Finite Group Databasebeta

← #36364522

Subgroups — #36364522

Order 22144.

394 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012046C211yesmaximal, characteristic, complemented
2#26012042C211yesmaximal, characteristic
3#26011870C211yesmaximal, characteristic, complemented
4#26011864C211yesmaximal, characteristic, complemented
5#26011839C211yesmaximal, characteristic, complemented
6#26011716C214yesmaximal, complemented
7#26011656C214yesmaximal, complemented
8#26011542C211yesmaximal, characteristic, complemented
9#487×C173C211yesmaximal, characteristic, complemented
10C2×#14583827C2212yes—
11C2×#14583816C2214yes—
12C2×#14583815C2214yes—
13#19149679C2211yescharacteristic
14#19149676C2211yescharacteristic
15#19149670—21no—
16#19149668C2212yes—
17#19149659C2212yes—
18#19149655—21no—
19#19149654—21no—
20#19149654—21no—
21Q8×Dic173—21no—
22#19149644C2212yes—
23#19149641—21no—
24#19149641—21no—
25#19149637—21no—
26C42×D173C2211yescharacteristic
27#19149608C2212yes—
28#19149599C2214yes—
29#19149591C2212yes—
30C4×#13063436C2211yescharacteristic
31C4⋊Q8×C173C2211yescharacteristic
32C4⋊Q8×C173C2211yescharacteristic
33#103×C173C2214yes—
34C4×C8×C173C2211yescharacteristic
35#14583827—44no—
36C2×Dic346C2312yes—
37C2×D692—22no—
38C2×D692—21no—
39C2×D692—21no—
40C2×D692C2312yes—
41C2×C4×D173—22no—
42C2×C4×D173D411yescharacteristic
43C2×C4×D173D411yescharacteristic
44C2×C4×D173C2311yescharacteristic
45#14583816—44no—
46#14583815—44no—
47C2×#13063436C2312yes—
48C4⋊Dic173—22no—
49C4⋊Dic173—21no—
50C4⋊Dic173—21no—
51#14583805—22no—
52#14583803—22no—
53#14583803—22no—
54#14583803—22no—
55C4×Dic173—22no—
56C4×Dic173D411yescharacteristic
57C4×Dic173D411yescharacteristic
58C4×Dic173C2311yescharacteristic
59C2×Q8×C173—21no—
60C2×Q8×C173C2312yes—
61C2×C8×C173C2312yes—
62C4⋊C4×C173—21no—
63C4⋊C4×C173—21no—
64C4⋊C4×C173—21no—
65C4⋊C4×C173C2312yes—
66C42×C173C2311yescharacteristic
67C22×D173—22no—
68C22×D173C2×D411yescharacteristic
69Dic346—24no—
70D692—24no—
71D692—44no—
72D692—42no—
73D692—42no—
74C2×Dic173—22no—
75C2×Dic173C2×D411yescharacteristic
76C2×Dic173C2×D411yescharacteristic
77C2×Dic173C2×D411yescharacteristic
78C4×D173—22no—
79C4×D173—22no—
80C4×D173—22no—
81C4×D173—44no—
82#13063436—22no—
83Q8×C173—24no—
84Q8×C173—42no—
85C2×C4×C173—22no—
86C2×C4×C173—22no—
87C2×C4×C173C2×D411yescharacteristic
88C2×C4×C173C2×D411yescharacteristic
89C2×C4×C173C2411yescharacteristic
90C8×C173—22no—
91C22×C173C22×D411yescharacteristic
92C2×D173—21no—
93C2×D173—21no—
94C2×D173—21no—
95C2×D173—42no—
96C2×D173—42no—
97C2×D173—42no—
98C4×C173—42no—
99C4×C173—42no—
100C4×C173#13512yes—
101C4×C173#13712yes—
102C4×C173H(5,𝔽2)12yes—
103Dic173—21no—
104Dic173—21no—
105Dic173—21no—
106Dic173—42no—
107C2×C173#53511yescharacteristic
108C2×C173C2×#13511yescharacteristic
109C2×C173C2×#13711yescharacteristic
