Finite Group Databasebeta

← #36353864

Subgroups — #36353864

Order 22144.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D173C211yesmaximal, characteristic, complemented
2#26012003C211yesmaximal, characteristic
3C2×#19149607C211yesmaximal, characteristic, complemented
4C2×#19149605C211yesmaximal, characteristic, complemented
5C4×#14583803C211yesmaximal, characteristic, complemented
6#26011250C211yesmaximal, characteristic, complemented
7#339×C173C211yesmaximal, characteristic, complemented
8C2×#14583826C2211yescharacteristic
9C22×C4×D173C2211yescharacteristic
10C2×#14583810C2211yescharacteristic
11C2×#14583809—21no—
12M16×D173—22no—
13M16×D173—22no—
14C2×C8×D173—21no—
15#19149663—21no—
16#19149659C411yescharacteristic, complemented
17#19149659C411yescharacteristic, complemented
18#19149654C411yescharacteristic, complemented
19#19149654C411yescharacteristic, complemented
20#19149649—21no—
21C4×D692—21no—
22C4×D692—21no—
23C2×#14583803—21no—
24C2×C4×Dic173C2211yescharacteristic
25#19149607—22no—
26#19149605—22no—
27#19149586—21no—
28C2×M16×C173C2211yescharacteristic
29#117×C173C2211yescharacteristic
30C2×C42×C173C2211yescharacteristic
31C23×D173D411yescharacteristic
32#14583826—21no—
33#14583826—21no—
34#14583826—21no—
35#14583826—21no—
36C2×Dic346C2×C411yescharacteristic
37C2×D692—21no—
38C2×D692—21no—
39C2×D692C2×C411yescharacteristic
40C2×C22⋊D173—21no—
41C22×Dic173—21no—
42C22×Dic173D411yescharacteristic
43C2×C4×D173—21no—
44C2×C4×D173—21no—
45C2×C4×D173—21no—
46C2×C4×D173—21no—
47C2×C4×D173—21no—
48C2×C4×D173C2×C411yescharacteristic
49C2×C4×D173C2×C411yescharacteristic
50C2×C4×D173D411yescharacteristic
51C2×C4×D173D411yescharacteristic
52#14583810—21no—
53#14583810D412yescomplemented
54#14583809—24no—
55C2×#13063436—21no—
56C8×D173—24no—
57C4⋊Dic173—21no—
58C4⋊Dic173—21no—
59#14583803—24no—
60#14583803—21no—
61#14583803—21no—
62#14583803—21no—
63#14583803—21no—
64C4×Dic173—21no—
65C4×Dic173—21no—
66C22×C4×C173—21no—
67C22×C4×C173C2311yescharacteristic
68M16×C173—21no—
69M16×C173D412yescomplemented
70C2×C8×C173—21no—
71C4⋊C4×C173C2×C411yescharacteristic
72C4⋊C4×C173C2×C411yescharacteristic
73C22⋊C4×C173—21no—
74C42×C173—21no—
75C42×C173—21no—
76C42×C173—21no—
77C22×D173—21no—
78C22×D173—21no—
79C22×D173—22no—
80C22×D173—21no—
81C22×D173—41no—
82C22×D173C22⋊C411yescharacteristic
83C22×D173C22⋊C411yescharacteristic
84Dic346—21no—
85Dic346—21no—
86D692—21no—
87D692—21no—
88D692—42no—
89D692—42no—
90C22⋊D173—41no—
91C22⋊D173—41no—
92C2×Dic173—21no—
93C2×Dic173—21no—
94C2×Dic173—22no—
95C2×Dic173—21no—
96C2×Dic173—41no—
97C2×Dic173—41no—
98C2×Dic173C22⋊C411yescharacteristic
99C2×Dic173C22⋊C411yescharacteristic
100C4×D173—21no—
101C4×D173—21no—
102C4×D173—21no—
103C4×D173—21no—
104C4×D173—21no—
105C4×D173—21no—
106C4×D173—21no—
107C4×D173—21no—
108C4×D173—41no—
109C4×D173—41no—
110#13063436—22no—
111C23×C173C2×D411yescharacteristic
112C2×C4×C173—22no—
113C2×C4×C173—21no—
114C2×C4×C173—21no—
115C2×C4×C173—41no—
116C2×C4×C173—41no—
117C2×C4×C173C22×C411yescharacteristic
118C2×C4×C173C2×D411yescharacteristic
119C2×C4×C173C2×D411yescharacteristic
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173#4111yescharacteristic
122C2×C4×C173#4111yescharacteristic
123C8×C173—22no—
124C22×C173—21no—
125C22×C173—21no—
126C22×C173C2×C22⋊C411yescharacteristic
127C22×C173#12111yescharacteristic
128C22×C173#12111yescharacteristic
129C2×D173—21no—
130C2×D173—21no—
131C2×D173—21no—
132C2×D173—21no—
133C2×D173—21no—
134C2×D173—42no—
135C2×D173—42no—
136C2×D173—41no—
137C2×D173—41no—
138C2×D173—41no—
139C2×D173#10412yescomplemented
140C4×C173—42no—
141C4×C173—41no—
142C4×C173—41no—
143C4×C173C4×D411yescharacteristic
144C4×C173C4×D411yescharacteristic
145C4×C173#12011yescharacteristic
146C4×C173#12311yescharacteristic
147Dic173—21no—
148Dic173—21no—
149Dic173—42no—
150Dic173—41no—
151C2×C173—21no—
152C2×C173—21no—
153C2×C173#38611yescharacteristic
