Finite Group Databasebeta

← #36362970

Subgroups — #36362970

Order 22144.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#26012327C211yesmaximal, characteristic
2#26012067C212yesmaximal, complemented
3#26012062C211yesmaximal, characteristic, complemented
4#26011682C218yesmaximal, complemented
5#26011464C212yesmaximal, complemented
6#410×C173C211yesmaximal, characteristic, complemented
7#19149727—21no—
8#19149726—22no—
9#19149725—22no—
10#19149724—21no—
11#19149724C2214yes—
12C2×#14583826—22no—
13C2×#14583826C2212yes—
14#41×D173—24no—
15C22×Dic346C2211yescharacteristic, complemented
16#19149679—22no—
17#19149677—21no—
18#19149667—22no—
19#19149666—21no—
20#19149663C2218yescomplemented
21#19149660C2214yescomplemented
22#19149659—24no—
23#19149658—22no—
24#19149638C2211yescharacteristic, complemented
25#19149593C2218yescomplemented
26C2×#41×C173C2211yescharacteristic
27#117×C173C2212yescomplemented
28#99×C173C2214yescomplemented
29#14583827—24no—
30#14583826—24no—
31#14583826—24no—
32#14583826—22no—
33#14583826—22no—
34#14583826—24no—
35#14583826—22no—
36#14583826—22no—
37#14583825—24no—
38#14583825—28no—
39C2×Dic346—22no—
40C2×Dic346—21no—
41C2×Dic346—22no—
42C2×Dic346—21no—
43C2×Dic346—22no—
44C2×Dic346D412yes—
45Q8×D173—24no—
46C2×D692—22no—
47C2×D692D412yes—
48C2×C22⋊D173—24no—
49C2×C22⋊D173C2314yes—
50D4×D173—24no—
51D4×D173—28no—
52C22×Dic173C2312yes—
53C2×C4×D173—24no—
54C2×C4×D173D414yes—
55C4⋊Dic173—28no—
56C4⋊Dic173—21no—
57#14583805—24no—
58C22⋊Dic173—24no—
59C22⋊Dic173C2312yes—
60#14583803—28no—
61C4×Dic173—21no—
62#41×C173—22no—
63#41×C173—22no—
64C2×Q8×C173D411yescharacteristic
65C2×D4×C173D411yescharacteristic
66C2×D4×C173C2312yes—
67C22×C4×C173D412yes—
68C22×C4×C173C2311yescharacteristic
69C4⋊C4×C173—22no—
70C22⋊C4×C173C2314yes—
71C42×C173—22no—
72C22×D173—24no—
73C22×D173C2×D414yes—
74Dic346—24no—
75Dic346—22no—
76Dic346—22no—
77Dic346—41no—
78Dic346—41no—
79Dic346—42no—
80D692—24no—
81D692—22no—
82D692—22no—
83C22⋊D173—28no—
84C22⋊D173—24no—
85C22⋊D173—24no—
86C2×Dic173—24no—
87C2×Dic173—24no—
88C2×Dic173—22no—
89C2×Dic173—22no—
90C2×Dic173C2×D414yes—
91C4×D173—28no—
92C4×D173—24no—
93C4×D173—24no—
94C23×C173C2×D412yes—
95C23×C173C2411yescharacteristic
96Q8×C173—22no—
97D4×C173—24no—
98D4×C173—22no—
99C2×C4×C173—22no—
100C2×C4×C173—22no—
101C2×C4×C173—24no—
102C2×C4×C173C2×D412yes—
103C2×C4×C173C2×D412yes—
104C2×C4×C173C2×D412yes—
105C2×C4×C173C2×D411yescharacteristic
106C2×C4×C173C2×D411yescharacteristic
107C22×C173—22no—
108C22×C173—41no—
109C22×C173—42no—
110C22×C173C22×D412yes—
111C22×C173C22×D411yescharacteristic
112C2×D173—24no—
113C2×D173—44no—
114C2×D173—84no—
115C4×C173—22no—
116C4×C173—44no—
117C4×C173#12011yescharacteristic
118C4×C173#12011yescharacteristic
119C4×C173#12012yes—
120Dic173—24no—
121Dic173—42no—
122Dic173—42no—
123C2×C173—21no—
124C2×C173—22no—
125C2×C173—42no—
126C2×C173C2×#12011yescharacteristic
127D173—44no—
128C173#302811yescharacteristic, complemented, Sylow
129#3028—1731nomaximal, Sylow
130#585—1731no—
