Finite Group Databasebeta

← #34036766

Subgroups — #34036766

Order 18752.

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×#17989801C211yesmaximal, characteristic, complemented
2C22×C22⋊D293C211yesmaximal, characteristic, complemented
3#22627610C211yesmaximal, characteristic, complemented
4#22627608C211yesmaximal, characteristic, complemented
5#22627606C211yesmaximal, characteristic, complemented
6#22627601C211yesmaximal, characteristic, complemented
7#22627596C211yesmaximal, characteristic, complemented
8#22627596C211yesmaximal, characteristic, complemented
9#22627595C211yesmaximal, characteristic, complemented
10#22627592C211yesmaximal, characteristic, complemented
11D4×Dic293C211yesmaximal, characteristic, complemented
12#22627584C211yesmaximal, characteristic, complemented
13C2×C22⋊Dic293C211yesmaximal, characteristic, complemented
14C22⋊C4×D293C211yesmaximal, characteristic, complemented
15#120×C293C211yesmaximal, characteristic, complemented
16C23×D293C2211yescharacteristic
17#17989801—24no—
18C2×Dic586C2211yescharacteristic, complemented
19C2×C22⋊D293—24no—
20C2×C22⋊D293C2211yescharacteristic
21C2×C22⋊D293C2211yescharacteristic
22C2×C22⋊D293C2211yescharacteristic
23C2×C22⋊D293C2211yescharacteristic
24C2×C22⋊D293C2211yescharacteristic, complemented
25C2×C22⋊D293C2211yescharacteristic, complemented
26C22×Dic293C2211yescharacteristic
27C22×Dic293C2211yescharacteristic, complemented
28C22×Dic293C2211yescharacteristic, complemented
29C22×Dic293C2211yescharacteristic, complemented
30C2×C4×D293C2211yescharacteristic, complemented
31C2×C4×D293C2211yescharacteristic, complemented
32C4⋊Dic293C2211yescharacteristic, complemented
33C4⋊Dic293C2211yescharacteristic, complemented
34#17989781C2211yescharacteristic, complemented
35#17989781C2211yescharacteristic, complemented
36C22⋊Dic293C2211yescharacteristic
37C22⋊Dic293C2211yescharacteristic
38C22⋊Dic293C2211yescharacteristic, complemented
39C22⋊Dic293C2211yescharacteristic, complemented
40C22⋊Dic293C2211yescharacteristic, complemented
41#17989779C2211yescharacteristic, complemented
42#17989779C2211yescharacteristic, complemented
43#17989779C2211yescharacteristic, complemented
44#17989779C2211yescharacteristic, complemented
45C4×Dic293C2211yescharacteristic, complemented
46C24×C293C2211yescharacteristic
47C2×D4×C293C2211yescharacteristic
48C2×D4×C293C2211yescharacteristic, complemented
49C2×D4×C293C2211yescharacteristic, complemented
50C22⋊C4×C293C2211yescharacteristic
51C22⋊C4×C293C2211yescharacteristic, complemented
52C22⋊C4×C293C2211yescharacteristic, complemented
53C22×D293—22no—
54C22×D293—22no—
55C22×D293—41no—
56C22×D293C2311yescharacteristic
57C22×D293C2311yescharacteristic
58Dic586—22no—
59C22⋊D293—22no—
60C22⋊D293—22no—
61C22⋊D293—22no—
62C22⋊D293—22no—
63C22⋊D293—41no—
64C22⋊D293D414yescomplemented
65C2×Dic293—22no—
66C2×Dic293—22no—
67C2×Dic293—22no—
68C2×Dic293—41no—
69C2×Dic293C2311yescharacteristic
70C2×Dic293C2311yescharacteristic
71C2×Dic293C2311yescharacteristic
72C2×Dic293C2311yescharacteristic
73C2×Dic293C2311yescharacteristic
74C2×Dic293C2311yescharacteristic
75C4×D293—22no—
76C4×D293—41no—
77C23×C293—22no—
78C23×C293—22no—
79C23×C293—41no—
80C23×C293C2311yescharacteristic
81C23×C293C2311yescharacteristic
82C23×C293C2311yescharacteristic
83C23×C293C2311yescharacteristic
84D4×C293—22no—
85D4×C293—22no—
86D4×C293—41no—
87C2×C4×C293C2311yescharacteristic
88C2×C4×C293C2311yescharacteristic
89C2×C4×C293C2311yescharacteristic
90C22×C293—21no—
91C22×C293—21no—
92C22×C293—21no—
93C22×C293—21no—
94C22×C293—21no—
95C22×C293—21no—
96C22×C293—21no—
97C22×C293—21no—
98C22×C293—21no—
99C22×C293—21no—
100C22×C293—44no—
101C22×C293C2×D412yes—
102C22×C293#4112yescomplemented
103C22×C293C2411yescharacteristic
104C2×D293—21no—
105C2×D293—21no—
106C2×D293—21no—
107C2×D293—21no—
108C2×D293—21no—
109C2×D293—44no—
110C2×D293C2×D412yes—
111C4×C293—21no—
112C4×C293—21no—
113C4×C293—21no—
114Dic293—21no—
115Dic293—21no—
116Dic293—21no—
117Dic293—21no—
118Dic293—21no—
119Dic293C2×D412yes—
120C2×C293—22no—
121C2×C293—22no—
122C2×C293—41no—
123C2×C293—41no—
124C2×C293C22×D411yescharacteristic
125C2×C293C2×#4111yescharacteristic
126C2×C293H(5,𝔽2)11yescharacteristic
127D293—22no—
