Finite Group Databasebeta

← #36356852

Subgroups — #36356852

Order 22144.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#19149644C211yesmaximal, characteristic, complemented
2C2×#19149641C211yesmaximal, characteristic
3#26011429C211yesmaximal, characteristic, complemented
4#26011420C212yesmaximal, complemented
5#26011409C211yesmaximal, characteristic
6#26011408C211yesmaximal, characteristic
7#26011404C212yesmaximal, complemented
8#26011338C212yesmaximal, complemented
9#26011338C212yesmaximal, complemented
10#26011336C211yesmaximal, characteristic, complemented
11#400×C173C211yesmaximal, characteristic, complemented
12C22×Dic346C2211yescharacteristic
13C22×D692C2211yescharacteristic
14C22×C4×D173C2212yes—
15#19149644—24no—
16#19149641—24no—
17C2×#14583805C2211yescharacteristic
18C2×#14583805C2211yescharacteristic
19C2×#14583803C2212yes—
20C2×#14583803C2212yes—
21C2×#14583803C2212yes—
22C2×#14583803C2212yes—
23C2×#14583803C2212yes—
24C2×C4×Dic173C2211yescharacteristic
25C2×C4×Dic173C2211yescharacteristic
26C2×C4×Dic173C2212yes—
27C2×C4×Dic173C2211yescharacteristic
28#19149586C2212yes—
29#19149586C2212yes—
30#19149586C2211yescharacteristic
31#19149586C2212yes—
32C2×C4⋊C4×C173C2211yescharacteristic
33C2×C4⋊C4×C173C2211yescharacteristic
34#95×C173C2212yes—
35#95×C173C2211yescharacteristic
36#95×C173C2212yes—
37C23×D173C2312yes—
38C2×Dic346—22no—
39C2×Dic346—22no—
40C2×Dic346—22no—
41C2×D692—24no—
42C2×D692—22no—
43C22×Dic173C2312yes—
44C22×Dic173C2311yescharacteristic
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2311yescharacteristic
48C2×C4×D173—24no—
49C2×C4×D173—24no—
50C2×C4×D173—24no—
51#14583805—22no—
52#14583805—22no—
53#14583803—24no—
54#14583803—24no—
55#14583803—24no—
56#14583803—24no—
57#14583803—24no—
58C4×Dic173—22no—
59C4×Dic173—22no—
60C4×Dic173—24no—
61C4×Dic173D414yes—
62C22×C4×C173C2312yes—
63C22×C4×C173C2312yes—
64C22×C4×C173C2311yescharacteristic
65C22×C4×C173C2311yescharacteristic
66C22×C4×C173C2311yescharacteristic
67C4⋊C4×C173—22no—
68C4⋊C4×C173—22no—
69C22×D173—22no—
70C22×D173—22no—
71C22×D173—22no—
72C22×D173—22no—
73C22×D173—22no—
74C22×D173—22no—
75C22×D173#4114yes—
76Dic346—28no—
77D692—28no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—21no—
81C2×Dic173—21no—
82C2×Dic173—21no—
83C2×Dic173—22no—
84C2×Dic173—22no—
85C2×Dic173—22no—
86C2×Dic173—22no—
87C2×Dic173C2×D412yes—
88C2×Dic173C2×D412yes—
89C2×Dic173#4112yes—
90C2×Dic173#4114yes—
91C4×D173—48no—
92C23×C173C2411yescharacteristic
93C2×C4×C173—21no—
94C2×C4×C173—22no—
95C2×C4×C173—21no—
96C2×C4×C173—22no—
97C2×C4×C173—21no—
98C2×C4×C173—22no—
99C2×C4×C173—22no—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173C2×D412yes—
105C2×C4×C173#4112yes—
106C22×C173C22×D411yescharacteristic
107C22×C173C2×#4112yes—
108C22×C173C2×#4111yescharacteristic
109C22×C173C2×#4111yescharacteristic
110C22×C173C2×#4112yes—
111C2×D173—24no—
112C2×D173—24no—
113C2×D173—24no—
114C2×D173—42no—
115C2×D173—42no—
116C2×D173—42no—
117C2×D173—42no—
118C4×C173—22no—
119C4×C173—22no—
120C4×C173—41no—
121C4×C173—42no—
122C4×C173—42no—
123Dic173—22no—
124Dic173—22no—
125Dic173—41no—
126Dic173—42no—
127Dic173#12414yes—
128C2×C173#52511yescharacteristic
