Finite Group Databasebeta

← #36361891

Subgroups — #36361891

Order 22144.

688 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19149663C211yesmaximal, characteristic, complemented
2C2×#19149660C211yesmaximal, characteristic, complemented
3C2×#19149651C211yesmaximal, characteristic
4C22×C4×Dic173C211yesmaximal, characteristic, complemented
5#26011391C212yesmaximal, complemented
6#26011388C212yesmaximal, complemented
7#26011384C212yesmaximal, complemented
8#26011380C212yesmaximal, complemented
9#26011370C211yesmaximal, characteristic, complemented
10#26011349C211yesmaximal, characteristic, complemented
11#386×C173C211yesmaximal, characteristic, complemented
12C22×C22⋊D173C2211yescharacteristic
13C22×C22⋊D173C2211yescharacteristic
14C23×Dic173C2211yescharacteristic
15C23×Dic173C2211yescharacteristic
16#19149663—24no—
17#19149660—24no—
18#19149651—24no—
19C2×C4⋊Dic173C2212yes—
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2212yes—
22C2×#14583805C2212yes—
23C2×C22⋊Dic173C2211yescharacteristic
24C2×C22⋊Dic173C2211yescharacteristic
25C2×C22⋊Dic173C2212yes—
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2212yes—
28C2×#14583803C2212yes—
29C2×C4×Dic173—24no—
30C2×C4×Dic173C2212yes—
31C2×C4×Dic173C2212yes—
32#19149586C2212yes—
33#19149586C2212yes—
34C22×D4×C173C2211yescharacteristic
35C23×C4×C173C2211yescharacteristic
36C2×C22⋊C4×C173C2211yescharacteristic
37C2×C22⋊C4×C173C2212yes—
38#95×C173C2212yes—
39C23×D173C2311yescharacteristic
40C2×C22⋊D173—22no—
41C2×C22⋊D173—24no—
42C2×C22⋊D173—22no—
43C2×C22⋊D173—22no—
44C2×C22⋊D173—22no—
45C22×Dic173—24no—
46C22×Dic173—22no—
47C22×Dic173—22no—
48C22×Dic173D414yes—
49C22×Dic173D414yescomplemented
50C22×Dic173C2312yes—
51C22×Dic173C2311yescharacteristic
52C22×Dic173C2311yescharacteristic
53C22×Dic173C2312yes—
54C22×Dic173C2311yescharacteristic
55C4⋊Dic173—24no—
56#14583805—22no—
57#14583805—24no—
58#14583805—24no—
59C22⋊Dic173—22no—
60C22⋊Dic173—24no—
61C22⋊Dic173—22no—
62#14583803—22no—
63#14583803—24no—
64#14583803—24no—
65C4×Dic173—24no—
66C4×Dic173—24no—
67C24×C173C2311yescharacteristic
68C24×C173C2311yescharacteristic
69C2×D4×C173—22no—
70C2×D4×C173—22no—
71C2×D4×C173—22no—
72C22×C4×C173—22no—
73C22×C4×C173—24no—
74C22×C4×C173C2311yescharacteristic
75C22×C4×C173C2312yes—
76C22×C4×C173C2312yes—
77C22⋊C4×C173—22no—
78C22⋊C4×C173—24no—
79C22×D173—21no—
80C22×D173—21no—
81C22×D173—22no—
82C22×D173—21no—
83C22×D173—22no—
84C22⋊D173—44no—
85C22⋊D173—44no—
86C2×Dic173—28no—
87C2×Dic173—22no—
88C2×Dic173—22no—
89C2×Dic173—22no—
90C2×Dic173—28no—
91C2×Dic173—21no—
92C2×Dic173—21no—
93C2×Dic173—22no—
94C2×Dic173—21no—
95C2×Dic173—22no—
96C2×Dic173—22no—
97C2×Dic173C2×D412yes—
98C2×Dic173C2×D414yes—
99C2×Dic173C2×D412yes—
100C2×Dic173#4114yes—
101C23×C173—22no—
102C23×C173—22no—
103C23×C173—21no—
104C23×C173—21no—
105C23×C173—22no—
106C23×C173—22no—
107C23×C173—21no—
108C23×C173C2×D412yes—
109C23×C173C2×D412yes—
110C23×C173#4112yes—
111C23×C173C2411yescharacteristic
112D4×C173—44no—
113C2×C4×C173—28no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—22no—
117C2×C4×C173—21no—
118C2×C4×C173—21no—
119C2×C4×C173—22no—
120C2×C4×C173—21no—
121C2×C4×C173—22no—
122C2×C4×C173#4114yes—
123C22×C173—22no—
124C22×C173—22no—
125C22×C173—24no—
126C22×C173—24no—
127C22×C173—41no—
128C22×C173—41no—
129C22×C173—41no—
130C22×C173—42no—
131C22×C173—42no—
132C22×C173#12114yes—
133C22×C173C22×D411yescharacteristic
134C22×C173C22×D411yescharacteristic
135C22×C173C2×#4111yescharacteristic
136C22×C173C2×#4112yes—
137C22×C173C2×#4112yes—
138C2×D173—41no—
139C2×D173—41no—
140C2×D173—41no—
141C2×D173—42no—
142C2×D173—42no—
143C4×C173—24no—
144C4×C173—41no—
145C4×C173—42no—
146Dic173—24no—
147Dic173—22no—
148Dic173—41no—
149Dic173—42no—
