Finite Group Databasebeta

← #36354629

Subgroups — #36354629

Order 22144.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012289C211yesmaximal, characteristic, complemented
2#26012003C211yesmaximal, characteristic, complemented
3#26011998C211yesmaximal, characteristic, complemented
4#26011507C211yesmaximal, characteristic, complemented
5#26011454C211yesmaximal, characteristic, complemented
6#26011247C211yesmaximal, characteristic, complemented
7#421×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C22×C4×D173C2211yescharacteristic
10#19149710—21no—
11#19149709—21no—
12#19149705—21no—
13#19149704—21no—
14#19149699—22no—
15C2×#14583810C2211yescharacteristic
16#19149663—21no—
17#19149660—21no—
18#19149659—21no—
19#19149655—21no—
20#19149654—21no—
21#19149650—21no—
22#19149649—21no—
23#19149647—21no—
24C4×D692—22no—
25C4×C22⋊D173—22no—
26#19149638C2211yescharacteristic
27#19149605—22no—
28#19149591—22no—
29C2×#41×C173C2211yescharacteristic
30C2×M16×C173C2211yescharacteristic
31#117×C173C2211yescharacteristic
32#104×C173—22no—
33C23×D173D411yescharacteristic
34#14583826—21no—
35#14583826—21no—
36#14583826—21no—
37#14583826—21no—
38C2×Dic346D411yescharacteristic, complemented
39C2×D692—22no—
40C2×D692D411yescharacteristic, complemented
41C2×C22⋊D173—22no—
42C2×C22⋊D173—21no—
43C22×Dic173D411yescharacteristic
44C2×C4×D173—21no—
45C2×C4×D173—21no—
46C2×C4×D173—21no—
47C2×C4×D173—21no—
48C2×C4×D173—21no—
49C2×C4×D173—21no—
50C2×C4×D173—21no—
51#14583816—22no—
52#14583814—22no—
53#14583813—22no—
54#14583811—22no—
55#14583810—21no—
56#14583810—21no—
57C2×#13063436D412yescomplemented
58C4⋊Dic173—22no—
59C4⋊Dic173—21no—
60#14583805—22no—
61C22⋊Dic173—22no—
62C22⋊Dic173—21no—
63#14583803—22no—
64#14583803—22no—
65#14583803—22no—
66C4×Dic173D412yescomplemented
67#41×C173—21no—
68#41×C173—21no—
69#41×C173—21no—
70#41×C173—21no—
71C2×Q8×C173D411yescharacteristic, complemented
72C2×D4×C173—21no—
73C2×D4×C173D411yescharacteristic, complemented
74C22×C4×C173—21no—
75C22×C4×C173C2311yescharacteristic
76M16×C173—21no—
77M16×C173—21no—
78C2×C8×C173—21no—
79C4⋊C4×C173—21no—
80C22⋊C4×C173—21no—
81C42×C173D412yescomplemented
82C22×D173—21no—
83C22×D173—21no—
84C22×D173—21no—
85C22×D173—21no—
86C22×D173—42no—
87Dic346—21no—
88Dic346—21no—
89D692—21no—
90D692—21no—
91D692—44no—
92C22⋊D173—44no—
93C22⋊D173—41no—
94C22⋊D173—41no—
95C2×Dic173—21no—
96C2×Dic173—21no—
97C2×Dic173—21no—
98C2×Dic173—22no—
99C2×Dic173—21no—
100C4×D173—41no—
101C4×D173—41no—
102C4×D173—41no—
103C4×D173—41no—
104C4×D173—41no—
105C4×D173—41no—
106#13063436—22no—
107C23×C173—21no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—21no—
110Q8×C173—21no—
111D4×C173—21no—
112D4×C173—21no—
113D4×C173—41no—
114D4×C173—41no—
115C2×C4×C173—22no—
116C2×C4×C173—21no—
117C2×C4×C173—41no—
118C2×C4×C173—41no—
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2×D411yescharacteristic
122C2×C4×C173C2×D411yescharacteristic
123C2×C4×C173C2×D411yescharacteristic
124C2×C4×C173#4111yescharacteristic
125C8×C173—41no—
126C22×C173—41no—
127C22×C173—42no—
128C22×C173—41no—
129C22×C173#12011yescharacteristic
130C22×C173#12011yescharacteristic
131C22×C173#12111yescharacteristic
132C2×D173—22no—
133C2×D173—22no—
134C2×D173—22no—
135C2×D173—41no—
136C2×D173—41no—
137C2×D173—41no—
138C2×D173—41no—
139C2×D173—42no—
140C2×D173—41no—
141C4×C173—42no—
142C4×C173—41no—
143C4×C173#12011yescharacteristic
144C4×C173#12111yescharacteristic
145C4×C173#12111yescharacteristic
146C4×C173#12711yescharacteristic
147Dic173—42no—
148Dic173—41no—
149Dic173—41no—
150C2×C173—21no—
151C2×C173—21no—
152C2×C173—21no—
153C2×C173—81no—
154C2×C173#39211yescharacteristic
155D173—42no—
156D173—81no—
157C173#201011yescharacteristic, complemented, Sylow
