Finite Group Databasebeta

← #44400604

Subgroups — #44400604

Order 29056.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#16813874C211yesmaximal, characteristic
2C2×#20300219C211yesmaximal, characteristic
3C2×#20300212C211yesmaximal, characteristic
4C2×#20300211C211yesmaximal, characteristic
5C2×#20300199C211yesmaximal, characteristic
6#28408938C211yesmaximal, characteristic
7#391×C227C211yesmaximal, characteristic
8C22×Dic454C2211yescharacteristic
9C2×#16813874—22no—
10C2×#16813874—22no—
11C2×#16813874C414yescomplemented
12C22×C227⋊C8C2211yescharacteristic
13C22×C227⋊C8C2211yescharacteristic
14C2×C4⋊Dic227C2211yescharacteristic
15#20300219—22no—
16#20300212—22no—
17#20300211—22no—
18#20300199—22no—
19C227⋊#95—22no—
20C22×Q8×C227C2211yescharacteristic
21C2×C4⋊C4×C227C2211yescharacteristic
22C2×C42×C227C2211yescharacteristic
23#95×C227—22no—
24C2×Dic454—21no—
25C2×Dic454—21no—
26C2×Dic454—21no—
27C2×Dic454—21no—
28C2×Dic454—21no—
29C2×Dic454C2×C412yes—
30C22×Dic227—21no—
31C22×Dic227—21no—
32#16813874—28no—
33C2×C227⋊C8—21no—
34C2×C227⋊C8—21no—
35C2×C227⋊C8C2×C412yes—
36C2×C227⋊C8D412yes—
37C2×C227⋊C8D412yes—
38C2×C227⋊C8D412yes—
39C4⋊Dic227—21no—
40C4⋊Dic227—21no—
41C2×Q8×C227—21no—
42C2×Q8×C227—21no—
43C2×Q8×C227—21no—
44C2×Q8×C227—21no—
45C2×Q8×C227—21no—
46C2×Q8×C227C2×C412yes—
47C22×C4×C227—21no—
48C22×C4×C227—21no—
49C22×C4×C227D412yes—
50C22×C4×C227C2311yescharacteristic
51C4⋊C4×C227—21no—
52C4⋊C4×C227—21no—
53C42×C227—21no—
54C42×C227—21no—
55Dic454—22no—
56Dic454—22no—
57Dic454—41no—
58Dic454—41no—
59C2×Dic227—22no—
60C2×Dic227—41no—
61C2×Dic227—41no—
62C2×Dic227—41no—
63C2×Dic227—42no—
64C227⋊C8—22no—
65C227⋊C8C22⋊C414yes—
66C23×C227C2×D411yescharacteristic
67Q8×C227—22no—
68Q8×C227—22no—
69Q8×C227—41no—
70Q8×C227—41no—
71C2×C4×C227—22no—
72C2×C4×C227—22no—
73C2×C4×C227—22no—
74C2×C4×C227—22no—
75C2×C4×C227—41no—
76C2×C4×C227—41no—
77C2×C4×C227—41no—
78C2×C4×C227—42no—
79C2×C4×C227C22×C411yescharacteristic
80C2×C4×C227C2×D411yescharacteristic
81C2×C4×C227C2×D411yescharacteristic
82C2×C4×C227C2×D411yescharacteristic
83C2×C4×C227#4111yescharacteristic
84C2×C4×C227#4111yescharacteristic
85C22×C227C4×D411yescharacteristic
86C22×C227#12111yescharacteristic
87C22×C227#12711yescharacteristic
88C22×C227#13712yes—
89C22×C227#13712yes—
90C4×C227—42no—
91C4×C227—42no—
92C4×C227—81no—
93C4×C227C2×C22⋊C411yescharacteristic
94C4×C227C4×D411yescharacteristic
95C4×C227#12111yescharacteristic
96C4×C227#12411yescharacteristic
97Dic227—42no—
98Dic227—81no—
99C2×C227#39011yescharacteristic
100C2×C227#44112yes—
101C2×C227#47012yes—
102C2×C227#49711yescharacteristic
103C2×C227#49711yescharacteristic
104C227#197111yescharacteristic, complemented, Sylow
105#1971—2271nomaximal, Sylow
106C22×Q16—2271no—
107C2×#106—2271no—
108C2×#103—2271no—
109C2×#103—2271no—
110C2×#97—2271no—
111#391—2271no—
112#391D22711yescharacteristic
113C22×Q8—2271no—
114C22×Q8C2×D22711yescharacteristic
115C2×Q16—4542no—
116C2×Q16—4542no—
117C2×Q16—2274no—
118C22×C8—2271no—
119C22×C8—2271no—
120C2×C4⋊C4—2271no—
121C2×C4⋊C4C2×D22711yescharacteristic
122C2×C42C2×D22711yescharacteristic
123#106—4542no—
124#103—4542no—
125#103—4542no—
126#97—4542no—
127#95—22no—
128#95—4542no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—4541no—
135C2×Q8—4541no—
136C2×Q8—4541no—
137C2×Q8—4541no—
138C2×Q8—4541no—
139C2×Q8—2272no—
140C2×Q8C4×D22712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—4541no—
144C22×C4—4541no—
145C22×C4C22⋊D22712yes—
146C22×C4C22×D22711yescharacteristic
147Q16—4548no—
148C2×C8—4541no—
149C2×C8—4541no—
150C2×C8—2272no—
151C2×C8—2272no—
152C2×C8—2272no—
153C2×C8—2272no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—4541no—
157C4⋊C4—4541no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D22711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—9081no—
166Q8—9081no—
167Q8—4542no—
168Q8—4542no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—9082no—
178C2×C4—9081no—
179C2×C4—9081no—
180C2×C4—9081no—
181C2×C4—4542no—
182C2×C4C2×C4×D22711yescharacteristic
183C2×C4D4×D22711yescharacteristic
184C2×C4D4×D22711yescharacteristic
185C2×C4D4×D22711yescharacteristic
186C2×C4#1681388311yescharacteristic
187C2×C4#1681388411yescharacteristic
188C8—4542no—
189C8—2274no—
190C22C4×C22⋊D22711yescharacteristic
191C22#2030026011yescharacteristic
192C22#2030028411yescharacteristic
193C22#2030031812yes—
194C22#2030031912yes—
195C4—42no—
196C4—42no—
197C4—81no—
198C4—18161no—
199C4—9082no—
200C4C22⋊C4×D22711yescharacteristic
201C4(C22⋊D227)⋊C411yescharacteristic
202C4#2030026111yescharacteristic
203C4#2030027411yescharacteristic
204C2#2840893211yescharacteristic
205C2#2840913712yes—
206C2#2840928812yes—
207C2#2840942411yescharacteristic
208C2#2840942711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D227
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#391×C227C2
socle, prime radicalC23×C227C2×D4
cosocleC22×C4×C227C23
hypercenter#391D227
hypocenterC227#1971
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#16813874C4semidirect
104. C227#1971semidirect

Maximal subgroups

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

subgroupindex
C22×#168138742
C2×#203002192
C2×#203002122
C2×#203002112
C2×#203001992
#284089382
#391×C2272
#1971227