Finite Group Databasebeta

← #20295666

Subgroups — #20295666

Order 6976.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#13182029C211yesmaximal, characteristic, complemented
2C22×C22⋊D109C211yesmaximal, characteristic, complemented
3#16812685C211yesmaximal, characteristic, complemented
4#16812682C211yesmaximal, characteristic, complemented
5#16812681C211yesmaximal, characteristic, complemented
6#16812676C211yesmaximal, characteristic, complemented
7#16812673C211yesmaximal, characteristic, complemented
8#16812672C211yesmaximal, characteristic, complemented
9#16812672C211yesmaximal, characteristic, complemented
10#16812669C211yesmaximal, characteristic, complemented
11#16812668C211yesmaximal, characteristic, complemented
12C4×C22⋊D109C211yesmaximal, characteristic, complemented
13#16812659C211yesmaximal, characteristic, complemented
14C22⋊C4×D109C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C109C211yesmaximal, characteristic, complemented
16C23×D109C2211yescharacteristic
17#13182029—24no—
18C2×Dic218C2211yescharacteristic, complemented
19C2×D436C2211yescharacteristic, complemented
20C2×D436C2211yescharacteristic, complemented
21C2×C22⋊D109—24no—
22C2×C22⋊D109C2211yescharacteristic
23C2×C22⋊D109C2211yescharacteristic
24C2×C22⋊D109C2211yescharacteristic
25C2×C22⋊D109C2211yescharacteristic
26C2×C22⋊D109C2211yescharacteristic
27C2×C22⋊D109C2211yescharacteristic, complemented
28C2×C22⋊D109C2211yescharacteristic, complemented
29C22×Dic109C2211yescharacteristic
30C2×C4×D109C2211yescharacteristic, complemented
31C2×C4×D109C2211yescharacteristic, complemented
32C2×C4×D109C2211yescharacteristic, complemented
33C4⋊Dic109C2211yescharacteristic, complemented
34#13182008C2211yescharacteristic, complemented
35#13182008C2211yescharacteristic, complemented
36#13182008C2211yescharacteristic, complemented
37C22⋊Dic109C2211yescharacteristic
38C22⋊Dic109C2211yescharacteristic, complemented
39C22⋊Dic109C2211yescharacteristic, complemented
40#13182006C2211yescharacteristic, complemented
41#13182006C2211yescharacteristic, complemented
42#13182006C2211yescharacteristic, complemented
43#13182006C2211yescharacteristic, complemented
44#13182006C2211yescharacteristic, complemented
45C4×Dic109C2211yescharacteristic, complemented
46C24×C109C2211yescharacteristic
47C22×C4×C109C2211yescharacteristic, complemented
48C22×C4×C109C2211yescharacteristic, complemented
49C22⋊C4×C109C2211yescharacteristic
50C22⋊C4×C109C2211yescharacteristic
51C22⋊C4×C109C2211yescharacteristic, complemented
52C22⋊C4×C109C2211yescharacteristic, complemented
53C23×C109—22no—
54C23×C109—22no—
55C23×C109—41no—
56C23×C109C2311yescharacteristic
57C23×C109C2311yescharacteristic
58C23×C109C2311yescharacteristic
59C22×D109—22no—
60C22×D109—22no—
61C22×D109—41no—
62C22×D109C2311yescharacteristic
63C22×D109C2311yescharacteristic
64C22×D109C2311yescharacteristic
65C2×C4×C109—22no—
66C2×C4×C109—41no—
67C2×C4×C109C2311yescharacteristic
68C2×C4×C109C2311yescharacteristic
69C2×C4×C109C2311yescharacteristic
70C2×C4×C109C2311yescharacteristic
71C22⋊D109—22no—
72C22⋊D109—22no—
73C22⋊D109—22no—
74C22⋊D109—22no—
75C22⋊D109—22no—
76C22⋊D109—41no—
77C22⋊D109D414yescomplemented
78C2×Dic109—22no—
79C2×Dic109C2311yescharacteristic
80C2×Dic109C2311yescharacteristic
81C2×Dic109C2311yescharacteristic
82C2×Dic109C2311yescharacteristic
83C2×Dic109C2311yescharacteristic
84D436—22no—
85D436—41no—
86C4×D109—22no—
87C4×D109—22no—
88C4×D109—41no—
89Dic218—22no—
90C22×C109—21no—
91C22×C109—21no—
92C22×C109—21no—
93C22×C109—21no—
94C22×C109—21no—
95C22×C109—21no—
96C22×C109—21no—
97C22×C109—44no—
98C22×C109C2×D412yes—
99C22×C109#4112yescomplemented
100C22×C109C2411yescharacteristic
101C2×D109—21no—
102C2×D109—21no—
103C2×D109—21no—
104C2×D109—21no—
105C2×D109—21no—
106C2×D109—21no—
107C2×D109—21no—
108C2×D109—21no—
109C2×D109—44no—
110C2×D109C2×D412yes—
111C4×C109—21no—
112C4×C109—21no—
113C4×C109—21no—
114C4×C109—21no—
115Dic109—21no—
116Dic109—21no—
117Dic109—21no—
118Dic109—21no—
119Dic109C2×D412yes—
120C2×C109—22no—
121C2×C109—22no—
122C2×C109—41no—
123C2×C109C22×D411yescharacteristic
