Finite Group Databasebeta

← #46847825

Subgroups — #46847825

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#31774193C212yesmaximal, complemented
3#31774178C212yesmaximal, complemented
4#31774143C212yesmaximal, complemented
5#31774138C211yesmaximal, characteristic, complemented
6#31774133C211yesmaximal, characteristic
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#22576094C2211yescharacteristic
24#22576094C2211yescharacteristic
25#22576094C2212yes—
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×C271D411yescharacteristic
61C4⋊C4×C271D411yescharacteristic
62C22⋊C4×C271D412yes—
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—21no—
83C8×C271—21no—
84C8×C271—22no—
85C22×C271—21no—
86C22×C271—21no—
87C22×C271—21no—
88C22×C271C22×D411yescharacteristic
89C2×D271—42no—
90C2×D271—42no—
91C2×D271—42no—
92C4×C271—21no—
93C4×C271—22no—
94C4×C271—22no—
95C4×C271H(5,𝔽2)11yescharacteristic
96C4×C2712−1+411yescharacteristic
97Dic271—42no—
98Dic271—42no—
99Dic271—42no—
100C2×C271—41no—
101C2×C271#53711yescharacteristic
102C2×C271#57611yescharacteristic
103C2×C271#57811yescharacteristic
104D271—82no—
105C271#326211yescharacteristic, complemented, Sylow
106#3262—2711nomaximal, Sylow
107#542—2712no—
108#477—2712no—
109#474—2712no—
110#468—2712no—
111#467—2711no—
112#466—2711no—
113#464—2712no—
114#463—2712no—
115#424D27111yescharacteristic, complemented
116C2×Q16—2711no—
117C2×SD16—2712no—
118C2×D8—2711no—
119C2×M16C2×D27111yescharacteristic
120C22×C8C2×D27111yescharacteristic
121C4⋊Q8—2712no—
122#126—5422no—
123#124—2712no—
124#124—5422no—
125#123—5422no—
126C22⋊Q8—2712no—
127C22⋊Q8—5422no—
128#121—2712no—
129C4×Q8—2712no—
130C4×D4—2712no—
131#117C2×D27111yescharacteristic
132#107—2711no—
133#107—2711no—
134#107—2712no—
135C4⋊C8C2×D27112yes—
136C4⋊C8C2×D27112yes—
137#103—2712no—
138#103—2712no—
139#102—2712no—
140#102—2712no—
141C2×Q8—2712no—
142C2×Q8—5422no—
143C2×D4—2712no—
144C2×D4—5422no—
145C22×C4—5422no—
146C22×C4C22×D27111yescharacteristic
147Q16—10841no—
148SD16—10842no—
149D8—10841no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D27111yescharacteristic
153C2×C8C22×D27112yes—
154C2×C8C22×D27111yescharacteristic
155C4⋊C4—2712no—
156C4⋊C4—2712no—
157C4⋊C4—5422no—
158C4⋊C4—5422no—
159C4⋊C4—5422no—
160C4⋊C4D108411yescharacteristic
161C4⋊C4D108411yescharacteristic
162C22⋊C4—5422no—
163C22⋊C4—5422no—
164C22⋊C4—5422no—
165C22⋊C4—5422no—
166C22⋊C4—5422no—
167C22⋊C4D108412yes—
168C42—5422no—
169C42C22×D27112yes—
170C23—5422no—
171C23C2×D108411yescharacteristic
172Q8—5422no—
173Q8—5422no—
174Q8—21682no—
175D4—5422no—
176D4—5422no—
177D4—21682no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—5422no—
181C2×C4—5422no—
182C2×C4—5422no—
183C2×C4—21682no—
184C2×C4C2×D108412yes—
185C2×C4C2×D108412yes—
186C2×C4C2×D108411yescharacteristic
187C2×C4C23×D27111yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—10842no—
195C22—10842no—
196C22—10842no—
197C22C22×D108411yescharacteristic
198C4—21no—
199C4—22no—
200C4—22no—
201C4—10842no—
202C4—10842no—
203C4—10842no—
204C4#2257620511yescharacteristic
205C4#2257620811yescharacteristic
206C2—41no—
207C2—21682no—
208C2#3177447811yescharacteristic
209C2#3177470211yescharacteristic
210C2#3177471411yescharacteristic
subgroupG/H
centerC22C22×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#424×C271C2
socleC22×C271C22×D4
prime radical#31774133C2
hypercenter#424D271
hypocenterC271#3262
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. #31774193C2semidirect
3. #31774178C2semidirect
4. #31774143C2semidirect
5. #31774138C2semidirect
7. #31774120C2semidirect
8. #31774113C2semidirect
9. #424×C271C2semidirect
105. C271#3262semidirect
115. #424D271semidirect

Maximal subgroups

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

subgroupindex
#317745182
#317741932
#317741782
#317741432
#317741382
#317741332
#317741202
#317741132
#424×C2712
#3262271