Finite Group Databasebeta

← #46847826

Subgroups — #46847826

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#31774190C212yesmaximal, complemented
3#31774176C212yesmaximal, complemented
4#31774145C212yesmaximal, complemented
5#31774133C211yesmaximal, characteristic
6#31774121C212yesmaximal, complemented
7#31774115C212yesmaximal, complemented
8#31773923C211yesmaximal, characteristic, complemented
9#467×C271C211yesmaximal, characteristic, complemented
10C2×#16864641C2212yes—
11C2×D2168C2211yescharacteristic
12C2×#16864638C2211yescharacteristic
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#22576094C2211yescharacteristic
24#22576093C2212yes—
25Dic271⋊C8C2212yes—
26Dic271⋊C8C2212yes—
27#22576084C2212yes—
28#22576082C2212yes—
29#22576080C2212yes—
30C2×Q16×C271C2211yescharacteristic
31C22×C8×C271C2211yescharacteristic
32C22⋊Q8×C271C2212yes—
33#107×C271C2211yescharacteristic
34#103×C271C2212yes—
35C2×Dic542—22no—
36C2×D1084C2312yes—
37C2×C22⋊D271—22no—
38C2×C4×D271—22no—
39#16864641—42no—
40D2168—41no—
41#16864638—41no—
42C2×C271⋊C8C2312yes—
43C4⋊Dic271—22no—
44C4⋊Dic271D411yescharacteristic
45C4⋊Dic271D411yescharacteristic
46C4⋊Dic271C2312yes—
47Dic271⋊C4—22no—
48C22⋊Dic271D412yes—
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—
58C2×C8×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—21no—
87C22×C271—21no—
88C22×C271C22×D411yescharacteristic
89C2×D271—42no—
90C2×D271—42no—
91C2×D271—42no—
92C4×C271—21no—
93C4×C271—42no—
94C4×C271—42no—
95C4×C271H(5,𝔽2)11yescharacteristic
96C4×C2712−1+411yescharacteristic
97Dic271—22no—
98Dic271—22no—
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#467D27111yescharacteristic, complemented
112#466—2711no—
113#464—2712no—
114#463—2712no—
115#424—2711no—
116C2×Q16C2×D27111yescharacteristic
117C2×SD16—2712no—
118C2×D8—2711no—
119C2×M16—2711no—
120C22×C8C2×D27111yescharacteristic
121C4⋊Q8—2712no—
122#126—5422no—
123#124—2712no—
124#124—5422no—
125#123—5422no—
126C22⋊Q8—5422no—
127C22⋊Q8C2×D27112yes—
128#121—2712no—
129C4×Q8—2712no—
130C4×D4—2712no—
131#117—2711no—
132#107—2711no—
133#107—2712no—
134#107C2×D27111yescharacteristic
135C4⋊C8—2712no—
136C4⋊C8—2712no—
137#103—2712no—
138#103C2×D27112yes—
139#102—2712no—
140#102—2712no—
141C2×Q8—5422no—
142C2×Q8C22×D27112yes—
143C2×D4—2712no—
144C2×D4—5422no—
145C22×C4—5422no—
146C22×C4C22×D27111yescharacteristic
147Q16—41no—
148SD16—10842no—
149D8—10841no—
150M16—10841no—
151C2×C8—41no—
152C2×C8—2712no—
153C2×C8C22×D27111yescharacteristic
154C2×C8C22×D27111yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—2711no—
157C4⋊C4—2711no—
158C4⋊C4—2712no—
159C4⋊C4—5422no—
160C4⋊C4—5422no—
161C4⋊C4C22×D27112yes—
162C22⋊C4—22no—
163C22⋊C4—2712no—
164C22⋊C4—5422no—
165C22⋊C4—5422no—
166C22⋊C4—5422no—
167C22⋊C4—5422no—
168C42—2712no—
169C42—5422no—
170C23—5422no—
171C23D4×D27111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—21682no—
175D4—5422no—
176D4—5422no—
177D4—21682no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2712no—
183C2×C4—2712no—
184C2×C4—5422no—
185C2×C4—21682no—
186C2×C4D4×D27111yescharacteristic
187C2×C4C23×D27111yescharacteristic
188C8—21no—
189C8—21no—
190C8—5422no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—10842no—
195C22—10842no—
196C22—10842no—
197C22C2×D4×D27111yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—5422no—
202C4—5422no—
203C4—10842no—
204C4#2257620511yescharacteristic
205C4#2257620711yescharacteristic
206C2—41no—
207C2—21682no—
208C2#3177447411yescharacteristic
209C2#3177469911yescharacteristic
210C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#467×C271C2
socleC22×C271C22×D4
prime radical#31774133C2
hypercenter#467D271
hypocenterC271#3262
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #31774520C2semidirect
2. #31774190C2semidirect
3. #31774176C2semidirect
4. #31774145C2semidirect
6. #31774121C2semidirect
7. #31774115C2semidirect
8. #31773923C2semidirect
9. #467×C271C2semidirect
105. C271#3262semidirect
111. #467D271semidirect

Maximal subgroups

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

subgroupindex
#317745202
#317741902
#317741762
#317741452
#317741332
#317741212
#317741152
#317739232
#467×C2712
#3262271