Finite Group Databasebeta

← #36374726

Subgroups — #36374726

Order 22336.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#26013634C211yesmaximal, characteristic, complemented
2#26013632C211yesmaximal, characteristic, complemented
3#26013626C211yesmaximal, characteristic, complemented
4#26013625C211yesmaximal, characteristic, complemented
5#26013622C211yesmaximal, characteristic, complemented
6#26013620C211yesmaximal, characteristic, complemented
7#26013617C211yesmaximal, characteristic, complemented
8#26013611C211yesmaximal, characteristic, complemented
9#26013610C211yesmaximal, characteristic, complemented
10#26013607C211yesmaximal, characteristic, complemented
11#26013607C211yesmaximal, characteristic, complemented
12C4×C22⋊D349C211yesmaximal, characteristic, complemented
13#26013599C211yesmaximal, characteristic, complemented
14C22⋊C4×D349C211yesmaximal, characteristic, complemented
15#121×C349C211yesmaximal, characteristic, complemented
16C23×D349C2211yescharacteristic
17C2×Dic698C2211yescharacteristic, complemented
18C2×C22⋊D349C2211yescharacteristic
19C2×C22⋊D349C2211yescharacteristic
20C2×C22⋊D349C2211yescharacteristic
21C2×C22⋊D349C2211yescharacteristic
22C22×Dic349C2211yescharacteristic, complemented
23C2×C4×D349C2211yescharacteristic
24C2×C4×D349C2211yescharacteristic
25C4⋊Dic349C2211yescharacteristic, complemented
26#19150212C2211yescharacteristic, complemented
27#19150212C2211yescharacteristic, complemented
28#19150212C2211yescharacteristic, complemented
29#19150212C2211yescharacteristic, complemented
30C22⋊Dic349C2211yescharacteristic
31C22⋊Dic349C2211yescharacteristic
32C22⋊Dic349C2211yescharacteristic
33C22⋊Dic349C2211yescharacteristic, complemented
34C22⋊Dic349C2211yescharacteristic, complemented
35C22⋊Dic349C2211yescharacteristic, complemented
36#19150210C2211yescharacteristic
37#19150210C2211yescharacteristic
38#19150210C2211yescharacteristic
39#19150210C2211yescharacteristic
40#19150210C2211yescharacteristic
41#19150210C2211yescharacteristic
42C4×Dic349C2211yescharacteristic, complemented
43C4×Dic349C2211yescharacteristic, complemented
44C2×D4×C349C2211yescharacteristic
45C2×D4×C349C2211yescharacteristic
46C2×D4×C349C2211yescharacteristic, complemented
47C22×C4×C349C2211yescharacteristic, complemented
48C4⋊C4×C349C2211yescharacteristic, complemented
49C22⋊C4×C349C2211yescharacteristic
50C22⋊C4×C349C2211yescharacteristic, complemented
51C22×D349—22no—
52C22×D349—41no—
53C22×D349—41no—
54C22×D349C2311yescharacteristic
55C22×D349C2311yescharacteristic
56Dic698—41no—
57C22⋊D349—22no—
58C22⋊D349—22no—
59C22⋊D349—41no—
60C22⋊D349—41no—
61C2×Dic349—41no—
62C2×Dic349C2311yescharacteristic
63C2×Dic349C2311yescharacteristic
64C2×Dic349C2311yescharacteristic
65C2×Dic349C2311yescharacteristic
66C2×Dic349C2311yescharacteristic
67C2×Dic349C2311yescharacteristic
68C4×D349—41no—
69C4×D349—41no—
70C23×C349C2311yescharacteristic
71C23×C349C2311yescharacteristic
72C23×C349C2311yescharacteristic
73D4×C349—41no—
74D4×C349—41no—
75D4×C349—41no—
76C2×C4×C349—41no—
77C2×C4×C349C2311yescharacteristic
78C2×C4×C349C2311yescharacteristic
79C2×C4×C349C2311yescharacteristic
80C2×C4×C349C2311yescharacteristic
81C22×C349—21no—
82C22×C349—21no—
83C22×C349—21no—
84C22×C349—21no—
85C22×C349—21no—
86C22×C349—21no—
87C22×C349—21no—
88C22×C349—21no—
89C22×C349—21no—
90C22×C349C2411yescharacteristic
91C2×D349—21no—
92C2×D349—21no—
93C2×D349—21no—
94C2×D349—21no—
95C2×D349—21no—
96C2×D349—44no—
97C2×D349#4112yes—
98C4×C349—21no—
99C4×C349—21no—
100C4×C349—21no—
101C4×C349—21no—
102Dic349—21no—
103Dic349—21no—
104Dic349—21no—
105Dic349—21no—
106Dic349—21no—
107Dic349—21no—
108C2×C349—41no—
109C2×C349—41no—
110C2×C349—41no—
111C2×C349C2×#4111yescharacteristic