158#2010—1731nomaximal, Sylow
159#575—1731no—
160#525—1731no—
161#525—1731no—
162#421—1731no—
163#421D17311yescharacteristic, complemented
164#407—1731no—
165H(3,ℤ/4)—1731no—
166C2×#41—1731no—
167C2×#41C2×D17311yescharacteristic
168C23×C4—1731no—
169#137—3461no—
170#137—3461no—
171AGL(1,ℤ/8)—3461no—
172AGL(1,ℤ/8)—3461no—
173#135—3462no—
174C2×M16—1731no—
175C2×M16C2×D17311yescharacteristic
176#123—3461no—
177#123—3461no—
178C22⋊Q8—3461no—
179C22⋊Q8—3461no—
180#121—3461no—
181#121—3461no—
182#120—3461no—
183#120—3461no—
184C4×D4—3462no—
185C4×D4—3462no—
186#117—1731no—
187#117C2×D17311yescharacteristic
188#104—22no—
189#104—3462no—
190C22⋊C8—3462no—
191C24—1731no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—3461no—
197#41—3461no—
198#41—3461no—
199#41—3461no—
200C2×Q8—1731no—
201C2×Q8D69211yescharacteristic, complemented
202C2×D4—21no—
203C2×D4—3462no—
204C2×D4—3462no—
205C2×D4—3461no—
206C2×D4—1731no—
207C2×D4D69211yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—3461no—
210C22×C4—3461no—
211C22×C4—3461no—
212C22×C4—3461no—
213C22×C4—3461no—
214C22×C4—3461no—
215C22×C4—3461no—
216C22×C4—1731no—
217C22×C4C22×D17311yescharacteristic
218Q16—3462no—
219SD16—3462no—
220SD16—3462no—
221D8—3462no—
222M16—21no—
223M16—21no—
224M16—3461no—
225M16—3461no—
226C2×C8—21no—
227C2×C8—1732no—
228C4⋊C4—21no—
229C4⋊C4—3462no—
230C4⋊C4—3462no—
231C4⋊C4—3461no—
232C22⋊C4—21no—
233C22⋊C4—3462no—
234C22⋊C4—3462no—
235C22⋊C4—3462no—
236C22⋊C4—3462no—
237C22⋊C4—3461no—
238C42—1732no—
239C42C22⋊D17312yescomplemented
240C23—21no—
241C23—3461no—
242C23—3461no—
243C23—3461no—
244C23—3461no—
245C23—6922no—
246C23D4×D17311yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—3461no—
250Q8—3461no—
251D4—21no—
252D4—21no—
253D4—41no—
254D4—41no—
255D4—3461no—
256D4—3461no—
257D4—6924no—
258D4—6924no—
259D4—6921no—
260D4—6921no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—3461no—
266C2×C4—3461no—
267C2×C4—3462no—
268C2×C4—3461no—
269C2×C4—3461no—
270C2×C4—6921no—
271C2×C4—6921no—
272C2×C4—6921no—
273C2×C4—6921no—
274C2×C4—6921no—
275C2×C4—6921no—
276C2×C4D4×D17311yescharacteristic
277C2×C4D4×D17311yescharacteristic
278C2×C4D4×D17311yescharacteristic
279C2×C4C2×C22⋊D17311yescharacteristic
280C2×C4C2×D69211yescharacteristic
281C2×C4#1458382611yescharacteristic
282C8—41no—
283C8—3462no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—3462no—
288C22—3462no—
289C22—3462no—
290C22—6922no—
291C22—6921no—
292C22—6921no—
293C22—6921no—
294C22—6921no—
295C22—6921no—
296C22#1914964711yescharacteristic
297C22#1914964911yescharacteristic
298C22#1914965211yescharacteristic
299C4—42no—
300C4—41no—
301C4—6922no—
302C4—6921no—
303C4—6921no—
304C4#1914964911yescharacteristic
305C4#1914965211yescharacteristic
306C4#1914965311yescharacteristic
307C4#1914967511yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2—13841no—
313C2—6922no—
314C2#2601140011yescharacteristic
subgroupG/H
centerC4#19149675
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#421×C173C2
socleC2×C173#392
hypercenter#421D173
hypocenterC173#2010
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #26012289C2semidirect
2. #26012003C2semidirect
3. #26011998C2semidirect
4. #26011507C2semidirect
5. #26011454C2semidirect
6. #26011247C2semidirect
7. #421×C173C2semidirect
38. C2×Dic346D4semidirect
40. C2×D692D4semidirect
57. C2×#13063436D4semidirect
66. C4×Dic173D4semidirect
71. C2×Q8×C173D4semidirect
73. C2×D4×C173D4semidirect
81. C42×C173D4semidirect
157. C173#2010semidirect
163. #421D173semidirect
201. C2×Q8D692semidirect
207. C2×D4D692semidirect
239. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260122892
#260120032
#260119982
#260115072
#260114542
#260112472
#421×C1732
#2010173