Finite Group Databasebeta

← #46847834

Subgroups — #46847834

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#31774514C212yesmaximal, complemented
2#31774185C212yesmaximal, complemented
3#31774181C212yesmaximal, complemented
4#31774155C211yesmaximal, characteristic, complemented
5#31774124C212yesmaximal, complemented
6#31774118C212yesmaximal, complemented
7#31774110C212yesmaximal, complemented
8#31773925C211yesmaximal, characteristic, complemented
9#469×C271C211yesmaximal, characteristic
10C2×Dic1084C2211yescharacteristic
11C2×#16864642C2212yes—
12C22×C271⋊C8C2211yescharacteristic
13#22576159C2212yes—
14#22576153—22no—
15#22576147—22no—
16#22576147C2212yes—
17#22576141—22no—
18#22576139C2212yes—
19#22576137—22no—
20#22576125C2212yes—
21D4×Dic271C2212yes—
22#22576119C2211yescharacteristic
23#22576092C2211yescharacteristic
24#22576091C2212yes—
25Dic271⋊C8C2212yes—
26Dic271⋊C8C2212yes—
27#22576085C2212yes—
28#22576083C2212yes—
29#22576081C2212yes—
30C2×D8×C271C2211yescharacteristic
31C2×M16×C271C2211yescharacteristic
32#121×C271C2212yes—
33#106×C271C2211yescharacteristic
34#102×C271C2212yes—
35C2×Dic542—22no—
36C2×Dic542C2312yes—
37C22×Dic271—22no—
38Dic1084—41no—
39#16864642—42no—
40C2×C271⋊C8—41no—
41C2×C271⋊C8C2312yes—
42C4⋊Dic271D412yes—
43C4⋊Dic271C2312yes—
44Dic271⋊C4—22no—
45Dic271⋊C4—22no—
46Dic271⋊C4—22no—
47C22⋊Dic271—22no—
48C22⋊Dic271—22no—
49C22⋊Dic271—22no—
50C22⋊Dic271—22no—
51C22⋊Dic271D411yescharacteristic
52C22⋊Dic271D411yescharacteristic
53C4×Dic271—22no—
54C4×Dic271C2312yes—
55C2×D4×C271—22no—
56C2×D4×C271C2312yes—
57C22×C4×C271C2311yescharacteristic
58D8×C271—41no—
59M16×C271—41no—
60C2×C8×C271C2311yescharacteristic
61C2×C8×C271C2311yescharacteristic
62C4⋊C4×C271C2312yes—
63C22⋊C4×C271—22no—
64Dic542—22no—
65Dic542—22no—
66Dic542—82no—
67C2×Dic271—22no—
68C2×Dic271—22no—
69C2×Dic271—82no—
70C2×Dic271C2×D412yes—
71C2×Dic271C2×D412yes—
72C271⋊C8—22no—
73C23×C271—22no—
74C23×C271C2×D411yescharacteristic
75D4×C271—22no—
76D4×C271—22no—
77D4×C271—82no—
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—22no—
81C2×C4×C271C2×D411yescharacteristic
82C2×C4×C271C2411yescharacteristic
83C8×C271—21no—
84C8×C271—21no—
85C22×C271—21no—
86C22×C271—22no—
87C22×C271—42no—
88C22×C271—42no—
89C22×C271—42no—
90C22×C271C22×D411yescharacteristic
91C4×C271—21no—
92C4×C271—42no—
93C4×C271H(5,𝔽2)11yescharacteristic
94C4×C2712−1+411yescharacteristic
95Dic271—22no—
96Dic271—22no—
97Dic271—42no—
98Dic271—42no—
99C2×C271—41no—
100C2×C271—82no—
101C2×C271#53711yescharacteristic
102C2×C271#57712yes—
103C271#326311yescharacteristic, complemented, Sylow
104#3263—2711nomaximal, Sylow
105#542—2712no—
106#476—2712no—
107#475—2712no—
108#470—2711no—
109#469D27111yescharacteristic
110#465—2712no—
111#464—2712no—
112#463—2712no—
113#424—2711no—
114C2×Q16—2711no—
115C2×SD16—2712no—
116C2×D8C2×D27111yescharacteristic
117C2×M16C2×D27111yescharacteristic
118C22×C8—2711no—
119C4⋊Q8—2712no—
120#126—5422no—
121#124—2712no—
122#124—5422no—
123#123—5422no—
124C22⋊Q8—2712no—
125C22⋊Q8—5422no—
126#121C2×D27112yes—
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#103—2712no—
137#102—2712no—
138#102C2×D27112yes—
139C2×Q8—2712no—
140C2×Q8—5422no—
141C2×D4—22no—
142C2×D4C22×D27112yes—
143C22×C4—5422no—
144C22×C4C22×D27111yescharacteristic
145Q16—10841no—
146SD16—10842no—
147D8—41no—
148M16—41no—
149C2×C8—2712no—
150C2×C8—10841no—
151C2×C8C22×D27111yescharacteristic
152C2×C8C22×D27111yescharacteristic
153C4⋊C4—2712no—
154C4⋊C4—2712no—
155C4⋊C4—5422no—
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—22no—
169C23D4×D27111yescharacteristic
170Q8—5422no—
171Q8—5422no—
172Q8—21682no—
173D4—22no—
174D4—22no—
175D4—82no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—2712no—
180C2×C4—2712no—
181C2×C4—5422no—
182C2×C4—5422no—
183C2×C4—21682no—
184C2×C4D4×D27111yescharacteristic
185C2×C4C23×D27111yescharacteristic
186C8—21no—
187C8—21no—
188C8—5422no—
189C22—21no—
190C22—22no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22C2×D4×D27111yescharacteristic
195C4—21no—
196C4—42no—
197C4—5422no—
198C4—5422no—
199C4—10842no—
200C4—10842no—
201C4#2257620511yescharacteristic
202C4#2257620711yescharacteristic
203C2—41no—
204C2—82no—
205C2#3177447411yescharacteristic
206C2#3177470712yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting, prime radical#469×C271C2
socleC22×C271C22×D4
hypercenter#469D271
hypocenterC271#3263
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31774514C2semidirect
2. #31774185C2semidirect
3. #31774181C2semidirect
4. #31774155C2semidirect
5. #31774124C2semidirect
6. #31774118C2semidirect
7. #31774110C2semidirect
8. #31773925C2semidirect
103. C271#3263semidirect

Maximal subgroups

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

subgroupindex
#317745142
#317741852
#317741812
#317741552
#317741242
#317741182
#317741102
#317739252
#469×C2712
#3263271