Finite Group Databasebeta

← #44401421

Subgroups — #44401421

Order 29056.

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

normal subgroups only

This group has 230 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#28409606C211yesmaximal, characteristic
2C2×#20300207C211yesmaximal, characteristic, complemented
3#28408830C212yesmaximal, complemented
4#28408828C212yesmaximal, complemented
5C2×#101×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C411yescharacteristic
7C2×#16813868—21no—
8#20300286C411yescharacteristic
9#20300286C2211yescharacteristic
10#20300277C411yescharacteristic
11#20300277C2211yescharacteristic
12#20300271—21no—
13#20300271—21no—
14#20300264—21no—
15#20300264—21no—
16#20300247C411yescharacteristic
17#20300207—21no—
18#20300207C2212yes—
19C22×Q8×C227C2211yescharacteristic
20C2×M16×C227—21no—
21#101×C227—21no—
22#101×C227C2212yes—
23#16813884—41no—
24#16813884—41no—
25C2×Dic454C2×C411yescharacteristic
26C2×D908C2×C411yescharacteristic
27C2×C22⋊D227—21no—
28C2×C4×D227—21no—
29#16813868—22no—
30#16813868—41no—
31C2×C227⋊C8—41no—
32C4⋊Dic227—21no—
33Dic227⋊C4—21no—
34Dic227⋊C4—21no—
35Dic227⋊C4—21no—
36Dic227⋊C4—21no—
37C22⋊Dic227—21no—
38(C2×D227)⋊C4—21no—
39(C2×D227)⋊C4—21no—
40(C2×D227)⋊C4—21no—
41(C2×D227)⋊C4—21no—
42C4×Dic227C2×C411yescharacteristic
43C4×Dic227C2×C411yescharacteristic
44C2×Q8×C227—41no—
45C2×Q8×C227—41no—
46C2×Q8×C227C2×C411yescharacteristic
47C2×Q8×C227D411yescharacteristic
48C2×Q8×C227D411yescharacteristic
49C2×Q8×C227C2311yescharacteristic
50C22×C4×C227C2×C411yescharacteristic
51C22×C4×C227D411yescharacteristic
52C22×C4×C227D411yescharacteristic
53M16×C227—22no—
54M16×C227—41no—
55C2×C8×C227—41no—
56C22×D227—21no—
57Dic454—41no—
58D908—41no—
59C22⋊D227—41no—
60C22⋊D227—41no—
61C2×Dic227—21no—
62C2×Dic227—21no—
63C2×Dic227—21no—
64C4×D227—42no—
65C227⋊C8—42no—
66C23×C227C22⋊C411yescharacteristic
67Q8×C227—41no—
68Q8×C227—41no—
69Q8×C227—41no—
70Q8×C227—41no—
71C2×C4×C227—21no—
72C2×C4×C227—21no—
73C2×C4×C227—21no—
74C2×C4×C227—21no—
75C2×C4×C227—21no—
76C2×C4×C227—21no—
77C2×C4×C227C22⋊C411yescharacteristic
78C2×C4×C227C22⋊C411yescharacteristic
79C2×C4×C227C22⋊C411yescharacteristic
80C2×C4×C227C22×C411yescharacteristic
81C2×C4×C227C2×D411yescharacteristic
82C2×C4×C227C2×D411yescharacteristic
83C8×C227—42no—
84C22×C227—41no—
85C22×C227#9911yescharacteristic
86C22×C227#9911yescharacteristic
87C22×C227C2×C22⋊C411yescharacteristic
88C2×D227—42no—
89C2×D227—41no—
90C4×C227—21no—
91C4×C227—21no—
92C4×C227—21no—
93C4×C227—21no—
94C4×C227—21no—
95C4×C227—21no—
96Dic227—41no—
97Dic227—41no—
98Dic227—41no—
99C2×C227—21no—
100C2×C227—21no—
101C2×C227—21no—
102C2×C227C2×#9911yescharacteristic
103D227—81no—
104C227#213311yescharacteristic, complemented, Sylow
105#2133—2271nomaximal, Sylow
106#536—2271no—
107C2×#101—2271no—
108C2×#101D22711yescharacteristic, complemented
109#356—2272no—
110#355—2272no—
111C2×#41—2271no—
112C22×Q8C2×D22711yescharacteristic
113C2×M16—21no—
114C2×M16—4541no—
115C4⋊Q8—2271no—
116C4⋊Q8—2271no—
117#124—2271no—
118#124—2271no—
119#123—4541no—
120#123—4541no—
121C22⋊Q8—4541no—
122C22⋊Q8—4541no—
123#117—2271no—
124#101—21no—
125#101—4541no—
126#101—2272no—
127#101C2×D22712yes—
128#41—9081no—
129#41—9081no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—2271no—
133C2×Q8C4×D22711yescharacteristic
134C2×Q8C22⋊D22711yescharacteristic
135C2×Q8D90811yescharacteristic
136C2×Q8C22×D22711yescharacteristic
137C2×D4—4541no—
138C2×D4—2271no—
139C22×C4—4541no—
140C22×C4C4×D22711yescharacteristic
141C22×C4C22⋊D22711yescharacteristic
142C22×C4D90811yescharacteristic
143M16—22no—
144M16—41no—
145M16—9081no—
146M16—4542no—
147C2×C8—41no—
148C2×C8—9081no—
149C4⋊C4—4541no—
150C4⋊C4—4541no—
151C4⋊C4—4541no—
152C4⋊C4—4541no—
153C4⋊C4—4541no—
154C22⋊C4—4541no—
155C22⋊C4—4541no—
156C22⋊C4—4541no—
157C22⋊C4—4541no—
158C22⋊C4—4541no—
159C42—2271no—
160C42—2271no—
161C23—4541no—
162C23(C2×D227)⋊C411yescharacteristic
163Q8—41no—
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—9081no—
168D4—9081no—
169D4—9081no—
170D4—9081no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—9082no—
178C2×C4—4541no—
179C2×C4—4541no—
180C2×C4—4541no—
181C2×C4(C2×D227)⋊C411yescharacteristic
182C2×C4(C2×D227)⋊C411yescharacteristic
183C2×C4(C2×D227)⋊C411yescharacteristic
184C2×C4C2×C4×D22711yescharacteristic
185C2×C4C2×C22⋊D22711yescharacteristic
186C2×C4C2×D90811yescharacteristic
187C8—42no—
188C8—9082no—
189C22—41no—
190C22—9082no—
191C22—9081no—
192C22#2030020211yescharacteristic
193C22#2030020211yescharacteristic
194C22C2×(C2×D227)⋊C411yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—9081no—
202C4—9081no—
203C4—9081no—
204C2—21no—
205C2—21no—
206C2—21no—
207C2—18161no—
208C2C2×#2030020211yescharacteristic
subgroupG/H
centerC2C2×#20300202
commutatorC2×C4×C227C22×C4
FrattiniC2×Q8C22×D227
FittingC2×#101×C227C2
socleC2×C227C2×#99
prime radicalC2×#16813884C4
cosocleC2×Q8×C227C23
hypercenterC2×#101D227
hypocenterC227#2133
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#20300207C2semidirect
3. #28408830C2semidirect
4. #28408828C2semidirect
5. C2×#101×C227C2semidirect
104. C227#2133semidirect
108. C2×#101D227semidirect

Maximal subgroups

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

subgroupindex
#284096062
C2×#203002072
#284088302
#284088282
C2×#101×C2272
#2133227