129C2×C173#52511yescharacteristic
130C2×C173C2×#12411yescharacteristic
131C2×C173#52912yes—
132C2×C173#52912yes—
133D173—44no—
134C173#239811yescharacteristic, complemented, Sylow
135#2398—1731nomaximal, Sylow
136C2×C4×Q8—1731no—
137C2×C4×D4—1731no—
138#400—1731no—
139#400D17311yescharacteristic, complemented
140#397—1732no—
141#394—1731no—
142#394—1731no—
143#393—1732no—
144C4×C22⋊C4—1732no—
145C4×C22⋊C4—1732no—
146#376—1731no—
147C22×Q8—1731no—
148C22×D4—1731no—
149C23×C4—1732no—
150C4×Q8—3464no—
151C4×D4—3464no—
152C2×C4⋊C4—1731no—
153C2×C4⋊C4—1731no—
154C2×C4⋊C4C2×D17311yescharacteristic
155C2×C4⋊C4C2×D17311yescharacteristic
156C2×C22⋊C4—1732no—
157C2×C22⋊C4—1732no—
158C2×C22⋊C4—1732no—
159C2×C22⋊C4—1732no—
160C2×C22⋊C4—1732no—
161C2×C42—1731no—
162C2×C42—1731no—
163C2×C42—1732no—
164C2×C42—1731no—
165#95—1732no—
166#95—1731no—
167#95—1732no—
168#95—1732no—
169#95C2×D17311yescharacteristic
170#95C2×D17312yes—
171#95C2×D17312yes—
172C24—1732no—
173C2×Q8—3462no—
174C2×Q8—3462no—
175C2×Q8—3462no—
176C2×D4—3464no—
177C2×D4—3462no—
178C22×C4—3464no—
179C22×C4—3464no—
180C22×C4—3464no—
181C22×C4—1732no—
182C22×C4—1731no—
183C22×C4—1731no—
184C22×C4—1731no—
185C22×C4—1731no—
186C22×C4C22×D17312yes—
187C22×C4C22×D17312yes—
188C22×C4C22×D17311yescharacteristic
189C22×C4C22×D17311yescharacteristic
190C22×C4C22×D17311yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—3462no—
194C4⋊C4—3462no—
195C22⋊C4—3464no—
196C22⋊C4—3464no—
197C22⋊C4—3464no—
198C22⋊C4—3464no—
199C22⋊C4—3464no—
200C42—3462no—
201C42—3462no—
202C42—3464no—
203C42—1734no—
204C23—3462no—
205C23—3462no—
206C23—3462no—
207C23—3462no—
208C23—3462no—
209C23—3462no—
210C23—1734no—
211C23C23×D17311yescharacteristic
212Q8—3468no—
213D4—3468no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3461no—
226C2×C4—3461no—
227C2×C4—3461no—
228C2×C4—3461no—
229C2×C4—3461no—
230C2×C4—3462no—
231C2×C4—3462no—
232C2×C4—3462no—
233C2×C4—3462no—
234C2×C4—1732no—
235C2×C4—1732no—
236C2×C4—1732no—
237C2×C4—1734no—
238C2×C4—6928no—
239C2×C4D4×D17312yes—
240C2×C4#1458382712yes—
241C22—3464no—
242C22—3464no—
243C22—3464no—
244C22—6922no—
245C22—6922no—
246C22—6922no—
247C22—6922no—
248C22C2×D4×D17311yescharacteristic
249C22#41×D17312yes—
250C22#41×D17311yescharacteristic
251C22#41×D17312yes—
252C22C2×#1458382711yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—3462no—
259C4—3462no—
260C4—1734no—
261C4—6921no—
262C4—6922no—
263C2—6924no—
264C2#2601199911yescharacteristic
265C2#2601200411yescharacteristic
266C2#124×D17311yescharacteristic
267C2#2601202412yes—
268C2#2601202612yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#400×C173C2
prime radicalC22×D692C22
hypercenter#400D173
hypocenterC173#2398
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#19149644C2semidirect
3. #26011429C2semidirect
4. #26011420C2semidirect
7. #26011404C2semidirect
8. #26011338C2semidirect
9. #26011338C2semidirect
10. #26011336C2semidirect
11. #400×C173C2semidirect
134. C173#2398semidirect
139. #400D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496442
C2×#191496412
#260114292
#260114202
#260114092
#260114082
#260114042
#260113382
#260113362
#400×C1732
#2398173