Finite Group Databasebeta

← #31800641

Subgroups — #31800641

Order 17728.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17984562C211yesmaximal, characteristic, complemented
2C2×D4×D277C211yesmaximal, characteristic, complemented
3#22579866C211yesmaximal, characteristic, complemented
4#22579865C211yesmaximal, characteristic, complemented
5#22579861C211yesmaximal, characteristic, complemented
6#22579858C211yesmaximal, characteristic, complemented
7#22579853C211yesmaximal, characteristic, complemented
8#22579852C211yesmaximal, characteristic, complemented
9#22579848C211yesmaximal, characteristic, complemented
10#22579848C211yesmaximal, characteristic, complemented
11C4×C22⋊D277C211yesmaximal, characteristic, complemented
12#22579840C211yesmaximal, characteristic, complemented
13C2×#17984540C211yesmaximal, characteristic, complemented
14C22⋊C4×D277C211yesmaximal, characteristic, complemented
15#121×C277C211yesmaximal, characteristic, complemented
16C23×D277C2211yescharacteristic
17C23×D277C2211yescharacteristic
18#17984562—24no—
19C2×Dic554C2211yescharacteristic, complemented
20C2×D1108C2211yescharacteristic
21C2×C22⋊D277C2211yescharacteristic
22C2×C22⋊D277C2211yescharacteristic
23C2×C22⋊D277C2211yescharacteristic
24C2×C22⋊D277C2211yescharacteristic, complemented
25C2×C22⋊D277C2211yescharacteristic, complemented
26D4×D277—24no—
27C22×Dic277C2211yescharacteristic, complemented
28C22×Dic277C2211yescharacteristic, complemented
29C2×C4×D277C2211yescharacteristic
30C2×C4×D277C2211yescharacteristic, complemented
31C2×C4×D277C2211yescharacteristic, complemented
32#17984542C2211yescharacteristic, complemented
33#17984542C2211yescharacteristic, complemented
34#17984542C2211yescharacteristic, complemented
35C22⋊Dic277C2211yescharacteristic, complemented
36C22⋊Dic277C2211yescharacteristic, complemented
37C22⋊Dic277C2211yescharacteristic, complemented
38#17984540C2211yescharacteristic
39#17984540C2211yescharacteristic
40#17984540C2211yescharacteristic
41#17984540C2211yescharacteristic, complemented
42#17984540C2211yescharacteristic, complemented
43#17984540C2211yescharacteristic, complemented
44#17984540C2211yescharacteristic, complemented
45C4×Dic277C2211yescharacteristic, complemented
46C2×D4×C277C2211yescharacteristic
47C2×D4×C277C2211yescharacteristic, complemented
48C2×D4×C277C2211yescharacteristic, complemented
49C22×C4×C277C2211yescharacteristic, complemented
50C4⋊C4×C277C2211yescharacteristic, complemented
51C22⋊C4×C277C2211yescharacteristic, complemented
52C22⋊C4×C277C2211yescharacteristic, complemented
53C22×D277—22no—
54C22×D277—22no—
55C22×D277—22no—
56C22×D277—22no—
57C22×D277—41no—
58C22×D277—41no—
59C22×D277C2311yescharacteristic
60C22×D277C2311yescharacteristic
61C22×D277C2311yescharacteristic
62Dic554—22no—
63D1108—22no—
64C22⋊D277—22no—
65C22⋊D277—22no—
66C22⋊D277—22no—
67C22⋊D277—41no—
68C22⋊D277D414yescomplemented
69C2×Dic277—22no—
70C2×Dic277—22no—
71C2×Dic277C2311yescharacteristic
72C2×Dic277C2311yescharacteristic
73C2×Dic277C2311yescharacteristic
74C2×Dic277C2311yescharacteristic
75C2×Dic277C2311yescharacteristic
76C4×D277—22no—
77C4×D277—22no—
78C4×D277—41no—
79C23×C277C2311yescharacteristic
80C23×C277C2311yescharacteristic
81C23×C277C2311yescharacteristic
82D4×C277—22no—
83D4×C277—22no—
84D4×C277—41no—
85C2×C4×C277—41no—
86C2×C4×C277C2311yescharacteristic
87C2×C4×C277C2311yescharacteristic
88C2×C4×C277C2311yescharacteristic
89C2×C4×C277C2311yescharacteristic
90C22×C277—21no—
91C22×C277—21no—
92C22×C277—21no—
93C22×C277—21no—
94C22×C277—21no—
95C22×C277—21no—
96C22×C277—21no—
97C22×C277—21no—
98C22×C277C2×D412yes—
99C22×C277C2411yescharacteristic
100C2×D277—21no—
101C2×D277—21no—
102C2×D277—21no—
103C2×D277—21no—
104C2×D277—21no—
105C2×D277—21no—
106C2×D277—21no—
107C2×D277—44no—
108C2×D277—44no—
109C2×D277C2×D412yes—
110C2×D277#4112yescomplemented
111C4×C277—21no—
112C4×C277—21no—
113C4×C277—21no—
114C4×C277—21no—
115Dic277—21no—
116Dic277—21no—
117Dic277—21no—
118Dic277—21no—
119Dic277C2×D412yes—
120C2×C277—22no—
121C2×C277—41no—
122C2×C277—41no—
123C2×C277C22×D411yescharacteristic
124C2×C277C2×#4111yescharacteristic
125C2×C277H(5,𝔽2)11yescharacteristic
126D277—22no—
127D277—22no—
128D277—41no—
