Finite Group Databasebeta

← #44401397

Subgroups — #44401397

Order 29056.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#28409578C211yesmaximal, characteristic
2#28409006C211yesmaximal, characteristic, complemented
3#28408826C212yesmaximal, complemented
4#28408824C212yesmaximal, complemented
5#410×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C411yescharacteristic, complemented
7#20300286C411yescharacteristic, complemented
8#20300284C2211yescharacteristic
9#20300277C411yescharacteristic, complemented
10#20300275C2211yescharacteristic
11#20300260—21no—
12#20300260—21no—
13C4×C22⋊D227—21no—
14C4×C22⋊D227—21no—
15#20300247—21no—
16C2×C4×Dic227C411yescharacteristic, complemented
17#20300203—21no—
18#20300203C2212yes—
19C2×#41×C227C2211yescharacteristic
20#117×C227—21no—
21#99×C227—21no—
22#99×C227C2212yes—
23#16813884—41no—
24#16813884—41no—
25C2×Dic454C2×C411yescharacteristic
26C2×D908C2×C411yescharacteristic
27C2×C22⋊D227—21no—
28C2×C22⋊D227—21no—
29C2×C22⋊D227—21no—
30C22×Dic227—21no—
31C2×C4×D227—21no—
32C4⋊Dic227—41no—
33Dic227⋊C4—21no—
34Dic227⋊C4—21no—
35C22⋊Dic227—21no—
36C22⋊Dic227—21no—
37C22⋊Dic227—22no—
38(C2×D227)⋊C4—21no—
39(C2×D227)⋊C4—21no—
40C4×Dic227—21no—
41C4×Dic227—41no—
42C4×Dic227C2×C411yescharacteristic
43C4×Dic227C2×C411yescharacteristic
44#41×C227—41no—
45#41×C227—41no—
46C2×Q8×C227C2×C411yescharacteristic
47C2×D4×C227D411yescharacteristic
48C2×D4×C227D411yescharacteristic
49C2×D4×C227C2311yescharacteristic
50C22×C4×C227C2×C411yescharacteristic
51C22×C4×C227D411yescharacteristic
52C22×C4×C227D411yescharacteristic
53C4⋊C4×C227—41no—
54C22⋊C4×C227—22no—
55C42×C227—41no—
56C22×D227—21no—
57Dic454—41no—
58D908—41no—
59C22⋊D227—42no—
60C22⋊D227—42no—
61C22⋊D227—42no—
62C2×Dic227—21no—
63C2×Dic227—21no—
64C2×Dic227—21no—
65C2×Dic227—42no—
66C2×Dic227—42no—
67C4×D227—41no—
68C4×D227—41no—
69C23×C227C22⋊C411yescharacteristic
70C23×C227C2×D411yescharacteristic
71C23×C227C2×D411yescharacteristic
72Q8×C227—41no—
73D4×C227—41no—
74D4×C227—41no—
75D4×C227—41no—
76C2×C4×C227—21no—
77C2×C4×C227—21no—
78C2×C4×C227—21no—
79C2×C4×C227—21no—
80C2×C4×C227—21no—
81C2×C4×C227—21no—
82C2×C4×C227—42no—
83C2×C4×C227C22⋊C411yescharacteristic
84C2×C4×C227C22⋊C411yescharacteristic
85C2×C4×C227C22⋊C411yescharacteristic
86C2×C4×C227C22×C411yescharacteristic
87C22×C227—21no—
88C22×C227—21no—
89C22×C227—21no—
90C22×C227—41no—
91C22×C227—41no—
92C22×C227—41no—
93C22×C227C2×C22⋊C411yescharacteristic
94C2×D227—41no—
95C2×D227—41no—
96C2×D227—41no—
97C4×C227—21no—
98C4×C227—21no—
99C4×C227—21no—
100C4×C227—82no—
101C4×C227#9911yescharacteristic
102C4×C227#9911yescharacteristic
103Dic227—22no—
104Dic227—41no—
105Dic227—41no—
106Dic227—82no—
107C2×C227—21no—
108C2×C227—41no—
109C2×C227—41no—
110C2×C227—41no—
111C2×C227C2×#9911yescharacteristic
