Finite Group Databasebeta

← #36354690

Subgroups — #36354690

Order 22144.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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#26012053C211yesmaximal, characteristic, complemented
2C2×#19149649C212yesmaximal, complemented
3C2×#19149593C212yesmaximal, complemented
4#26011346C211yesmaximal, characteristic, complemented
5#342×C173C211yesmaximal, characteristic, complemented
6C24×D173C2211yescharacteristic
7C22×D692—22no—
8C22×C22⋊D173C2212yes—
9#19149663—22no—
10#19149653—22no—
11#19149649—24no—
12#19149649—24no—
13#19149649—22no—
14C2×C22⋊Dic173C2211yescharacteristic
15C2×#14583803—22no—
16C2×#14583803—22no—
17#19149593—24no—
18C2×C22⋊C4×C173C2211yescharacteristic
19C2×C22⋊C4×C173C2212yes—
20C23×D173—21no—
21C23×D173—22no—
22C23×D173—22no—
23C23×D173—21no—
24C23×D173—44no—
25C23×D173D412yescomplemented
26C23×D173D412yescomplemented
27C2×D692—28no—
28C2×D692—22no—
29C2×D692—48no—
30C2×C22⋊D173—22no—
31C2×C22⋊D173—22no—
32C2×C22⋊D173—22no—
33C2×C22⋊D173—22no—
34C2×C22⋊D173—22no—
35C2×C22⋊D173—22no—
36C22×Dic173—21no—
37C22×Dic173D412yescomplemented
38C4⋊Dic173—22no—
39C22⋊Dic173—22no—
40#14583803—28no—
41#14583803—24no—
42#14583803—44no—
43C24×C173C2311yescharacteristic
44C22×C4×C173—22no—
45C22×C4×C173D412yescomplemented
46C22⋊C4×C173—22no—
47C22⋊C4×C173—22no—
48C22⋊C4×C173D414yescomplemented
49C22×D173—28no—
50C22×D173—24no—
51C22×D173—24no—
52C22×D173—22no—
53C22×D173—22no—
54C22×D173—22no—
55C22×D173—22no—
56C22×D173—44no—
57C22×D173—48no—
58C22×D173—44no—
59C22×D173—42no—
60C22×D173—44no—
61C22×D173—41no—
62C22×D173—42no—
63C22×D173—41no—
64C22×D173—44no—
65D692—416no—
66D692—82no—
67C22⋊D173—42no—
68C22⋊D173—42no—
69C22⋊D173—42no—
70C22⋊D173—42no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—42no—
75C2×Dic173—41no—
76C23×C173—42no—
77C23×C173C2×D412yes—
78C23×C173C2×D411yescharacteristic
79C23×C173C2×D411yescharacteristic
80C23×C173C2×D412yes—
81C23×C173#4111yescharacteristic
82C2×C4×C173—24no—
83C2×C4×C173—22no—
84C2×C4×C173—44no—
85C2×C4×C173—42no—
86C22×C173—22no—
87C22×C173—24no—
88C22×C173—42no—
89C22×C173—41no—
90C22×C173—41no—
91C22×C173#12012yes—
92C22×C173#12011yescharacteristic
93C22×C173#12112yes—
94C22×C173#12111yescharacteristic
95C22×C173#12711yescharacteristic
96C2×D173—28no—
97C2×D173—24no—
98C2×D173—44no—
99C2×D173—44no—
100C2×D173—42no—
101C2×D173—48no—
102C2×D173—42no—
103C2×D173—44no—
104C2×D173—44no—
105C2×D173—42no—
106C2×D173—44no—
107C2×D173—82no—
108C4×C173—44no—
109C4×C173—42no—
110Dic173—42no—
111Dic173—81no—
112C2×C173—21no—
113C2×C173—22no—
114C2×C173—21no—
115C2×C173—22no—
116C2×C173#39211yescharacteristic
117C2×C173B(4,𝔽2)12yes—
118D173—44no—
119D173—82no—
120C173#202111yescharacteristic, complemented, Sylow
121#2021—1731nomaximal, Sylow
