Finite Group Databasebeta

← #46836766

Subgroups — #46836766

Order 34688.

344 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#22576148C211yesmaximal, characteristic
2#31774374C211yesmaximal, characteristic, complemented
3#31773903C211yesmaximal, characteristic
4#31773901C211yesmaximal, characteristic
5C2×#22576078C211yesmaximal, characteristic, complemented
6#31773657C211yesmaximal, characteristic, complemented
7#339×C271C211yesmaximal, characteristic, complemented
8C22×Dic542C2211yescharacteristic
9C22×D1084C2211yescharacteristic
10C2×#16864638C2211yescharacteristic
11#22576148—22no—
12#22576148C414yescomplemented
13C4×Dic542—22no—
14#22576125—22no—
15#22576119C2211yescharacteristic
16#22576116—22no—
17C2×(C2×D271)⋊C4—22no—
18#22576082—22no—
19#22576080—22no—
20#22576078—22no—
21C2×M16×C271C2211yescharacteristic
22#117×C271C2211yescharacteristic
23C2×C42×C271C2211yescharacteristic
24C23×D271—21no—
25C2×Dic542—21no—
26C2×Dic542—21no—
27C2×Dic542—21no—
28C2×Dic542—21no—
29C2×Dic542C2×C412yes—
30C2×Dic542D412yes—
31C2×D1084—21no—
32C2×D1084—21no—
33C2×D1084—21no—
34C2×D1084—21no—
35C2×D1084—21no—
36C2×D1084C2×C412yes—
37C22×Dic271D412yes—
38#16864638—21no—
39#16864638—21no—
40C2×C271⋊C8—21no—
41C4⋊Dic271D412yes—
42Dic271⋊C4—24no—
43C22⋊Dic271—21no—
44(C2×D271)⋊C4—28no—
45C4×Dic271—24no—
46C4×Dic271—21no—
47C22×C4×C271—21no—
48C22×C4×C271C2311yescharacteristic
49M16×C271—21no—
50M16×C271—21no—
51C2×C8×C271—21no—
52C4⋊C4×C271D412yes—
53C22⋊C4×C271—21no—
54C42×C271—21no—
55C42×C271—21no—
56C42×C271C2×C412yes—
57C22×D271—22no—
58C22×D271—44no—
59C22×D271—41no—
60C22×D271—41no—
61Dic542—22no—
62Dic542—22no—
63Dic542—22no—
64Dic542—22no—
65D1084—22no—
66D1084—22no—
67D1084—41no—
68D1084—41no—
69C2×Dic271—22no—
70C2×Dic271—22no—
71C2×Dic271—22no—
72C2×Dic271C22⋊C414yes—
73C271⋊C8—41no—
74C23×C271C2×D411yescharacteristic
75C2×C4×C271—22no—
76C2×C4×C271—22no—
77C2×C4×C271—42no—
78C2×C4×C271C22×C411yescharacteristic
79C2×C4×C271C2×D411yescharacteristic
80C2×C4×C271C2×D411yescharacteristic
81C2×C4×C271C2×D411yescharacteristic
82C2×C4×C271#4111yescharacteristic
83C2×C4×C271#4111yescharacteristic
84C8×C271—41no—
85C22×C271—22no—
86C22×C271C2×C22⋊C411yescharacteristic
87C22×C271#12011yescharacteristic
88C22×C271#12311yescharacteristic
89C2×D271—44no—
90C2×D271—42no—
91C2×D271—82no—
92C2×D271—82no—
93C4×C271—42no—
94C4×C271—42no—
95C4×C271C4×D411yescharacteristic
96C4×C271C4×D411yescharacteristic
97C4×C271#12111yescharacteristic
98C4×C271#12111yescharacteristic
99Dic271—24no—
100Dic271—42no—
101C2×C271—21no—
102C2×C271—21no—
103C2×C271#38611yescharacteristic
104C2×C271#45412yes—
105D271—82no—
106C271#188811yescharacteristic, complemented, Sylow
107#1888—2711nomaximal, Sylow
108C2×#124—2711no—
109#519—2711no—
110#418—2711no—
111#418—2711no—
112C2×#101—2711no—
113#339—2711no—
114#339D27111yescharacteristic, complemented
115C22×Q8—2711no—
116C22×D4—2711no—
117C2×M16—2711no—
118C2×M16C2×D27111yescharacteristic
119#124—2714no—
120#124—5422no—
121C4×Q8—5422no—
122C4×Q8—5422no—
123#117—2711no—
124#117—5422no—
125#117C2×D27111yescharacteristic
126C2×C22⋊C4—5422no—
127C2×C42C2×D27111yescharacteristic
128#102—5422no—
129#102—5422no—
130#101—5422no—
131C24—5421no—
132C2×Q8—2712no—
133C2×Q8—2712no—
134C2×Q8—5421no—
135C2×Q8—5421no—
136C2×Q8—5421no—
137C2×Q8—5421no—
138C2×D4—2712no—
139C2×D4—5421no—
140C2×D4—5421no—
141C2×D4—5421no—
142C2×D4—5421no—
143C2×D4—5421no—
144C22×C4—21no—
145C22×C4—2712no—
146C22×C4C22×D27111yescharacteristic
147M16—21no—
148M16—21no—
149M16—5421no—
150M16—5421no—
151C2×C8—21no—
152C2×C8—5421no—
153C4⋊C4—2712no—
154C4⋊C4—5424no—
155C4⋊C4D108412yes—
156C22⋊C4—21no—
157C22⋊C4—5428no—
158C22⋊C4—5421no—
159C42—21no—
160C42—21no—
161C42—5424no—
162C42—5421no—
163C42C4×D27112yes—
164C23—5422no—
165C23—10844no—
166C23—10841no—
167C23—10841no—
168C23D4×D27111yescharacteristic
169Q8—5422no—
170Q8—5422no—
171Q8—5422no—
172Q8—5422no—
173D4—5422no—
174D4—5422no—
175D4—10841no—
176D4—10841no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—2714no—
181C2×C4—5422no—
182C2×C4—5422no—
183C2×C4—5422no—
184C2×C4C2×C4×D27111yescharacteristic
185C2×C4D4×D27111yescharacteristic
186C2×C4D4×D27111yescharacteristic
187C2×C4C2×D108411yescharacteristic
188C2×C4#1686465411yescharacteristic
189C2×C4#1686465511yescharacteristic
190C8—41no—
191C8—10841no—
192C22—22no—
193C22—10844no—
194C22—10842no—
195C22—21682no—
196C22—21682no—
197C22C22⋊C4×D27111yescharacteristic
198C22#2257613011yescharacteristic
199C22#2257614511yescharacteristic
200C4—42no—
201C4—42no—
202C4—5424no—
203C4—10842no—
204C4#2257612211yescharacteristic
205C4C4×D108411yescharacteristic
206C4#2257613311yescharacteristic
207C4#2257613511yescharacteristic
208C2—21no—
209C2—21no—
210C2—21682no—
211C2#3177377511yescharacteristic
212C2#3177407112yes—
subgroupG/H
centerC22#22576130
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#339×C271C2
socleC22×C271#120
prime radicalC22×D1084C22
cosocleC22×C4×C271C23
hypercenter#339D271
hypocenterC271#1888
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #31774374C2semidirect
5. C2×#22576078C2semidirect
6. #31773657C2semidirect
7. #339×C271C2semidirect
12. #22576148C4semidirect
106. C271#1888semidirect
114. #339D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761482
#317743742
#317739032
#317739012
C2×#225760782
#317736572
#339×C2712
#1888271