112C2×C349H(5,𝔽2)11yescharacteristic
113C2×C349H(5,𝔽2)11yescharacteristic
114D349—22no—
115D349—41no—
116C349#53811yescharacteristic, complemented, Sylow
117#538—3491nomaximal, Sylow
118#126—3491no—
119#126—3491no—
120#124—3491no—
121#124—3491no—
122#123—3491no—
123#123—3491no—
124C22⋊Q8—3491no—
125#121—3491no—
126#121—3491no—
127#121D34911yescharacteristic, complemented
128#120—3491no—
129#120—3491no—
130C4×D4—3491no—
131C4×D4—3491no—
132C2×C22⋊C4—3491no—
133C24—3491no—
134C2×Q8—3491no—
135C2×D4—3491no—
136C2×D4—3491no—
137C2×D4—3491no—
138C2×D4—3491no—
139C2×D4C2×D34911yescharacteristic
140C2×D4C2×D34911yescharacteristic
141C2×D4C2×D34911yescharacteristic, complemented
142C22×C4—3491no—
143C22×C4—3491no—
144C22×C4—3491no—
145C22×C4C2×D34911yescharacteristic, complemented
146C4⋊C4—3491no—
147C4⋊C4—3491no—
148C4⋊C4—3491no—
149C4⋊C4—3491no—
150C4⋊C4—3491no—
151C4⋊C4C2×D34911yescharacteristic, complemented
152C22⋊C4—3491no—
153C22⋊C4—3491no—
154C22⋊C4—3491no—
155C22⋊C4—3491no—
156C22⋊C4—3491no—
157C22⋊C4—3491no—
158C22⋊C4—3491no—
159C22⋊C4—3491no—
160C22⋊C4—3491no—
161C22⋊C4—3491no—
162C22⋊C4—3491no—
163C22⋊C4—3491no—
164C22⋊C4C2×D34911yescharacteristic
165C22⋊C4C2×D34911yescharacteristic, complemented
166C42—3491no—
167C42—3491no—
168C23—3491no—
169C23—3491no—
170C23—13961no—
171C23—13961no—
172C23—6982no—
173C23C22×D34911yescharacteristic
174C23C22×D34911yescharacteristic
175C23C22×D34911yescharacteristic
176Q8—13961no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—13961no—
181D4—13961no—
182D4—6982no—
183D4—6982no—
184C2×C4—41no—
185C2×C4—3491no—
186C2×C4—3491no—
187C2×C4—3491no—
188C2×C4—3491no—
189C2×C4—3491no—
190C2×C4—3491no—
191C2×C4—13961no—
192C2×C4—13961no—
193C2×C4—13961no—
194C2×C4C22×D34911yescharacteristic
195C2×C4C22×D34911yescharacteristic
196C2×C4C22×D34911yescharacteristic
197C2×C4C22×D34911yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—3492no—
208C22—13964no—
209C22—6981no—
210C22—6981no—
211C22—6981no—
212C22—6981no—
213C22—6981no—
214C22C23×D34911yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—6981no—
220C4—6981no—
221C4—6981no—
222C4—6981no—
223C4—6981no—
224C4—6981no—
225C2—41no—
226C2—41no—
227C2—41no—
228C2—13961no—
229C2—6982no—
230C2#41×D34911yescharacteristic
231C2#2601368411yescharacteristic
232C2#2601368411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#121×C349C2
hypercenter#121D349
hypocenterC349#538
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26013634C2semidirect
2. #26013632C2semidirect
3. #26013626C2semidirect
4. #26013625C2semidirect
5. #26013622C2semidirect
6. #26013620C2semidirect
7. #26013617C2semidirect
8. #26013611C2semidirect
9. #26013610C2semidirect
10. #26013607C2semidirect
11. #26013607C2semidirect
12. C4×C22⋊D349C2semidirect
13. #26013599C2semidirect
14. C22⋊C4×D349C2semidirect
15. #121×C349C2semidirect
17. C2×Dic698C22semidirect
22. C22×Dic349C22semidirect
25. C4⋊Dic349C22semidirect
26. #19150212C22semidirect
27. #19150212C22semidirect
28. #19150212C22semidirect
29. #19150212C22semidirect
33. C22⋊Dic349C22semidirect
34. C22⋊Dic349C22semidirect
35. C22⋊Dic349C22semidirect
42. C4×Dic349C22semidirect
43. C4×Dic349C22semidirect
46. C2×D4×C349C22semidirect
47. C22×C4×C349C22semidirect
48. C4⋊C4×C349C22semidirect
50. C22⋊C4×C349C22semidirect
116. C349#538semidirect
127. #121D349semidirect
141. C2×D4C2×D349semidirect
145. C22×C4C2×D349semidirect
151. C4⋊C4C2×D349semidirect
165. C22⋊C4C2×D349semidirect

Maximal subgroups

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

subgroupindex
#260136342
#260136322
#260136262
#260136252
#260136222
#260136202
#260136172
#260136112
#260136102
#260136072
C4×C22⋊D3492
#260135992
C22⋊C4×D3492
#121×C3492
#538349