122#534—1731no—
123C2×#120—1732no—
124C2×#99—1732no—
125#379—1731no—
126#342D17311yescharacteristic, complemented
127C25—1731no—
128C22×D4—3462no—
129C22×D4—1732no—
130#123—3462no—
131#121—3462no—
132#120—3464no—
133#120—3462no—
134#120—3464no—
135C2×C22⋊C4—3462no—
136C2×C22⋊C4—3462no—
137C2×C22⋊C4—1731no—
138C2×C22⋊C4C2×D17311yescharacteristic
139C2×C22⋊C4C2×D17312yes—
140#99—3464no—
141C24—3462no—
142C24—3461no—
143C24—3462no—
144C24—3461no—
145C24—1732no—
146C24—1732no—
147C24—6924no—
148C24C22×D17311yescharacteristic
149C2×D4—3462no—
150C2×D4—3468no—
151C2×D4—3462no—
152C2×D4—3462no—
153C2×D4—3462no—
154C2×D4—3462no—
155C2×D4—3462no—
156C2×D4—3462no—
157C2×D4—6928no—
158C22×C4—22no—
159C22×C4—3461no—
160C22×C4—1732no—
161C22×C4D69212yescomplemented
162C4⋊C4—3462no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3468no—
166C22⋊C4—3464no—
167C22⋊C4—3462no—
168C22⋊C4—6924no—
169C22⋊C4D69214yescomplemented
170C23—42no—
171C23—3464no—
172C23—3468no—
173C23—3464no—
174C23—3462no—
175C23—3462no—
176C23—3462no—
177C23—3462no—
178C23—6924no—
179C23—6922no—
180C23—6924no—
181C23—6924no—
182C23—6928no—
183C23—6924no—
184C23—6921no—
185C23—6922no—
186C23—6921no—
187C23D4×D17312yes—
188C23D4×D17311yescharacteristic
189C23C2×D69211yescharacteristic
190C23C2×D69212yes—
191C23#1458382711yescharacteristic
192D4—13842no—
193D4—69216no—
194D4—6922no—
195D4—6922no—
196D4—6922no—
197D4—6922no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—6922no—
206C2×C4—6921no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3468no—
213C22—3464no—
214C22—13842no—
215C22—6924no—
216C22—6924no—
217C22—6924no—
218C22—6922no—
219C22—6928no—
220C22—6922no—
221C22—6924no—
222C22—6922no—
223C22—6924no—
224C22#1914964912yes—
225C22#1914964911yescharacteristic
226C22#1914965412yes—
227C22#1914965411yescharacteristic
228C22#1914967611yescharacteristic
229C4—44no—
230C4—42no—
231C4—13841no—
232C4—6922no—
233C2—22no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—13842no—
238C2—6924no—
239C2#2601140111yescharacteristic
240C2#2601167912yes—
subgroupG/H
centerC22#19149649
commutator, cosocleC24×C173C23
FrattiniC24C22×D173
Fitting#342×C173C2
socleC22×C173#120
hypercenter#342D173
hypocenterC173#2021
LCSG ▷ C24×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #26012053C2semidirect
2. C2×#19149649C2semidirect
3. C2×#19149593C2semidirect
4. #26011346C2semidirect
5. #342×C173C2semidirect
25. C23×D173D4semidirect
26. C23×D173D4semidirect
37. C22×Dic173D4semidirect
45. C22×C4×C173D4semidirect
48. C22⋊C4×C173D4semidirect
120. C173#2021semidirect
126. #342D173semidirect
161. C22×C4D692semidirect
169. C22⋊C4D692semidirect

Maximal subgroups

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

subgroupindex
#260120532
C2×#191496492
C2×#191495932
#260113462
#342×C1732
#2021173