154C2×C173C2×#10411yescharacteristic
155C2×C173#42111yescharacteristic
156D173—22no—
157D173—41no—
158D173—81no—
159C173#187111yescharacteristic, complemented, Sylow
160#1871—1731nomaximal, Sylow
161C22×M16—1731no—
162#525—1731no—
163C2×#104—1731no—
164C2×#104—1731no—
165C4×C22⋊C4—1731no—
166#339—1731no—
167#339D17311yescharacteristic, complemented
168C2×#41—1731no—
169C23×C4—1731no—
170C2×M16—3461no—
171C2×M16—3462no—
172C2×M16—3462no—
173C2×M16—1731no—
174C2×M16C2×D17311yescharacteristic
175C22×C8—3461no—
176#123—3461no—
177C22⋊Q8—1731no—
178C22⋊Q8—1731no—
179#121—1731no—
180#121—1731no—
181#120—3461no—
182C4×D4—3461no—
183C4×D4—3461no—
184#117C2×D17311yescharacteristic
185C2×C22⋊C4—3461no—
186C2×C42—1731no—
187C2×C42C2×D17311yescharacteristic
188#104—3462no—
189#104—3462no—
190#95—3461no—
191C24—1731no—
192#41—3461no—
193#41—3461no—
194#41—3461no—
195#41—3461no—
196C2×Q8—1731no—
197C2×D4—3461no—
198C2×D4—3461no—
199C2×D4—3461no—
200C2×D4—1731no—
201C22×C4—21no—
202C22×C4—3461no—
203C22×C4—3461no—
204C22×C4—3461no—
205C22×C4—3461no—
206C22×C4—3461no—
207C22×C4—3461no—
208C22×C4—1731no—
209C22×C4—1731no—
210C22×C4—1731no—
211C22×C4—1731no—
212C22×C4—1731no—
213C22×C4C22×D17311yescharacteristic
214M16—21no—
215M16—3464no—
216M16—3461no—
217M16—1732no—
218M16D69212yescomplemented
219C2×C8—21no—
220C2×C8—3464no—
221C2×C8—3461no—
222C4⋊C4—3461no—
223C4⋊C4—3461no—
224C4⋊C4C4×D17311yescharacteristic
225C4⋊C4C4×D17311yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3464no—
228C22⋊C4—3461no—
229C22⋊C4—3461no—
230C22⋊C4—3461no—
231C22⋊C4—3461no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—3461no—
236C42—3461no—
237C23—3462no—
238C23—3461no—
239C23—3461no—
240C23—3461no—
241C23—1731no—
242C23—1731no—
243C23—6921no—
244C23D4×D17311yescharacteristic
245Q8—3461no—
246Q8—3461no—
247D4—3461no—
248D4—3461no—
249D4—6922no—
250D4—6922no—
251D4—6921no—
252D4—6921no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—3461no—
259C2×C4—3461no—
260C2×C4—3461no—
261C2×C4—3461no—
262C2×C4—3461no—
263C2×C4—3461no—
264C2×C4—3461no—
265C2×C4—3461no—
266C2×C4—3461no—
267C2×C4—3461no—
268C2×C4—3462no—
269C2×C4—3461no—
270C2×C4—1731no—
271C2×C4—1731no—
272C2×C4—6921no—
273C2×C4—6921no—
274C2×C4—6921no—
275C2×C4—6921no—
276C2×C4C2×C4×D17311yescharacteristic
277C2×C4D4×D17311yescharacteristic
278C2×C4D4×D17311yescharacteristic
279C2×C4C2×D69211yescharacteristic
280C2×C4#1458382611yescharacteristic
281C2×C4#1458382711yescharacteristic
282C8—22no—
283C8—3462no—
284C22—21no—
285C22—21no—
286C22—3461no—
287C22—3461no—
288C22—3461no—
289C22—3461no—
290C22—3461no—
291C22—1732no—
292C22—6922no—
293C22—6922no—
294C22—6921no—
295C22—6921no—
296C22—6921no—
297C22C22⋊C4×D17311yescharacteristic
298C22#1914965211yescharacteristic
299C22#1914965411yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—3461no—
304C4—3461no—
305C4—6922no—
306C4—6921no—
307C4#1914964111yescharacteristic
308C4C4×D69211yescharacteristic
309C4#1914964911yescharacteristic
310C4#1914966411yescharacteristic
311C2—21no—
312C2—21no—
313C2—3462no—
314C2—13841no—
315C2—6921no—
316C2#2601136911yescharacteristic
317C2#104×D17311yescharacteristic
318C2#2601150611yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting#339×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012003C2
cosocleC22×C4×C173C23
hypercenter#339D173
hypocenterC173#1871
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D173C2semidirect
3. C2×#19149607C2semidirect
4. C2×#19149605C2semidirect
5. C4×#14583803C2semidirect
6. #26011250C2semidirect
7. #339×C173C2semidirect
16. #19149659C4semidirect
17. #19149659C4semidirect
18. #19149654C4semidirect
19. #19149654C4semidirect
53. #14583810D4semidirect
69. M16×C173D4semidirect
139. C2×D173#104semidirect
159. C173#1871semidirect
167. #339D173semidirect
218. M16D692semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1732
#260120032
C2×#191496072
C2×#191496052
C4×#145838032
#260112502
#339×C1732
#1871173