Finite Group Databasebeta

← #44400783

Subgroups — #44400783

Order 29056.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20300321C212yesmaximal, complemented
2#28409613C211yesmaximal, characteristic
3C2×#20300214C212yesmaximal, complemented
4#28408937C211yesmaximal, characteristic, complemented
5#343×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C2212yes—
7C22×Dic454C2211yescharacteristic
8#20300321—24no—
9#20300321—24no—
10C2×Dic908—22no—
11C2×#16813873—22no—
12#20300268—22no—
13#20300263—22no—
14#20300262—22no—
15C2×C4×Dic227C2211yescharacteristic
16#20300214—24no—
17C227⋊#95—22no—
18C2×M16×C227C2212yes—
19C2×C4⋊C4×C227C2211yescharacteristic
20#16813884—22no—
21#16813884—22no—
22#16813884—22no—
23#16813884—22no—
24C2×Dic454—22no—
25C2×Dic454—22no—
26C2×Dic454—21no—
27C2×Dic454D412yes—
28C2×Dic454D412yescomplemented
29C2×D908—22no—
30C2×D908D412yes—
31C2×C22⋊D227—22no—
32C22×Dic227—21no—
33C22×Dic227—21no—
34C2×C4×D227—22no—
35Dic908—28no—
36#16813873—28no—
37C4⋊Dic227—22no—
38(C2×D227)⋊C4—24no—
39C4×Dic227—22no—
40C22×C4×C227D412yes—
41C22×C4×C227C2311yescharacteristic
42M16×C227—22no—
43M16×C227D414yescomplemented
44C2×C8×C227—22no—
45C4⋊C4×C227—22no—
46C22×D227—22no—
47Dic454—24no—
48Dic454—22no—
49Dic454—22no—
50Dic454—42no—
51D908—22no—
52D908—22no—
53D908—82no—
54C22⋊D227—42no—
55C22⋊D227—42no—
56C2×Dic227—22no—
57C2×Dic227—22no—
58C2×Dic227—22no—
59C2×Dic227—41no—
60C2×Dic227—41no—
61C2×Dic227—41no—
62C2×Dic227—41no—
63C4×D227—42no—
64C4×D227—42no—
65C23×C227C2×D411yescharacteristic
66C2×C4×C227—22no—
67C2×C4×C227—42no—
68C2×C4×C227C2×D411yescharacteristic
69C2×C4×C227C2×D412yes—
70C2×C4×C227C2×D412yes—
71C2×C4×C227#4111yescharacteristic
72C8×C227—24no—
73C22×C227—21no—
74C22×C227—21no—
75C22×C227#12011yescharacteristic
76C22×C227#12111yescharacteristic
77C22×C227#12711yescharacteristic
78C2×D227—42no—
79C2×D227—42no—
80C2×D227—42no—
81C4×C227—42no—
82C4×C227#12012yes—
83C4×C227#12112yes—
84Dic227—42no—
85Dic227—42no—
86Dic227—42no—
87C2×C227—21no—
88C2×C227—21no—
89C2×C227#39211yescharacteristic
90C2×C227#45511yescharacteristic
91C2×C227#45611yescharacteristic
92D227—82no—
93C227#200711yescharacteristic, complemented, Sylow
94#2007—2271nomaximal, Sylow
95C2×#137—2272no—
96#537—2271no—
97C2×#104—2272no—
98#391—2271no—
99#343D22711yescharacteristic, complemented
100C2×#41—2272no—
101C22×Q8—2271no—
102#137—4544no—
103#137—4544no—
104C2×Q16—4542no—
105C2×SD16—4542no—
106C2×M16C2×D22712yes—
107C22⋊Q8—4542no—
108#121—4542no—
109#121—4542no—
110C2×C4⋊C4C2×D22711yescharacteristic
111C2×C42—2271no—
112#104—4544no—
113#95—4542no—
114#41—4542no—
115#41—4542no—
116#41—4542no—
117#41—4542no—
118C2×Q8—4542no—
119C2×Q8—4542no—
120C2×Q8—4541no—
121C2×Q8—2272no—
122C2×Q8—2272no—
123C2×D4—4542no—
124C2×D4—4542no—
125C2×D4—2272no—
126C22×C4—4541no—
127C22×C4—4541no—
128C22×C4—4542no—
129C22×C4D90812yes—
130C22×C4C22×D22711yescharacteristic
131Q16—4548no—
132SD16—4548no—
133M16—22no—
134M16D90814yescomplemented
135C2×C8—22no—
136C4⋊C4—22no—
137C4⋊C4—4542no—
138C22⋊C4—4544no—
139C42—4542no—
140C23—4542no—
141C23C2×D90811yescharacteristic
142Q8—9082no—
143Q8—4544no—
144Q8—4542no—
145Q8—4542no—
146D4—18162no—
147D4—9082no—
148D4—9082no—
149D4—4542no—
150D4—4542no—
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—9081no—
154C2×C4—9081no—
155C2×C4—9081no—
156C2×C4—9081no—
157C2×C4—9082no—
158C2×C4—9082no—
159C2×C4—4542no—
160C2×C4—4542no—
161C2×C4—4542no—
162C2×C4D4×D22711yescharacteristic
163C2×C4D4×D22712yes—
164C2×C4C2×D90812yes—
165C2×C4#1681388511yescharacteristic
166C8—24no—
167C22—21no—
168C22—21no—
169C22—9082no—
170C22—9082no—
171C22—9082no—
172C22#2030025811yescharacteristic
173C22#2030026311yescharacteristic
174C22#2030028511yescharacteristic
175C4—42no—
176C4—9082no—
177C4—9082no—
178C4—9082no—
179C4#2030025812yes—
180C4#2030026312yes—
181C2—21no—
182C2—21no—
183C2—18162no—
184C2#2840894211yescharacteristic
185C2#2840921211yescharacteristic
186C2#2840921711yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#343×C227C2
socleC22×C227#120
prime radical#28409613C2
hypercenter#343D227
hypocenterC227#2007
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
1. C2×#20300321C2semidirect
3. C2×#20300214C2semidirect
4. #28408937C2semidirect
5. #343×C227C2semidirect
28. C2×Dic454D4semidirect
43. M16×C227D4semidirect
93. C227#2007semidirect
99. #343D227semidirect
134. M16D908semidirect

Maximal subgroups

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

subgroupindex
C2×#203003212
#284096132
C2×#203002142
#284089372
#343×C2272
#2007227