no—
28#16864654—22no—
29#16864654C2×C414yes—
30#16864654D414yes—
31#16864653—28no—
32#16864653—24no—
33C2×Dic542—22no—
34C2×Dic542D412yes—
35Q8×D271—24no—
36C2×D1084—22no—
37C2×D1084D412yes—
38C2×C22⋊D271—24no—
39C2×C22⋊D271—22no—
40D4×D271—28no—
41D4×D271—24no—
42C2×C4×D271—24no—
43C2×C4×D271—22no—
44#16864638—22no—
45C2×C271⋊C8—21no—
46C2×C271⋊C8—22no—
47C2×C271⋊C8—21no—
48#41×C271—22no—
49#41×C271—22no—
50C2×Q8×C271C2×C411yescharacteristic
51C2×D4×C271C2×C411yescharacteristic
52C2×D4×C271D412yes—
53C22×C4×C271D412yes—
54C22×C4×C271C2311yescharacteristic
55C4⋊C4×C271—22no—
56C22⋊C4×C271—22no—
57C42×C271—22no—
58C22×D271—24no—
59C22×D271—22no—
60Dic542—24no—
61Dic542—22no—
62Dic542C22⋊C414yes—
63D1084—24no—
64D1084—22no—
65D1084C22⋊C414yes—
66C22⋊D271—28no—
67C22⋊D271—24no—
68C22⋊D271—24no—
69C2×Dic271—24no—
70C2×Dic271—22no—
71C4×D271—28no—
72C4×D271—24no—
73C4×D271—24no—
74C271⋊C8—41no—
75C271⋊C8—41no—
76C23×C271C22⋊C412yes—
77C23×C271C2×D411yescharacteristic
78Q8×C271—22no—
79D4×C271—24no—
80D4×C271—22no—
81C2×C4×C271—22no—
82C2×C4×C271—22no—
83C2×C4×C271—42no—
84C2×C4×C271C22⋊C412yes—
85C2×C4×C271C22×C411yescharacteristic
86C2×C4×C271C2×D412yes—
87C2×C4×C271C2×D412yes—
88C2×C4×C271C2×D411yescharacteristic
89C22×C271—22no—
90C22×C271—41no—
91C22×C271—42no—
92C22×C271C2×C22⋊C411yescharacteristic
93C22×C271#12012yes—
94C2×D271—24no—
95C2×D271—82no—
96C2×D271—84no—
97C4×C271—22no—
98C4×C271—82no—
99C4×C271C2×C22⋊C412yes—
100C4×C271#12011yescharacteristic
101C4×C271#12011yescharacteristic
102Dic271—24no—
103C2×C271—22no—
104C2×C271—21no—
105C2×C271—42no—
106C2×C271#37911yescharacteristic
107D271—44no—
108C271#179411yescharacteristic, complemented, Sylow
109#1794—2711nomaximal, Sylow
110#585—2711no—
111#421—2712no—
112#416—2712no—
113#410D27111yescharacteristic, complemented
114#408—2711no—
1152−1+4—2712no—
1162−1+4—5422no—
117H(5,𝔽2)—2712no—
118H(5,𝔽2)—5422no—
119H(5,𝔽2)—5422no—
120C2×#41—2712no—
121C2×#41—5424no—
122C2×#41—5422no—
123C2×#41C2×D27111yescharacteristic
124C2×M16—2711no—
125C22×C8—2711no—
126#117C2×D27112yescomplemented
127#104—5424no—
128#103—5422no—
129#102—5422no—
130#99—22no—
131C22⋊C8—5422no—
132#41—22no—
133#41—22no—
134#41—2714no—
135#41—2714no—
136#41—5424no—
137#41—5424no—
138#41—5424no—
139#41—5428no—
140#41—5424no—
141#41—5424no—
142#41—5422no—
143#41—5422no—
144C2×Q8—2712no—
145C2×Q8—5422no—
146C2×Q8—5424no—
147C2×Q8C4×D27111yescharacteristic
148C2×D4—2712no—
149C2×D4—5422no—
150C2×D4—5424no—
151C2×D4—5428no—
152C2×D4—5424no—
153C2×D4—5422no—
154C2×D4C4×D27111yescharacteristic
155C2×D4C22⋊D27112yes—
156C22×C4—5424no—
157C22×C4—5422no—
158C22×C4D108412yes—
159C22×C4C22×D27111yescharacteristic
160M16—5422no—
161C2×C8—5422no—
162C2×C8—5421no—
163C2×C8—5421no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—5424no—
168C23—5422no—
169C23(C2×D271)⋊C412yes—
170C23C2×C22⋊D27111yescharacteristic
171Q8—22no—
172Q8—2714no—
173Q8—5424no—
174Q8—5422no—
175D4—24no—
176D4—22no—
177D4—2714no—
178D4—5424no—
179D4—5428no—
180D4—5424no—
181D4—5424no—
182D4—5422no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—5428no—
187C2×C4—5424no—
188C2×C4—5422no—
189C2×C4—5424no—
190C2×C4—5424no—
191C2×C4(C2×D271)⋊C412yes—
192C2×C4C2×C4×D27111yescharacteristic
193C2×C4D4×D27112yes—
194C2×C4D4×D27112yes—
195C2×C4C2×D108411yescharacteristic
196C8—10841no—
197C8—10841no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—5424no—
202C22—21682no—
203C22—21684no—
204C22C2×(C2×D271)⋊C411yescharacteristic
205C22#2257612812yes—
206C4—22no—
207C4—82no—
208C4—5424no—
209C4C22⋊C4×D27112yes—
210C4#2257613011yescharacteristic
211C4#2257613011yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—10844no—
216C2#3177375211yescharacteristic
subgroupG/H
centerC4#22576130
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#410×C271C2
socleC2×C271#379
prime radical#31774733C2
cosocleC22×C4×C271C23
hypercenter#410D271
hypocenterC271#1794
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #31773915C2semidirect
3. #31773891C2semidirect
4. #31773864C2semidirect
5. #410×C271C2semidirect
6. #22576208C4semidirect
10. #22576205C4semidirect
14. C2×#16864638C22semidirect
15. C22×C271⋊C8C22semidirect
21. #117×C271C22semidirect
108. C271#1794semidirect
113. #410D271semidirect
126. #117C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747332
#317739152
#317738912
#317738642
#410×C2712
#1794271