129C277#54611yescharacteristic, complemented, Sylow
130#546—2771nomaximal, Sylow
131C2×#41—2771no—
132C22×D4—2771no—
133#124—2771no—
134#123—2771no—
135#123—2771no—
136C22⋊Q8—2771no—
137#121—2771no—
138#121—2771no—
139#121D27711yescharacteristic, complemented
140#120—2771no—
141#120—2771no—
142C4×D4—2771no—
143C4×D4—2771no—
144C2×C22⋊C4—2771no—
145C2×C22⋊C4—2771no—
146C24—2771no—
147C24—2771no—
148#41—5544no—
149C2×Q8—2771no—
150C2×D4—2771no—
151C2×D4—2771no—
152C2×D4—2771no—
153C2×D4—2771no—
154C2×D4—2771no—
155C2×D4—2771no—
156C2×D4—5544no—
157C2×D4C2×D27711yescharacteristic
158C2×D4C2×D27711yescharacteristic, complemented
159C2×D4C2×D27711yescharacteristic, complemented
160C22×C4—2771no—
161C22×C4—2771no—
162C22×C4—2771no—
163C22×C4—2771no—
164C22×C4—2771no—
165C22×C4C2×D27711yescharacteristic, complemented
166C4⋊C4—2771no—
167C4⋊C4—2771no—
168C4⋊C4—2771no—
169C4⋊C4C2×D27711yescharacteristic, complemented
170C22⋊C4—2771no—
171C22⋊C4—2771no—
172C22⋊C4—2771no—
173C22⋊C4—2771no—
174C22⋊C4—2771no—
175C22⋊C4—2771no—
176C22⋊C4—2771no—
177C22⋊C4—2771no—
178C22⋊C4—2771no—
179C22⋊C4—2771no—
180C22⋊C4C2×D27711yescharacteristic, complemented
181C22⋊C4C2×D27711yescharacteristic, complemented
182C42—2771no—
183C23—2771no—
184C23—2771no—
185C23—2771no—
186C23—5542no—
187C23—5542no—
188C23—5542no—
189C23—5542no—
190C23—11081no—
191C23—11081no—
192C23C22×D27711yescharacteristic
193C23C22×D27711yescharacteristic
194C23C22×D27711yescharacteristic
195Q8—5542no—
196D4—22no—
197D4—22no—
198D4—41no—
199D4—2774no—
200D4—5542no—
201D4—5542no—
202D4—5542no—
203D4—5542no—
204D4—11081no—
205C2×C4—41no—
206C2×C4—2771no—
207C2×C4—2771no—
208C2×C4—2771no—
209C2×C4—2771no—
210C2×C4—2771no—
211C2×C4—5542no—
212C2×C4—5542no—
213C2×C4—5542no—
214C2×C4—5542no—
215C2×C4—11081no—
216C2×C4C22×D27711yescharacteristic
217C2×C4C22×D27711yescharacteristic
218C2×C4C22×D27711yescharacteristic
219C2×C4C22×D27711yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—2772no—
229C22—2772no—
230C22—5541no—
231C22—5541no—
232C22—5541no—
233C22—5541no—
234C22—5541no—
235C22—5541no—
236C22—5541no—
237C22—11084no—
238C22—11084no—
239C22D4×D27712yes—
240C22C23×D27711yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—2772no—
246C4—5541no—
247C4—5541no—
248C4—5541no—
249C4—5541no—
250C2—22no—
251C2—41no—
252C2—41no—
253C2—5542no—
254C2—5542no—
255C2—11081no—
256C2C2×D4×D27711yescharacteristic
257C2#41×D27711yescharacteristic
258C2#2257992511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D277
commutator, socle, cosocleC22×C277C24
Fitting#121×C277C2
hypercenter#121D277
hypocenterC277#546
LCSG ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#17984562C2semidirect
2. C2×D4×D277C2semidirect
3. #22579866C2semidirect
4. #22579865C2semidirect
5. #22579861C2semidirect
6. #22579858C2semidirect
7. #22579853C2semidirect
8. #22579852C2semidirect
9. #22579848C2semidirect
10. #22579848C2semidirect
11. C4×C22⋊D277C2semidirect
12. #22579840C2semidirect
13. C2×#17984540C2semidirect
14. C22⋊C4×D277C2semidirect
15. #121×C277C2semidirect
19. C2×Dic554C22semidirect
24. C2×C22⋊D277C22semidirect
25. C2×C22⋊D277C22semidirect
27. C22×Dic277C22semidirect
28. C22×Dic277C22semidirect
30. C2×C4×D277C22semidirect
31. C2×C4×D277C22semidirect
32. #17984542C22semidirect
33. #17984542C22semidirect
34. #17984542C22semidirect
35. C22⋊Dic277C22semidirect
36. C22⋊Dic277C22semidirect
37. C22⋊Dic277C22semidirect
41. #17984540C22semidirect
42. #17984540C22semidirect
43. #17984540C22semidirect
44. #17984540C22semidirect
45. C4×Dic277C22semidirect
47. C2×D4×C277C22semidirect
48. C2×D4×C277C22semidirect
49. C22×C4×C277C22semidirect
50. C4⋊C4×C277C22semidirect
51. C22⋊C4×C277C22semidirect
52. C22⋊C4×C277C22semidirect
68. C22⋊D277D4semidirect
110. C2×D277#41semidirect
129. C277#546semidirect
139. #121D277semidirect
158. C2×D4C2×D277semidirect
159. C2×D4C2×D277semidirect
165. C22×C4C2×D277semidirect
169. C4⋊C4C2×D277semidirect
180. C22⋊C4C2×D277semidirect
181. C22⋊C4C2×D277semidirect

Maximal subgroups

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

subgroupindex
C2×#179845622
C2×D4×D2772
#225798662
#225798652
#225798612
#225798582
#225798532
#225798522
#225798482
C4×C22⋊D2772
#225798402
C2×#179845402
C22⋊C4×D2772
#121×C2772
#546277