Finite Group Databasebeta

← #46847830

Subgroups — #46847830

Order 34688.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#31774520C212yesmaximal, complemented
2#31774186C212yesmaximal, complemented
3#31774180C212yesmaximal, complemented
4#31774150C211yesmaximal, characteristic
5#31774128C212yesmaximal, complemented
6#31774121C212yesmaximal, complemented
7#31774115C212yesmaximal, complemented
8#31773925C211yesmaximal, characteristic, complemented
9#470×C271C211yesmaximal, characteristic, complemented
10C2×#16864641C2212yes—
11C2×D2168C2211yescharacteristic
12C22×C271⋊C8C2211yescharacteristic
13#22576159C2212yes—
14#22576155—22no—
15#22576149C2212yes—
16#22576146—22no—
17#22576142—22no—
18#22576140—22no—
19#22576134C2212yes—
20Q8×Dic271C2212yes—
21#22576122C2212yes—
22#22576119C2211yescharacteristic
23#22576092C2211yescharacteristic
24#22576091C2212yes—
25Dic271⋊C8C2212yes—
26Dic271⋊C8C2212yes—
27#22576084C2212yes—
28#22576082C2212yes—
29#22576080C2212yes—
30C2×Q16×C271C2211yescharacteristic
31C2×M16×C271C2211yescharacteristic
32C22⋊Q8×C271C2212yes—
33#106×C271C2211yescharacteristic
34#103×C271C2212yes—
35C2×Dic542—22no—
36C2×D1084C2312yes—
37C2×C22⋊D271—22no—
38C2×C4×D271—22no—
39#16864641—42no—
40D2168—41no—
41C2×C271⋊C8—41no—
42C2×C271⋊C8C2312yes—
43C4⋊Dic271—22no—
44C4⋊Dic271D412yes—
45C4⋊Dic271C2312yes—
46Dic271⋊C4—22no—
47C22⋊Dic271D411yescharacteristic
48C22⋊Dic271D411yescharacteristic
49(C2×D271)⋊C4—22no—
50(C2×D271)⋊C4—22no—
51(C2×D271)⋊C4—22no—
52(C2×D271)⋊C4—22no—
53C4×Dic271—22no—
54C4×Dic271C2312yes—
55C2×Q8×C271C2312yes—
56C22×C4×C271C2311yescharacteristic
57Q16×C271—41no—
58M16×C271—41no—
59C2×C8×C271C2311yescharacteristic
60C2×C8×C271C2311yescharacteristic
61C4⋊C4×C271—22no—
62C4⋊C4×C271C2312yes—
63C22⋊C4×C271—22no—
64C22×D271—22no—
65Dic542—82no—
66D1084—22no—
67D1084—22no—
68C22⋊D271—82no—
69C2×Dic271—22no—
70C2×Dic271C2×D412yes—
71C2×Dic271C2×D412yes—
72C4×D271—82no—
73C271⋊C8—22no—
74C23×C271C2×D411yescharacteristic
75Q8×C271—22no—
76Q8×C271—22no—
77C2×C4×C271—21no—
78C2×C4×C271—21no—
79C2×C4×C271—22no—
80C2×C4×C271—22no—
81C2×C4×C271C2×D411yescharacteristic
82C2×C4×C271C2411yescharacteristic
83C8×C271—21no—
84C8×C271—21no—
85C22×C271—21no—
86C22×C271—22no—
87C22×C271C22×D411yescharacteristic
88C2×D271—42no—
89C2×D271—42no—
90C2×D271—42no—
91C4×C271—21no—
92C4×C271—42no—
93C4×C271—42no—
94C4×C271H(5,𝔽2)11yescharacteristic
95C4×C2712−1+411yescharacteristic
96Dic271—22no—
97Dic271—22no—
98Dic271—42no—
99C2×C271—41no—
100C2×C271#53711yescharacteristic
101C2×C271#57712yes—
102D271—82no—
103C271#326311yescharacteristic, complemented, Sylow
104#3263—2711nomaximal, Sylow
105#542—2712no—
106#476—2712no—
107#475—2712no—
108#470D27111yescharacteristic, complemented
109#469—2711no—
110#465—2712no—
111#464—2712no—
112#463—2712no—
113#424—2711no—
114C2×Q16C2×D27111yescharacteristic
115C2×SD16—2712no—
116C2×D8—2711no—
117C2×M16C2×D27111yescharacteristic
118C22×C8—2711no—
119C4⋊Q8—2712no—
120#126—5422no—
121#124—2712no—
122#124—5422no—
123#123—5422no—
124C22⋊Q8—5422no—
125C22⋊Q8C2×D27112yes—
126#121—2712no—
127C4×Q8—2712no—
128C4×D4—2712no—
129#117—2711no—
130#106—2711no—
131#106—2712no—
132#106C2×D27111yescharacteristic
133C4⋊C8—2712no—
134C4⋊C8—2712no—
135#103—2712no—
136#103C2×D27112yes—
137#102—2712no—
138#102—2712no—
139C2×Q8—5422no—
140C2×Q8C22×D27112yes—
141C2×D4—2712no—
142C2×D4—5422no—
143C22×C4—5422no—
144C22×C4C22×D27111yescharacteristic
145Q16—41no—
146SD16—10842no—
147D8—10841no—
148M16—41no—
149C2×C8—2712no—
150C2×C8—10841no—
151C2×C8C22×D27111yescharacteristic
152C2×C8C22×D27111yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—2712no—
155C4⋊C4—2712no—
156C4⋊C4—5422no—
157C4⋊C4—5422no—
158C4⋊C4C22×D27112yes—
159C22⋊C4—22no—
160C22⋊C4—2711no—
161C22⋊C4—2711no—
162C22⋊C4—5422no—
163C22⋊C4—5422no—
164C22⋊C4—5422no—
165C22⋊C4—5422no—
166C42—2712no—
167C42—5422no—
168C23—5422no—
169C23D4×D27111yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—21682no—
173D4—5422no—
174D4—5422no—
175D4—21682no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—2712no—
181C2×C4—2712no—
182C2×C4—5422no—
183C2×C4—21682no—
184C2×C4D4×D27111yescharacteristic
185C2×C4C23×D27111yescharacteristic
186C8—21no—
187C8—21no—
188C8—5422no—
189C22—21no—
190C22—22no—
191C22—10842no—
192C22—10842no—
193C22—10842no—
194C22C2×D4×D27111yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—5422no—
199C4—5422no—
200C4—10842no—
201C4#2257620511yescharacteristic
202C4#2257620711yescharacteristic
203C2—41no—
204C2—21682no—
205C2#3177447411yescharacteristic
206C2#3177470612yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#470×C271C2
socleC22×C271C22×D4
prime radical#31774150C2
hypercenter#470D271
hypocenterC271#3263
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #31774520C2semidirect
2. #31774186C2semidirect
3. #31774180C2semidirect
5. #31774128C2semidirect
6. #31774121C2semidirect
7. #31774115C2semidirect
8. #31773925C2semidirect
9. #470×C271C2semidirect
103. C271#3263semidirect
108. #470D271semidirect

Maximal subgroups

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

subgroupindex
#317745202
#317741862
#317741802
#317741502
#317741282
#317741212
#317741152
#317739252
#470×C2712
#3263271