124C2×C109C2×#4111yescharacteristic
125C2×C109H(5,𝔽2)11yescharacteristic
126D109—22no—
127D109—41no—
128D109—41no—
129C109#54611yescharacteristic, complemented, Sylow
130#546—1091nomaximal, Sylow
131C2×#41—1091no—
132C22×D4—1091no—
133#124—1091no—
134#123—1091no—
135#123—1091no—
136C22⋊Q8—1091no—
137#121—1091no—
138#121—1091no—
139#121—1091no—
140#120—1091no—
141#120—1091no—
142C4×D4—1091no—
143C4×D4—1091no—
144C2×C22⋊C4—1091no—
145C2×C22⋊C4D10911yescharacteristic, complemented
146C24—1091no—
147C24C2×D10911yescharacteristic
148#41—2184no—
149C2×Q8—1091no—
150C2×D4—1091no—
151C2×D4—1091no—
152C2×D4—1091no—
153C2×D4—1091no—
154C2×D4—1091no—
155C2×D4—1091no—
156C2×D4—1091no—
157C2×D4—1091no—
158C2×D4—1091no—
159C2×D4—2184no—
160C22×C4—1091no—
161C22×C4—1091no—
162C22×C4—1091no—
163C22×C4—1091no—
164C22×C4C2×D10911yescharacteristic, complemented
165C22×C4C2×D10911yescharacteristic, complemented
166C4⋊C4—1091no—
167C4⋊C4—1091no—
168C4⋊C4—1091no—
169C4⋊C4—1091no—
170C22⋊C4—1091no—
171C22⋊C4—1091no—
172C22⋊C4—1091no—
173C22⋊C4—1091no—
174C22⋊C4—1091no—
175C22⋊C4—1091no—
176C22⋊C4—1091no—
177C22⋊C4—1091no—
178C22⋊C4C2×D10911yescharacteristic
179C22⋊C4C2×D10911yescharacteristic
180C22⋊C4C2×D10911yescharacteristic, complemented
181C22⋊C4C2×D10911yescharacteristic, complemented
182C42—1091no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—1091no—
187C23—1091no—
188C23—1091no—
189C23—4361no—
190C23—2182no—
191C23—2182no—
192C23C22×D10911yescharacteristic
193C23C22×D10911yescharacteristic
194C23C22×D10911yescharacteristic
195Q8—2182no—
196D4—1094no—
197D4—4361no—
198D4—4361no—
199D4—2182no—
200D4—2182no—
201D4—2182no—
202D4—2182no—
203D4—2182no—
204D4—2182no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—1091no—
208C2×C4—1091no—
209C2×C4—1091no—
210C2×C4—1091no—
211C2×C4—1091no—
212C2×C4—4361no—
213C2×C4—2182no—
214C2×C4—2182no—
215C2×C4—2182no—
216C2×C4C22×D10911yescharacteristic
217C2×C4C22×D10911yescharacteristic
218C2×C4C22×D10911yescharacteristic
219C2×C4C22×D10911yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—44no—
228C22—1092no—
229C22—4364no—
230C22—2181no—
231C22—2181no—
232C22—2181no—
233C22—2181no—
234C22—2181no—
235C22—2181no—
236C22—2181no—
237C22—2181no—
238C22D4×D10912yes—
239C22#1318202912yescomplemented
240C22C23×D10911yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—1092no—
246C4—2181no—
247C4—2181no—
248C4—2181no—
249C4—2181no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—4361no—
254C2—4361no—
255C2—2182no—
256C2C2×D4×D10911yescharacteristic
257C2C2×#1318202911yescharacteristic
258C2#1681274411yescharacteristic
subgroupG/H
center, FrattiniC22C23×D109
commutator, socleC22×C109C24
FittingC2×C22⋊C4×C109C2
hypercenterC2×C22⋊C4D109
hypocenterC109#546
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13182029C2semidirect
2. C22×C22⋊D109C2semidirect
3. #16812685C2semidirect
4. #16812682C2semidirect
5. #16812681C2semidirect
6. #16812676C2semidirect
7. #16812673C2semidirect
8. #16812672C2semidirect
9. #16812672C2semidirect
10. #16812669C2semidirect
11. #16812668C2semidirect
12. C4×C22⋊D109C2semidirect
13. #16812659C2semidirect
14. C22⋊C4×D109C2semidirect
15. C2×C22⋊C4×C109C2semidirect
18. C2×Dic218C22semidirect
19. C2×D436C22semidirect
20. C2×D436C22semidirect
27. C2×C22⋊D109C22semidirect
28. C2×C22⋊D109C22semidirect
30. C2×C4×D109C22semidirect
31. C2×C4×D109C22semidirect
32. C2×C4×D109C22semidirect
33. C4⋊Dic109C22semidirect
34. #13182008C22semidirect
35. #13182008C22semidirect
36. #13182008C22semidirect
38. C22⋊Dic109C22semidirect
39. C22⋊Dic109C22semidirect
40. #13182006C22semidirect
41. #13182006C22semidirect
42. #13182006C22semidirect
43. #13182006C22semidirect
44. #13182006C22semidirect
45. C4×Dic109C22semidirect
47. C22×C4×C109C22semidirect
48. C22×C4×C109C22semidirect
51. C22⋊C4×C109C22semidirect
52. C22⋊C4×C109C22semidirect
77. C22⋊D109D4semidirect
99. C22×C109#41semidirect
129. C109#546semidirect
145. C2×C22⋊C4D109semidirect
164. C22×C4C2×D109semidirect
165. C22×C4C2×D109semidirect
180. C22⋊C4C2×D109semidirect
181. C22⋊C4C2×D109semidirect
239. C22#13182029semidirect