131#536—1731no—
132#536—1732no—
133#458—1738no—
134#410—1732no—
135#410D17311yescharacteristic, complemented
1362−1+4—3462no—
1372−1+4—3461no—
138H(5,𝔽2)—3462no—
139H(5,𝔽2)—3461no—
140H(5,𝔽2)—1734no—
141C2×#41—3464no—
142C2×#41—3462no—
143C2×#41—1732no—
144C2×#41C2×D17311yescharacteristic
145C22×Q8—1731no—
146C4⋊Q8—3462no—
147C4⋊Q8—3461no—
148#124—3462no—
149#124—3461no—
150#123—1734no—
151#123—1738no—
152C22⋊Q8—3464no—
153C22⋊Q8—3462no—
154#117—1731no—
155#117C2×D17312yescomplemented
156#99—1738no—
157#99C2×D17314yescomplemented
158#41—22no—
159#41—22no—
160#41—3464no—
161#41—3464no—
162#41—3464no—
163#41—3464no—
164#41—3468no—
165#41—3462no—
166#41—3462no—
167#41—3462no—
168#41—3464no—
169#41—3462no—
170C2×Q8—3462no—
171C2×Q8—3464no—
172C2×Q8—3462no—
173C2×Q8—3462no—
174C2×Q8—3461no—
175C2×Q8—3461no—
176C2×Q8—1732no—
177C2×Q8D69211yescharacteristic
178C2×D4—3462no—
179C2×D4—3464no—
180C2×D4—3468no—
181C2×D4—3464no—
182C2×D4—1734no—
183C2×D4—1732no—
184C2×D4D69211yescharacteristic
185C2×D4C22×D17312yes—
186C22×C4—3464no—
187C22×C4—1732no—
188C22×C4—1734no—
189C22×C4D69212yes—
190C22×C4C22×D17311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3468no—
193C4⋊C4—3461no—
194C4⋊C4—3464no—
195C22⋊C4—3468no—
196C22⋊C4—3464no—
197C22⋊C4—1732no—
198C22⋊C4C22×D17314yes—
199C42—22no—
200C42—3461no—
201C23—3464no—
202C23—1734no—
203C23C2×D69212yes—
204C23C23×D17311yescharacteristic
205Q8—22no—
206Q8—3464no—
207Q8—3462no—
208Q8—3462no—
209Q8—6921no—
210Q8—6921no—
211Q8—6922no—
212D4—24no—
213D4—22no—
214D4—3464no—
215D4—3468no—
216D4—3464no—
217D4—3464no—
218D4—3462no—
219D4—3462no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—3468no—
224C2×C4—3464no—
225C2×C4—3464no—
226C2×C4—3464no—
227C2×C4—3464no—
228C2×C4—3462no—
229C2×C4—3462no—
230C2×C4—1734no—
231C2×C4D4×D17312yes—
232C2×C4D4×D17312yes—
233C2×C4C2×D69212yes—
234C2×C4C2×D69211yescharacteristic
235C2×C4C2×D69211yescharacteristic
236C22—22no—
237C22—41no—
238C22—42no—
239C22—3464no—
240C22—13844no—
241C22—6924no—
242C22C2×D4×D17312yes—
243C22C22×D69211yescharacteristic
244C4—22no—
245C4—44no—
246C4—3464no—
247C4—6922no—
248C4—6922no—
249C4#1914964911yescharacteristic
250C4#1914964911yescharacteristic
251C4#1914964912yes—
252C2—22no—
253C2—21no—
254C2—42no—
255C2—6924no—
256C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC23×C173C24
FrattiniC23C23×D173
Fitting#410×C173C2
socleC2×C173C2×#120
prime radical#26012327C2
hypercenter#410D173
hypocenterC173#3028
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #26012067C2semidirect
3. #26012062C2semidirect
4. #26011682C2semidirect
5. #26011464C2semidirect
6. #410×C173C2semidirect
15. C22×Dic346C22semidirect
20. #19149663C22semidirect
21. #19149660C22semidirect
24. #19149638C22semidirect
25. #19149593C22semidirect
27. #117×C173C22semidirect
28. #99×C173C22semidirect
128. C173#3028semidirect
135. #410D173semidirect
155. #117C2×D173semidirect
157. #99C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260120672
#260120622
#260116822
#260114642
#410×C1732
#3028173