150Dic173#12314yescomplemented
151C2×C173—24no—
152C2×C173—81no—
153C2×C173C2×#12111yescharacteristic
154C2×C173C2×#12311yescharacteristic
155C2×C173#52511yescharacteristic
156C2×C173#52912yes—
157C2×C173#53212yes—
158D173—81no—
159C173#285411yescharacteristic, complemented, Sylow
160#2854—1731nomaximal, Sylow
161C2×#123—1731no—
162C2×#123—1731no—
163C2×#121—1731no—
164C22×C42—1731no—
165#390—1732no—
166#389—1732no—
167#388—1732no—
168#388—1732no—
169#386—1731no—
170#386D17311yescharacteristic, complemented
171#380—1731no—
172C22×D4—1731no—
173C22×D4—1731no—
174C22×D4C2×D17311yescharacteristic
175C23×C4—1731no—
176C23×C4—1731no—
177C23×C4C2×D17311yescharacteristic
178#123—3464no—
179#123—3464no—
180#121—3464no—
181C2×C4⋊C4—1731no—
182C2×C4⋊C4—1732no—
183C2×C4⋊C4—1732no—
184C2×C4⋊C4—1732no—
185C2×C22⋊C4—1731no—
186C2×C22⋊C4—1731no—
187C2×C22⋊C4—1732no—
188C2×C22⋊C4—1731no—
189C2×C22⋊C4—1732no—
190C2×C22⋊C4—1732no—
191C2×C22⋊C4C2×D17311yescharacteristic
192C2×C22⋊C4C2×D17312yes—
193C2×C42—3464no—
194C2×C42—1732no—
195C2×C42—1732no—
196#95—1732no—
197#95—1732no—
198#95C2×D17312yes—
199C24—1731no—
200C24C22×D17311yescharacteristic
201C24C22×D17311yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—3462no—
206C2×D4—3462no—
207C2×D4—3462no—
208C2×D4—3462no—
209C2×D4—3464no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—3462no—
213C22×C4—3462no—
214C22×C4—3464no—
215C22×C4—1732no—
216C22×C4—1731no—
217C22×C4—1734no—
218C22×C4—1734no—
219C22×C4—1731no—
220C22×C4—1732no—
221C22×C4—1731no—
222C22×C4C22×D17312yes—
223C22×C4C22×D17311yescharacteristic
224C22×C4C22×D17312yes—
225C4⋊C4—3462no—
226C4⋊C4—3464no—
227C4⋊C4—3464no—
228C4⋊C4—3464no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—3462no—
232C22⋊C4—3464no—
233C22⋊C4—3462no—
234C22⋊C4—3462no—
235C22⋊C4—3464no—
236C22⋊C4—3464no—
237C42—3464no—
238C42—3464no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—3461no—
247C23—3461no—
248C23—3462no—
249C23—3462no—
250C23—3461no—
251C23D4×D17312yes—
252C23D4×D17312yes—
253C23#1458382612yes—
254C23C23×D17311yescharacteristic
255D4—44no—
256D4—6924no—
257D4—6924no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3468no—
268C2×C4—3462no—
269C2×C4—3462no—
270C2×C4—3462no—
271C2×C4—3468no—
272C2×C4—3462no—
273C2×C4—3461no—
274C2×C4—3461no—
275C2×C4—3462no—
276C2×C4—3462no—
277C2×C4—3461no—
278C2×C4—1732no—
279C2×C4—1734no—
280C2×C4—1732no—
281C2×C4—1734no—
282C2×C4#1458382514yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—6921no—
293C22—6921no—
294C22—6921no—
295C22—6922no—
296C22—6922no—
297C22#1914965214yes—
298C22C2×D4×D17311yescharacteristic
299C22C2×D4×D17311yescharacteristic
300C22#41×D17312yes—
301C22C2×#1458382512yes—
302C22C2×#1458382611yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—3464no—
307C4—3462no—
308C4—1734no—
309C4—6921no—
310C4—6922no—
311C2—24no—
312C2—81no—
313C2—13841no—
314C2C2×#1914965211yescharacteristic
315C2#123×D17311yescharacteristic
316C2#2601199711yescharacteristic
317C2#2601202112yes—
318C2#2601203612yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#386×C173C2
prime radicalC2×#19149651C2
hypercenter#386D173
hypocenterC173#2854
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#19149663C2semidirect
2. C2×#19149660C2semidirect
4. C22×C4×Dic173C2semidirect
5. #26011391C2semidirect
6. #26011388C2semidirect
7. #26011384C2semidirect
8. #26011380C2semidirect
9. #26011370C2semidirect
10. #26011349C2semidirect
11. #386×C173C2semidirect
49. C22×Dic173D4semidirect
150. Dic173#123semidirect
159. C173#2854semidirect
170. #386D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496632
C2×#191496602
C2×#191496512
C22×C4×Dic1732
#260113912
#260113882
#260113842
#260113802
#260113702
#260113492
#386×C1732
#2854173