110D173—41no—
111D173—82no—
112C173#321011yescharacteristic, complemented, Sylow
113#3210—1731nomaximal, Sylow
114#533—1731no—
115#532—1731no—
116#494—1731no—
117#492—1731no—
118#487D17311yescharacteristic, complemented
119#486—1731no—
120#464—1734no—
121#452—1734no—
122#431—1731no—
123C2×#41—1732no—
124C2×Q16—1734no—
125C2×SD16—1734no—
126C4⋊Q8—1731no—
127C4⋊Q8C2×D17311yescharacteristic
128C4⋊Q8C2×D17311yescharacteristic
129#127—1731no—
130#125—3461no—
131#124—1732no—
132C22⋊Q8—3461no—
133C22⋊Q8—1732no—
134#121—3461no—
135#121—3461no—
136C4×Q8—3461no—
137C4×Q8—1732no—
138C4×D4—3461no—
139C4×D4—3461no—
140#117—3461no—
141C2×C42—1731no—
142C4⋊C8—1732no—
143#103C2×D17314yes—
144#102—1734no—
145C22⋊C8—1732no—
146C4×C8—1731no—
147C4×C8C2×D17311yescharacteristic
148#41—6924no—
149C2×Q8—21no—
150C2×Q8—1732no—
151C2×Q8C22×D17312yes—
152C2×D4—3462no—
153C2×D4—3461no—
154C2×D4—3461no—
155C2×D4—1732no—
156C22×C4—3462no—
157C22×C4—1731no—
158C22×C4—1731no—
159C22×C4—1731no—
160Q16—6924no—
161SD16—6924no—
162C2×C8—1732no—
163C2×C8C22×D17312yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—3462no—
168C4⋊C4—3462no—
169C4⋊C4—3461no—
170C4⋊C4—3461no—
171C4⋊C4C22×D17312yes—
172C22⋊C4—3462no—
173C22⋊C4—3462no—
174C22⋊C4—3462no—
175C42—3462no—
176C42—1731no—
177C42—1731no—
178C42—1731no—
179C42C22×D17311yescharacteristic
180C23—3462no—
181C23—1731no—
182Q8—24no—
183Q8—42no—
184Q8—3464no—
185D4—3464no—
186D4—6924no—
187D4—6922no—
188D4—6922no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—3462no—
192C2×C4—3462no—
193C2×C4—3462no—
194C2×C4—3462no—
195C2×C4—1731no—
196C2×C4—1731no—
197C2×C4—1731no—
198C2×C4—6924no—
199C2×C4D4×D17311yescharacteristic
200C2×C4D4×D17311yescharacteristic
201C2×C4C23×D17311yescharacteristic
202C8—22no—
203C8—3462no—
204C22—3461no—
205C22—3461no—
206C22—3461no—
207C22—6922no—
208C22—6922no—
209C22—6922no—
210C22C2×D4×D17311yescharacteristic
211C4—42no—
212C4—42no—
213C4—3461no—
214C4—3461no—
215C4—3461no—
216C4—6922no—
217C4#1914970012yes—
218C4#1914971112yes—
219C4#1914972512yes—
220C2—13842no—
221C2—6921no—
222C2#2601205911yescharacteristic
223C2C2×#1914970011yescharacteristic
224C2C2×#1914971111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#487×C173C2
socleC22×C173C22×D4
prime radical#26012042C2
hypercenter#487D173
hypocenterC173#3210
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #26012046C2semidirect
3. #26011870C2semidirect
4. #26011864C2semidirect
5. #26011839C2semidirect
6. #26011716C2semidirect
7. #26011656C2semidirect
8. #26011542C2semidirect
9. #487×C173C2semidirect
112. C173#3210semidirect
118. #487D173semidirect

Maximal subgroups

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

subgroupindex
#260120462
#260120422
#260118702
#260118642
#260118392
#260117162
#260116562
#260115422
#487×C1732
#3210173