128D293—41no—
129C293#54611yescharacteristic, complemented, Sylow
130#546—2931nomaximal, Sylow
131C2×#41—2931no—
132C22×D4—2931no—
133#124—2931no—
134#123—2931no—
135#123—2931no—
136C22⋊Q8—2931no—
137#121—2931no—
138#121—2931no—
139#121—2931no—
140#120—2931no—
141#120D29311yescharacteristic, complemented
142C4×D4—2931no—
143C4×D4—2931no—
144C2×C22⋊C4—2931no—
145C2×C22⋊C4—2931no—
146C24—2931no—
147C24C2×D29311yescharacteristic
148#41—5864no—
149C2×Q8—2931no—
150C2×D4—2931no—
151C2×D4—2931no—
152C2×D4—2931no—
153C2×D4—2931no—
154C2×D4—2931no—
155C2×D4—2931no—
156C2×D4—5864no—
157C2×D4C2×D29311yescharacteristic
158C2×D4C2×D29311yescharacteristic, complemented
159C2×D4C2×D29311yescharacteristic, complemented
160C22×C4—2931no—
161C22×C4—2931no—
162C22×C4—2931no—
163C22×C4—2931no—
164C22×C4—2931no—
165C22×C4—2931no—
166C4⋊C4—2931no—
167C4⋊C4—2931no—
168C4⋊C4—2931no—
169C4⋊C4—2931no—
170C22⋊C4—2931no—
171C22⋊C4—2931no—
172C22⋊C4—2931no—
173C22⋊C4—2931no—
174C22⋊C4—2931no—
175C22⋊C4—2931no—
176C22⋊C4—2931no—
177C22⋊C4—2931no—
178C22⋊C4—2931no—
179C22⋊C4C2×D29311yescharacteristic
180C22⋊C4C2×D29311yescharacteristic, complemented
181C22⋊C4C2×D29311yescharacteristic, complemented
182C42—2931no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—2931no—
187C23—2931no—
188C23—5862no—
189C23—5862no—
190C23—11721no—
191C23C22×D29311yescharacteristic
192C23C22×D29311yescharacteristic
193C23C22×D29311yescharacteristic
194C23C22×D29311yescharacteristic
195Q8—5862no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—2934no—
200D4—5862no—
201D4—5862no—
202D4—5862no—
203D4—5862no—
204D4—11721no—
205C2×C4—2931no—
206C2×C4—2931no—
207C2×C4—2931no—
208C2×C4—2931no—
209C2×C4—2931no—
210C2×C4—2931no—
211C2×C4—5862no—
212C2×C4—5862no—
213C2×C4—5862no—
214C2×C4—5862no—
215C2×C4—11721no—
216C2×C4—11721no—
217C2×C4C22×D29311yescharacteristic
218C2×C4C22×D29311yescharacteristic
219C2×C4C22×D29311yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—44no—
231C22—2932no—
232C22—5861no—
233C22—5861no—
234C22—5861no—
235C22—5861no—
236C22—5861no—
237C22—11724no—
238C22D4×D29312yes—
239C22#1798980112yescomplemented
240C22C23×D29311yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—2932no—
245C4—5861no—
246C4—5861no—
247C4—5861no—
248C4—5861no—
249C4—5861no—
250C2—22no—
251C2—22no—
252C2—41no—
253C2—41no—
254C2—5862no—
255C2—11721no—
256C2C2×D4×D29311yescharacteristic
257C2C2×#1798980111yescharacteristic
258C2#2262766911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D293
commutator, socle, cosocleC22×C293C24
Fitting#120×C293C2
hypercenter#120D293
hypocenterC293#546
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#17989801C2semidirect
2. C22×C22⋊D293C2semidirect
3. #22627610C2semidirect
4. #22627608C2semidirect
5. #22627606C2semidirect
6. #22627601C2semidirect
7. #22627596C2semidirect
8. #22627596C2semidirect
9. #22627595C2semidirect
10. #22627592C2semidirect
11. D4×Dic293C2semidirect
12. #22627584C2semidirect
13. C2×C22⋊Dic293C2semidirect
14. C22⋊C4×D293C2semidirect
15. #120×C293C2semidirect
18. C2×Dic586C22semidirect
24. C2×C22⋊D293C22semidirect
25. C2×C22⋊D293C22semidirect
27. C22×Dic293C22semidirect
28. C22×Dic293C22semidirect
29. C22×Dic293C22semidirect
30. C2×C4×D293C22semidirect
31. C2×C4×D293C22semidirect
32. C4⋊Dic293C22semidirect
33. C4⋊Dic293C22semidirect
34. #17989781C22semidirect
35. #17989781C22semidirect
38. C22⋊Dic293C22semidirect
39. C22⋊Dic293C22semidirect
40. C22⋊Dic293C22semidirect
41. #17989779C22semidirect
42. #17989779C22semidirect
43. #17989779C22semidirect
44. #17989779C22semidirect
45. C4×Dic293C22semidirect
48. C2×D4×C293C22semidirect
49. C2×D4×C293C22semidirect
51. C22⋊C4×C293C22semidirect
52. C22⋊C4×C293C22semidirect
64. C22⋊D293D4semidirect
102. C22×C293#41semidirect
129. C293#546semidirect
141. #120D293semidirect
158. C2×D4C2×D293semidirect
159. C2×D4C2×D293semidirect
180. C22⋊C4C2×D293semidirect
181. C22⋊C4C2×D293semidirect
239. C22#17989801semidirect

Maximal subgroups

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

subgroupindex
C2×#179898012
C22×C22⋊D2932
#226276102
#226276082
#226276062
#226276012
#226275962
#226275952
#226275922
D4×Dic2932
#226275842
C2×C22⋊Dic2932
C22⋊C4×D2932
#120×C2932
#546293