Finite Group Databasebeta

← #46847835

Subgroups — #46847835

Order 34688.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#31774518C212yesmaximal, complemented
2#31774188C212yesmaximal, complemented
3#31774183C212yesmaximal, complemented
4#31774155C211yesmaximal, characteristic, complemented
5#31774150C211yesmaximal, characteristic
6#31774126C212yesmaximal, complemented
7#31774120C212yesmaximal, complemented
8#31774113C212yesmaximal, complemented
9#424×C271C211yesmaximal, characteristic, complemented
10C2×Dic1084C2211yescharacteristic
11C2×#16864643C2212yes—
12C2×D2168C2211yescharacteristic
13#22576160C2212yes—
14#22576155—22no—
15#22576148C2212yes—
16#22576146—22no—
17#22576142—22no—
18#22576139C2212yes—
19#22576138—22no—
20#22576134C2212yes—
21C4×Dic542C2212yes—
22C4×D1084C2212yes—
23#22576092C2212yes—
24#22576092C2211yescharacteristic
25#22576092C2211yescharacteristic
26#22576085C2212yes—
27#22576085C2212yes—
28#22576082C2212yes—
29#22576082C2212yes—
30C2×M16×C271C2211yescharacteristic
31C22×C8×C271C2211yescharacteristic
32#117×C271C2211yescharacteristic
33C4⋊C8×C271C2212yes—
34C4⋊C8×C271C2212yes—
35C2×Dic542—22no—
36C2×Dic542C2312yes—
37C2×D1084C2312yes—
38C2×C22⋊D271—22no—
39C2×C4×D271—22no—
40Dic1084—41no—
41#16864643—42no—
42D2168—41no—
43C4⋊Dic271—22no—
44C4⋊Dic271C2312yes—
45C4⋊Dic271C2312yes—
46Dic271⋊C4—22no—
47Dic271⋊C4—22no—
48C22⋊Dic271—22no—
49(C2×D271)⋊C4—22no—
50(C2×D271)⋊C4—22no—
51(C2×D271)⋊C4—22no—
52(C2×D271)⋊C4—22no—
53C4×Dic271—22no—
54C22×C4×C271C2311yescharacteristic
55M16×C271—41no—
56C2×C8×C271—41no—
57C2×C8×C271C2311yescharacteristic
58C2×C8×C271C2312yes—
59C2×C8×C271C2311yescharacteristic
60C4⋊C4×C271D412yes—
61C22⋊C4×C271D411yescharacteristic
62C22⋊C4×C271D411yescharacteristic
63C42×C271C2312yes—
64C22×D271—22no—
65Dic542—22no—
66Dic542—22no—
67Dic542—82no—
68D1084—22no—
69D1084—22no—
70C22⋊D271—82no—
71C2×Dic271—22no—
72C2×Dic271—22no—
73C2×Dic271—22no—
74C4×D271—82no—
75C23×C271C2×D411yescharacteristic
76C2×C4×C271—21no—
77C2×C4×C271—21no—
78C2×C4×C271C2×D412yes—
79C2×C4×C271C2×D412yes—
80C2×C4×C271C2×D411yescharacteristic
81C2×C4×C271C2411yescharacteristic
82C8×C271—22no—
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—22no—
93C4×C271—22no—
94C4×C271H(5,𝔽2)11yescharacteristic
95C4×C2712−1+411yescharacteristic
96Dic271—42no—
97Dic271—42no—
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#470—2711no—
109#469—2711no—
110#465—2712no—
111#464—2712no—
112#463—2712no—
113#424D27111yescharacteristic, complemented
114C2×Q16—2711no—
115C2×SD16—2712no—
116C2×D8—2711no—
117C2×M16C2×D27111yescharacteristic
118C22×C8C2×D27111yescharacteristic
119C4⋊Q8—2712no—
120#126—5422no—
121#124—2712no—
122#124—5422no—
123#123—5422no—
124C22⋊Q8—2712no—
125C22⋊Q8—5422no—
126#121—2712no—
127C4×Q8—2712no—
128C4×D4—2712no—
129#117C2×D27111yescharacteristic
130#106—2712no—
131#106—2711no—
132#106—2711no—
133C4⋊C8C2×D27112yes—
134C4⋊C8C2×D27112yes—
135#103—2712no—
136#103—2712no—
137#102—2712no—
138#102—2712no—
139C2×Q8—2712no—
140C2×Q8—5422no—
141C2×D4—2712no—
142C2×D4—5422no—
143C22×C4—5422no—
144C22×C4C22×D27111yescharacteristic
145Q16—10841no—
146SD16—10842no—
147D8—10841no—
148M16—41no—
149C2×C8—41no—
150C2×C8C22×D27111yescharacteristic
151C2×C8C22×D27112yes—
152C2×C8C22×D27111yescharacteristic
153C4⋊C4—2712no—
154C4⋊C4—2712no—
155C4⋊C4—5422no—
156C4⋊C4—5422no—
157C4⋊C4—5422no—
158C4⋊C4D108412yes—
159C22⋊C4—5422no—
160C22⋊C4—5422no—
161C22⋊C4—5422no—
162C22⋊C4—5422no—
163C22⋊C4—5422no—
164C22⋊C4D108411yescharacteristic
165C22⋊C4D108411yescharacteristic
166C42—5422no—
167C42C22×D27112yes—
168C23—5422no—
169C23C2×D108411yescharacteristic
170Q8—5422no—
171Q8—5422no—
172Q8—21682no—
173D4—5422no—
174D4—5422no—
175D4—21682no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—5422no—
179C2×C4—5422no—
180C2×C4—5422no—
181C2×C4—21682no—
182C2×C4C2×D108412yes—
183C2×C4C2×D108412yes—
184C2×C4C2×D108411yescharacteristic
185C2×C4C23×D27111yescharacteristic
186C8—22no—
187C8—21no—
188C8—21no—
189C22—21no—
190C22—22no—
191C22—10842no—
192C22—10842no—
193C22—10842no—
194C22C22×D108411yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—10842no—
199C4—10842no—
200C4—10842no—
201C4#2257620511yescharacteristic
202C4#2257620811yescharacteristic
203C2—41no—
204C2—21682no—
205C2#3177447811yescharacteristic
206C2#3177470912yes—
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#424×C271C2
socleC22×C271C22×D4
prime radical#31774150C2
hypercenter#424D271
hypocenterC271#3263
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #31774518C2semidirect
2. #31774188C2semidirect
3. #31774183C2semidirect
4. #31774155C2semidirect
6. #31774126C2semidirect
7. #31774120C2semidirect
8. #31774113C2semidirect
9. #424×C271C2semidirect
103. C271#3263semidirect
113. #424D271semidirect

Maximal subgroups

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

subgroupindex
#317745182
#317741882
#317741832
#317741552
#317741502
#317741262
#317741202
#317741132
#424×C2712
#3263271