112D227—81no—
113C227#212611yescharacteristic, complemented, Sylow
114#2126—2271nomaximal, Sylow
115#532—2271no—
116#410—2271no—
117#410D22711yescharacteristic, complemented
118#354—2272no—
119#353—2272no—
120C2×#41—2271no—
121C2×#41C2×D22711yescharacteristic
122C4⋊Q8—2271no—
123#127—2271no—
124#124—2271no—
125#124—2271no—
126#121—4541no—
127#121—4541no—
128C4×D4—4541no—
129C4×D4—4541no—
130#117—21no—
131#117—4541no—
132C2×C42—2271no—
133#99—21no—
134#99—4541no—
135#99—2272no—
136#99C2×D22712yes—
137#41—41no—
138#41—41no—
139#41—9081no—
140#41—9081no—
141C2×Q8—2271no—
142C2×Q8C4×D22711yescharacteristic
143C2×D4—4541no—
144C2×D4—4541no—
145C2×D4—4541no—
146C2×D4—2271no—
147C2×D4C22⋊D22711yescharacteristic
148C2×D4D90811yescharacteristic
149C2×D4C22×D22711yescharacteristic
150C22×C4—4541no—
151C22×C4—4541no—
152C22×C4C4×D22711yescharacteristic
153C22×C4C22⋊D22711yescharacteristic
154C22×C4D90811yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—9081no—
157C4⋊C4—4541no—
158C4⋊C4—4541no—
159C22⋊C4—22no—
160C22⋊C4—4542no—
161C22⋊C4—4541no—
162C22⋊C4—4541no—
163C22⋊C4—4541no—
164C22⋊C4—4541no—
165C42—41no—
166C42—9081no—
167C42—4541no—
168C42—2271no—
169C42—2271no—
170C23—4541no—
171C23(C2×D227)⋊C411yescharacteristic
172C23C2×C22⋊D22711yescharacteristic
173C23C2×D90811yescharacteristic
174Q8—41no—
175Q8—9081no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—9081no—
180D4—9082no—
181D4—9082no—
182D4—9082no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—9082no—
191C2×C4—9082no—
192C2×C4—9081no—
193C2×C4—9081no—
194C2×C4—4541no—
195C2×C4—4541no—
196C2×C4—4541no—
197C2×C4(C2×D227)⋊C411yescharacteristic
198C2×C4(C2×D227)⋊C411yescharacteristic
199C2×C4(C2×D227)⋊C411yescharacteristic
200C2×C4C2×C4×D22711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—9081no—
208C22—9081no—
209C22—9081no—
210C22C2×(C2×D227)⋊C411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—18162no—
216C4—9081no—
217C4—9081no—
218C4—4542no—
219C4#2030020211yescharacteristic
220C4#2030020211yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—18161no—
226C2C2×#2030020211yescharacteristic
subgroupG/H
centerC4#20300202
commutatorC2×C4×C227C22×C4
FrattiniC2×D4C22×D227
Fitting#410×C227C2
socleC2×C227C2×#99
prime radical#28409578C2
cosocleC2×D4×C227C23
hypercenter#410D227
hypocenterC227#2126
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #28409006C2semidirect
3. #28408826C2semidirect
4. #28408824C2semidirect
5. #410×C227C2semidirect
6. C2×#16813884C4semidirect
7. #20300286C4semidirect
9. #20300277C4semidirect
16. C2×C4×Dic227C4semidirect
113. C227#2126semidirect
117. #410D227semidirect

Maximal subgroups

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

subgroupindex
#284095782
#284090062
#284088262
#284088242
#410×C2272
#2126227