Finite Group Databasebeta

← #353×D227 #44401390

Subgroups — #353×D227 #44401390

Order 29056.

368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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#127×D227C211yesmaximal, characteristic
2#99×D227C212yesmaximal, complemented
3#28408825C211yesmaximal, characteristic, complemented
4#28408824C212yesmaximal, complemented
5#353×C227C211yesmaximal, characteristic, complemented
6C2×D4×D227—22no—
7C2×D4×D227C411yescharacteristic, complemented
8C2×D4×D227C2211yescharacteristic
9#20300285C411yescharacteristic, complemented
10#20300284—22no—
11#20300284C411yescharacteristic, complemented
12#20300284C2211yescharacteristic
13C22⋊C4×D227—22no—
14C42×D227C411yescharacteristic, complemented
15#20300203C2212yescomplemented
16#20300202—22no—
17#127×C227C2211yescharacteristic
18#99×C227C2212yescomplemented
19C23×D227—21no—
20C23×D227D412yes—
21C2×D908—22no—
22C2×D908C2×C411yescharacteristic
23C2×D908C2×C411yescharacteristic, complemented
24C2×C22⋊D227—22no—
25C2×C22⋊D227—21no—
26C2×C22⋊D227—22no—
27C2×C22⋊D227D412yes—
28D4×D227—41no—
29D4×D227—41no—
30D4×D227—41no—
31D4×D227—41no—
32D4×D227—48no—
33C2×C4×D227—21no—
34C2×C4×D227—42no—
35C2×C4×D227C2×C411yescharacteristic
36C22⋊Dic227—22no—
37(C2×D227)⋊C4—42no—
38C4×Dic227—21no—
39C4×Dic227C2×C411yescharacteristic, complemented
40C2×D4×C227—22no—
41C2×D4×C227C2×C411yescharacteristic, complemented
42C2×D4×C227C2311yescharacteristic
43C22⋊C4×C227—22no—
44C42×C227C2×C411yescharacteristic, complemented
45C22×D227—22no—
46C22×D227—22no—
47C22×D227—21no—
48C22×D227—44no—
49C22×D227—44no—
50C22×D227—41no—
51C22×D227—41no—
52C22×D227C22⋊C411yescharacteristic
53C22×D227C22⋊C412yes—
54D908—41no—
55D908—44no—
56D908—41no—
57C22⋊D227—42no—
58C22⋊D227—44no—
59C22⋊D227—44no—
60C22⋊D227—42no—
61C2×Dic227—21no—
62C2×Dic227—42no—
63C2×Dic227C22⋊C411yescharacteristic
64C4×D227—24no—
65C4×D227—21no—
66C4×D227—21no—
67C4×D227—84no—
68C23×C227—21no—
69C23×C227C2×D412yes—
70D4×C227—41no—
71D4×C227—44no—
72D4×C227—41no—
73C2×C4×C227—21no—
74C2×C4×C227—42no—
75C2×C4×C227C22×C411yescharacteristic
76C22×C227—22no—
77C22×C227—42no—
78C22×C227—42no—
79C22×C227—41no—
80C22×C227C2×C22⋊C411yescharacteristic
81C2×D227—22no—
82C2×D227—22no—
83C2×D227—44no—
84C2×D227—42no—
85C2×D227—42no—
86C2×D227—41no—
87C2×D227—82no—
88C2×D227—82no—
89C2×D227—82no—
90C2×D227#9911yescharacteristic
91C2×D227#9911yescharacteristic
92C4×C227—22no—
93C4×C227—21no—
94C4×C227—82no—
95Dic227—22no—
96Dic227—21no—
97Dic227—82no—
98C2×C227—21no—
99C2×C227—42no—
100C2×C227—81no—
101C2×C227C2×#9911yescharacteristic
102D227—21no—
103D227—42no—
104D227—81no—
105D227#35312yesdirect product factor
106C227C2×#35311yescharacteristic, complemented, Sylow
107C2×#353—2271nomaximal, Sylow
108C2×#127—2271no—
109C2×#99—2272no—
110#353—2271no—
111#353—2272no—
112#353D22711yescharacteristic, direct product factor
113C22×D4—4542no—
114C22×D4—2271no—
115C22×D4—2271no—
116#127—4542no—
117#127—2271no—
118#127—2271no—
119#127—2271no—
120#127C2×D22711yescharacteristic
121C2×C22⋊C4—4542no—
122C2×C42—2271no—
123#99—4542no—
124#99—2272no—
125#99C2×D22712yescomplemented
126C24—4541no—
127C24—2272no—
128C2×D4—22no—
129C2×D4—9081no—
130C2×D4—9081no—
131C2×D4—9088no—
132C2×D4—9081no—
133C2×D4—9081no—
134C2×D4—4542no—
135C2×D4—4542no—
136C2×D4—4542no—
137C2×D4—4541no—
138C2×D4—2272no—
139C2×D4—2271no—
140C2×D4—2271no—
141C2×D4C4×D22711yescharacteristic, complemented
142C2×D4C22×D22711yescharacteristic
143C22×C4—9082no—
144C22×C4—4541no—
145C22×C4—2271no—
146C22⋊C4—22no—
147C22⋊C4—9082no—
148C22⋊C4—4542no—
149C42—4541no—
150C42—2271no—
151C42C4×D22711yescharacteristic, complemented
152C23—21no—
153C23—9084no—
154C23—9081no—
155C23—9081no—
156C23—9084no—
157C23—4542no—
158C23—4542no—
159C23—4541no—
160C23—2272no—
161C23—2271no—
162C23D4×D22712yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—9081no—
167D4—9084no—
168D4—9081no—
169D4—9082no—
170D4—9084no—
171D4—9084no—
172D4—9082no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—18164no—
176C2×C4—9082no—
177C2×C4—4544no—
178C2×C4—4541no—
179C2×C4—4541no—
180C2×C4—4541no—
181C2×C4—2271no—
182C2×C4C2×C4×D22711yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—18162no—
188C22—18162no—
189C22—18162no—
190C22—9082no—
191C22—9082no—
192C22—9084no—
193C22—9081no—
194C22—4542no—
195C22—4542no—
196C22—2271no—
197C22—2271no—
198C22C22⋊C4×D22711yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—18162no—
203C4—4542no—
204C4—4541no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—18161no—
209C2—9082no—
210C2—4541no—
211C2—2272no—
212C2#99×D22711yescharacteristic
subgroupG/H
centerC2#99×D227
commutatorC2×C4×C227C22×C4
FrattiniC2×D4C22×D227
Fitting#353×C227C2
socleC2×C227C2×#99
prime radical#127×D227C2
cosocleC2×D4×C227C23
hypercenter#353D227
hypocenterC227C2×#353
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D227C2semidirect
3. #28408825C2semidirect
4. #28408824C2semidirect
5. #353×C227C2semidirect
7. C2×D4×D227C4semidirect
9. #20300285C4semidirect
11. #20300284C4semidirect
14. C42×D227C4semidirect
15. #20300203C22semidirect
18. #99×C227C22semidirect
23. C2×D908C2×C4semidirect
39. C4×Dic227C2×C4semidirect
41. C2×D4×C227C2×C4semidirect
44. C42×C227C2×C4semidirect
105. D227#353direct (N × H)
106. C227C2×#353semidirect
112. #353D227direct (N × H)
125. #99C2×D227semidirect
141. C2×D4C4×D227semidirect
151. C42C4×D227semidirect

Maximal subgroups

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

subgroupindex
#127×D2272
#99×D2272
#284088252
#284088242
#353×C2272
C2×#353227