Finite Group Databasebeta

← #36355555

Subgroups — #36355555

Order 22144.

258 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012172C211yesmaximal, characteristic, complemented
2#26011682C211yesmaximal, characteristic, complemented
3#26011677C211yesmaximal, characteristic
4#26011671C211yesmaximal, characteristic, complemented
5#26011285C211yesmaximal, characteristic, complemented
6#26011282C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149712—21no—
10#19149672—21no—
11#19149666C2211yescharacteristic
12#19149663—21no—
13#19149660—21no—
14#19149660C2211yescharacteristic
15#19149656—21no—
16#19149648—21no—
17D4×Dic173—21no—
18#19149635—21no—
19C2×C22⋊Dic173—21no—
20#19149605—21no—
21#19149594C2211yescharacteristic
22#19149593—21no—
23#19149593—21no—
24#120×C173C2211yescharacteristic
25#100×C173C2211yescharacteristic
26#99×C173C2211yescharacteristic
27#14583826—21no—
28#14583825—41no—
29C2×Dic346D411yescharacteristic, complemented
30C2×C22⋊D173—21no—
31C2×C22⋊D173—21no—
32C2×C22⋊D173—41no—
33D4×D173—41no—
34C22×Dic173—21no—
35C22×Dic173—21no—
36C22×Dic173D411yescharacteristic, complemented
37Dic692—41no—
38#14583815—41no—
39C4⋊Dic173—21no—
40C4⋊Dic173—41no—
41#14583805—21no—
42#14583805—21no—
43#14583805—21no—
44C22⋊Dic173—21no—
45C22⋊Dic173—21no—
46C22⋊Dic173—21no—
47C22⋊Dic173—21no—
48C22⋊Dic173—21no—
49C22⋊Dic173—21no—
50C22⋊Dic173—41no—
51C22⋊Dic173D411yescharacteristic, complemented
52#14583803—41no—
53C4×Dic173—21no—
54C4×Dic173D411yescharacteristic, complemented
55C24×C173D411yescharacteristic
56C2×D4×C173—21no—
57C2×D4×C173C2311yescharacteristic
58M16×C173—21no—
59C22⋊C4×C173—21no—
60C22⋊C4×C173—21no—
61C22⋊C4×C173D411yescharacteristic
62C22×D173—21no—
63C22×D173—21no—
64Dic346—21no—
65Dic346—41no—
66D692—21no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C22⋊D173—41no—
70C22⋊D173—82no—
71C2×Dic173—21no—
72C2×Dic173—21no—
73C2×Dic173—21no—
74C2×Dic173—21no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—42no—
78C2×Dic173—41no—
79C2×Dic173—41no—
80C2×Dic173—81no—
81C4×D173—41no—
82C23×C173—21no—
83C23×C173—41no—
84C23×C173—42no—
85C23×C173C2×D411yescharacteristic
86C23×C173C2×D411yescharacteristic
87D4×C173—41no—
88D4×C173—42no—
89C2×C4×C173—21no—
90C2×C4×C173—41no—
91C2×C4×C173C2×D411yescharacteristic
92C8×C173—41no—
93C22×C173—22no—
94C22×C173—21no—
95C22×C173—21no—
96C22×C173—41no—
97C22×C173—41no—
98C22×C173—41no—
99C22×C173—41no—
100C22×C173—42no—
101C22×C173—81no—
102C22×C173#12011yescharacteristic
103C2×D173—41no—
104C2×D173—81no—
105C2×D173—81no—
106C4×C173—21no—
107C4×C173—41no—
108C4×C173—81no—
109Dic173—41no—
110Dic173—41no—
111Dic173—41no—
112Dic173—41no—
113Dic173—41no—
114Dic173—81no—
115C2×C173—21no—
116C2×C173—41no—
117C2×C173—41no—
118C2×C173—42no—
119C2×C173B(4,𝔽2)11yescharacteristic
120D173—81no—
121C173#219711yescharacteristic, complemented, Sylow
122#2197—1731nomaximal, Sylow
123#551—1731no—
124#458—1731no—
125B(4,𝔽2)—1731no—
126#455—1731no—
127#354—1731no—
128#352—1731no—
129AΣL(1,𝔽16)D17311yescharacteristic, complemented
130H(5,𝔽2)—1731no—
131#137—3461no—
132#126—3461no—
133#124—1731no—
134#123—3461no—
135#123—3461no—
136#123—1731no—
137C22⋊Q8—3461no—
138#120—3461no—
139#120C2×D17311yescharacteristic
140C4×D4—3461no—
141#117—3461no—
142C2×C22⋊C4—3461no—
143#104—3461no—
144#100C2×D17311yescharacteristic
145#99—3461no—
146#99—3461no—
147#99—1731no—
148#99C2×D17311yescharacteristic
149C24D69211yescharacteristic
150#41—3461no—
151#41—6921no—
152C2×Q8—1731no—
153C2×D4—21no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—6921no—
157C2×D4—6921no—
158C2×D4C22×D17311yescharacteristic
159C22×C4—3461no—
160C22×C4—3461no—
161C22×C4—1731no—
162Q16—6921no—
163SD16—6921no—
164M16—21no—
165C4⋊C4—3461no—
166C4⋊C4—3461no—
167C4⋊C4—3461no—
168C4⋊C4—3461no—
169C4⋊C4—6921no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3461no—
173C22⋊C4—3461no—
174C22⋊C4—3461no—
175C22⋊C4—3461no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—1731no—
179C22⋊C4—6921no—
180C22⋊C4—6921no—
181C22⋊C4D69211yescharacteristic
182C42—3461no—
183C42—1731no—
184C23—21no—
185C23—41no—
186C23—42no—
187C23—3461no—
188C23—3461no—
189C23D4×D17311yescharacteristic
190C23C2×D69211yescharacteristic
191Q8—3461no—
192Q8—6921no—
193D4—41no—
194D4—42no—
195D4—3461no—
196D4—13842no—
197D4—6921no—
198D4—6921no—
199D4—6921no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—3461no—
203C2×C4—3461no—
204C2×C4—3461no—
205C2×C4—3461no—
206C2×C4—3461no—
207C2×C4—3461no—
208C2×C4—13841no—
209C2×C4—6921no—
210C2×C4—6921no—
211C2×C4—6922no—
212C2×C4—6921no—
213C2×C4D4×D17311yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—13841no—
225C22—13841no—
226C22—6921no—
227C22#1914964911yescharacteristic
228C4—21no—
229C4—41no—
230C4—81no—
231C4—13841no—
232C4—6921no—
233C4—6921no—
234C4—6921no—
235C4—6921no—
236C4—6921no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—13841no—
242C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
FittingAΣL(1,𝔽16)×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011677C2
hypercenterAΣL(1,𝔽16)D173
hypocenterC173#2197
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #26012172C2semidirect
2. #26011682C2semidirect
4. #26011671C2semidirect
5. #26011285C2semidirect
6. #26011282C2semidirect
7. AΣL(1,𝔽16)×C173C2semidirect
29. C2×Dic346D4semidirect
36. C22×Dic173D4semidirect
51. C22⋊Dic173D4semidirect
54. C4×Dic173D4semidirect
121. C173#2197semidirect
129. AΣL(1,𝔽16)D173semidirect

Maximal subgroups

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

subgroupindex
#260121722
#260116822
#260116772
#260116712
#260112852
#260112822
AΣL(1,𝔽16)×C1732
#2197173