Finite Group Databasebeta

← #44400136

Subgroups — #44400136

Order 29056.

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×#20300276C211yesmaximal, characteristic
2#28409509C211yesmaximal, characteristic, complemented
3#28409038C211yesmaximal, characteristic
4#28409036C211yesmaximal, characteristic
5C2×#20300206C211yesmaximal, characteristic, complemented
6#28408792C211yesmaximal, characteristic, complemented
7#339×C227C211yesmaximal, characteristic, complemented
8C22×Dic454C2211yescharacteristic
9C22×D908C2211yescharacteristic
10C2×#16813868C2211yescharacteristic
11#20300276—22no—
12#20300276C414yescomplemented
13C4×Dic454—22no—
14#20300253—22no—
15#20300247C2211yescharacteristic
16#20300244—22no—
17C2×(C2×D227)⋊C4—22no—
18#20300210—22no—
19#20300208—22no—
20#20300206—22no—
21C2×M16×C227C2211yescharacteristic
22#117×C227C2211yescharacteristic
23C2×C42×C227C2211yescharacteristic
24C23×D227—21no—
25C2×Dic454—21no—
26C2×Dic454—21no—
27C2×Dic454—21no—
28C2×Dic454—21no—
29C2×Dic454C2×C412yes—
30C2×Dic454D412yes—
31C2×D908—21no—
32C2×D908—21no—
33C2×D908—21no—
34C2×D908—21no—
35C2×D908—21no—
36C2×D908C2×C412yes—
37C22×Dic227D412yes—
38#16813868—21no—
39#16813868—21no—
40C2×C227⋊C8—21no—
41C4⋊Dic227D412yes—
42Dic227⋊C4—24no—
43C22⋊Dic227—21no—
44(C2×D227)⋊C4—28no—
45C4×Dic227—24no—
46C4×Dic227—21no—
47C22×C4×C227—21no—
48C22×C4×C227C2311yescharacteristic
49M16×C227—21no—
50M16×C227—21no—
51C2×C8×C227—21no—
52C4⋊C4×C227D412yes—
53C22⋊C4×C227—21no—
54C42×C227—21no—
55C42×C227—21no—
56C42×C227C2×C412yes—
57C22×D227—22no—
58C22×D227—44no—
59C22×D227—41no—
60C22×D227—41no—
61Dic454—22no—
62Dic454—22no—
63Dic454—22no—
64Dic454—22no—
65D908—22no—
66D908—22no—
67D908—41no—
68D908—41no—
69C2×Dic227—22no—
70C2×Dic227—22no—
71C2×Dic227—22no—
72C2×Dic227C22⋊C414yes—
73C227⋊C8—41no—
74C23×C227C2×D411yescharacteristic
75C2×C4×C227—22no—
76C2×C4×C227—22no—
77C2×C4×C227—42no—
78C2×C4×C227C22×C411yescharacteristic
79C2×C4×C227C2×D411yescharacteristic
80C2×C4×C227C2×D411yescharacteristic
81C2×C4×C227C2×D411yescharacteristic
82C2×C4×C227#4111yescharacteristic
83C2×C4×C227#4111yescharacteristic
84C8×C227—41no—
85C22×C227—22no—
86C22×C227C2×C22⋊C411yescharacteristic
87C22×C227#12011yescharacteristic
88C22×C227#12311yescharacteristic
89C2×D227—44no—
90C2×D227—42no—
91C2×D227—82no—
92C2×D227—82no—
93C4×C227—42no—
94C4×C227—42no—
95C4×C227C4×D411yescharacteristic
96C4×C227C4×D411yescharacteristic
97C4×C227#12111yescharacteristic
98C4×C227#12111yescharacteristic
99Dic227—24no—
100Dic227—42no—
101C2×C227—21no—
102C2×C227—21no—
103C2×C227#38611yescharacteristic
104C2×C227#45412yes—
105D227—82no—
106C227#188811yescharacteristic, complemented, Sylow
107#1888—2271nomaximal, Sylow
108C2×#124—2271no—
109#519—2271no—
110#418—2271no—
111#418—2271no—
112C2×#101—2271no—
113#339—2271no—
114#339D22711yescharacteristic, complemented
115C22×Q8—2271no—
116C22×D4—2271no—
117C2×M16—2271no—
118C2×M16C2×D22711yescharacteristic
119#124—4542no—
120#124—2274no—
121C4×Q8—4542no—
122C4×Q8—4542no—
123#117—4542no—
124#117—2271no—
125#117C2×D22711yescharacteristic
126C2×C22⋊C4—4542no—
127C2×C42C2×D22711yescharacteristic
128#102—4542no—
129#102—4542no—
130#101—4542no—
131C24—4541no—
132C2×Q8—4541no—
133C2×Q8—4541no—
134C2×Q8—4541no—
135C2×Q8—4541no—
136C2×Q8—2272no—
137C2×Q8—2272no—
138C2×D4—4541no—
139C2×D4—4541no—
140C2×D4—4541no—
141C2×D4—4541no—
142C2×D4—4541no—
143C2×D4—2272no—
144C22×C4—21no—
145C22×C4—2272no—
146C22×C4C22×D22711yescharacteristic
147M16—21no—
148M16—21no—
149M16—4541no—
150M16—4541no—
151C2×C8—21no—
152C2×C8—4541no—
153C4⋊C4—4544no—
154C4⋊C4—2272no—
155C4⋊C4D90812yes—
156C22⋊C4—21no—
157C22⋊C4—4548no—
158C22⋊C4—4541no—
159C42—21no—
160C42—21no—
161C42—4544no—
162C42—4541no—
163C42C4×D22712yes—
164C23—9084no—
165C23—9081no—
166C23—9081no—
167C23—4542no—
168C23D4×D22711yescharacteristic
169Q8—4542no—
170Q8—4542no—
171Q8—4542no—
172Q8—4542no—
173D4—9081no—
174D4—9081no—
175D4—4542no—
176D4—4542no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—4542no—
181C2×C4—4542no—
182C2×C4—4542no—
183C2×C4—2274no—
184C2×C4C2×C4×D22711yescharacteristic
185C2×C4D4×D22711yescharacteristic
186C2×C4D4×D22711yescharacteristic
187C2×C4C2×D90811yescharacteristic
188C2×C4#1681388411yescharacteristic
189C2×C4#1681388511yescharacteristic
190C8—41no—
191C8—9081no—
192C22—22no—
193C22—18162no—
194C22—18162no—
195C22—9084no—
196C22—9082no—
197C22C22⋊C4×D22711yescharacteristic
198C22#2030025811yescharacteristic
199C22#2030027311yescharacteristic
200C4—42no—
201C4—42no—
202C4—9082no—
203C4—4544no—
204C4#2030025011yescharacteristic
205C4C4×D90811yescharacteristic
206C4#2030026111yescharacteristic
207C4#2030026311yescharacteristic
208C2—21no—
209C2—21no—
210C2—18162no—
211C2#2840891011yescharacteristic
212C2#2840920612yes—
subgroupG/H
centerC22#20300258
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#339×C227C2
socleC22×C227#120
prime radicalC22×D908C22
cosocleC22×C4×C227C23
hypercenter#339D227
hypocenterC227#1888
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #28409509C2semidirect
5. C2×#20300206C2semidirect
6. #28408792C2semidirect
7. #339×C227C2semidirect
12. #20300276C4semidirect
106. C227#1888semidirect
114. #339D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002762
#284095092
#284090382
#284090362
C2×#203002062
#284087922
#339×C2272
#1888227