Finite Group Databasebeta

← #36355553

Subgroups — #36355553

Order 22144.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#26011680C211yesmaximal, characteristic, complemented
2#26011677C211yesmaximal, characteristic
3#26011668C211yesmaximal, characteristic, complemented
4#26011286C211yesmaximal, characteristic, complemented
5#26011281C211yesmaximal, characteristic, complemented
6#26011278C211yesmaximal, characteristic, complemented
7#551×C173C211yesmaximal, characteristic, complemented
8#19149724C2211yescharacteristic
9#19149710—21no—
10#19149662—21no—
11#19149648—21no—
12#19149607—21no—
13#19149596C2211yescharacteristic
14#19149594C2211yescharacteristic
15#19149594C2211yescharacteristic
16#19149593—21no—
17#19149593—21no—
18#126×C173—21no—
19#124×C173C2211yescharacteristic
20#123×C173—21no—
21#123×C173C2211yescharacteristic
22C22⋊Q8×C173—21no—
23#120×C173C2211yescharacteristic
24C4×D4×C173—21no—
25#117×C173—21no—
26C2×C22⋊C4×C173—21no—
27#14583826—21no—
28#14583825—41no—
29C2×C22⋊D173—21no—
30C2×C22⋊D173—21no—
31C2×C22⋊D173—41no—
32D4×D173—41no—
33#14583816—41no—
34#14583813—41no—
35#14583810—21no—
36#14583805—41no—
37C22⋊Dic173—21no—
38C22⋊Dic173—21no—
39C22⋊Dic173—41no—
40#14583803—41no—
41C24×C173D411yescharacteristic
42C2×Q8×C173D411yescharacteristic, complemented
43C2×D4×C173—21no—
44C2×D4×C173C2311yescharacteristic
45C22×C4×C173—21no—
46C22×C4×C173—21no—
47C22×C4×C173D411yescharacteristic, complemented
48C4⋊C4×C173—21no—
49C4⋊C4×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173—21no—
52C22⋊C4×C173—21no—
53C22⋊C4×C173—21no—
54C22⋊C4×C173—21no—
55C22⋊C4×C173—21no—
56C22⋊C4×C173—21no—
57C22⋊C4×C173—21no—
58C22⋊C4×C173D411yescharacteristic
59C22⋊C4×C173D411yescharacteristic, complemented
60C42×C173—21no—
61C42×C173D411yescharacteristic, complemented
62C22×D173—21no—
63C22×D173—21no—
64Dic346—21no—
65D692—21no—
66C22⋊D173—41no—
67C22⋊D173—41no—
68C22⋊D173—41no—
69C22⋊D173—82no—
70C2×Dic173—21no—
71C2×Dic173—21no—
72C2×Dic173—41no—
73C2×Dic173—41no—
74C4×D173—41no—
75#13063436—41no—
76C23×C173—21no—
77C23×C173—41no—
78C23×C173—42no—
79C23×C173C2×D411yescharacteristic
80C23×C173C2×D411yescharacteristic
81Q8×C173—41no—
82D4×C173—41no—
83D4×C173—42no—
84C2×C4×C173—21no—
85C2×C4×C173—21no—
86C2×C4×C173—21no—
87C2×C4×C173—21no—
88C2×C4×C173—21no—
89C2×C4×C173—42no—
90C2×C4×C173—41no—
91C2×C4×C173—81no—
92C2×C4×C173C2×D411yescharacteristic
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—41no—
109C4×C173—41no—
110C4×C173—41no—
111C4×C173—41no—
112Dic173—41no—
113Dic173—81no—
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#551D17311yescharacteristic, complemented
124#458—1731no—
125B(4,𝔽2)—1731no—
126#455—1731no—
127#354—1731no—
128#352—1731no—
129AΣL(1,𝔽16)—1731no—
130H(5,𝔽2)—1731no—
131#137—3461no—
132#126—21no—
133#124C2×D17311yescharacteristic
134#123—21no—
135#123—3461no—
136#123C2×D17311yescharacteristic
137C22⋊Q8—21no—
138#120—3461no—
139#120C2×D17311yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—3461no—
144#100—1731no—
145#99—3461no—
146#99—3461no—
147#99—1731no—
148#99—1731no—
149C24C22⋊D17311yescharacteristic
150#41—3461no—
151#41—6921no—
152C2×Q8C22⋊D17311yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—3461no—
155C2×D4—3461no—
156C2×D4—6921no—
157C2×D4—6921no—
158C2×D4C22×D17311yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D17311yescharacteristic, complemented
162Q16—6921no—
163SD16—6921no—
164M16—3461no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—6921no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—3461no—
177C22⋊C4—3461no—
178C22⋊C4—6921no—
179C22⋊C4—6921no—
180C22⋊C4C22⋊D17311yescharacteristic
181C22⋊C4C22⋊D17311yescharacteristic, complemented
182C42—21no—
183C42C22⋊D17311yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—3461no—
188C23—3461no—
189C23C2×C22⋊D17311yescharacteristic
190C23C2×C22⋊D17311yescharacteristic
191Q8—41no—
192Q8—3461no—
193D4—41no—
194D4—42no—
195D4—3461no—
196D4—13842no—
197D4—6921no—
198D4—6921no—
199D4—6921no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—3461no—
209C2×C4—3461no—
210C2×C4—6921no—
211C2×C4—6921no—
212C2×C4—6921no—
213C2×C4C2×C22⋊D17311yescharacteristic
214C8—6921no—
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#1914964811yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—13841no—
235C4—13841no—
236C4—6921no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—13841no—
242C2#2601167811yescharacteristic
subgroupG/H
centerC2#26011678
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#551×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011677C2
hypercenter#551D173
hypocenterC173#2197
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #26011680C2semidirect
3. #26011668C2semidirect
4. #26011286C2semidirect
5. #26011281C2semidirect
6. #26011278C2semidirect
7. #551×C173C2semidirect
42. C2×Q8×C173D4semidirect
47. C22×C4×C173D4semidirect
59. C22⋊C4×C173D4semidirect
61. C42×C173D4semidirect
121. C173#2197semidirect
123. #551D173semidirect
152. C2×Q8C22⋊D173semidirect
161. C22×C4C22⋊D173semidirect
181. C22⋊C4C22⋊D173semidirect
183. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260116802
#260116772
#260116682
#260112862
#260112812
#260112782
#551×C1732
#2197173