Finite Group Databasebeta

← #17989261

Subgroups — #17989261

Order 4672.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#14556586C211yesmaximal, characteristic, complemented
2#14556585C211yesmaximal, characteristic, complemented
3#14556579C211yesmaximal, characteristic, complemented
4#14556576C211yesmaximal, characteristic, complemented
5#14556576C211yesmaximal, characteristic, complemented
6#14556574C211yesmaximal, characteristic, complemented
7#14556569C211yesmaximal, characteristic, complemented
8#14556568C211yesmaximal, characteristic, complemented
9#14556561C211yesmaximal, characteristic
10C4×D292C211yesmaximal, characteristic, complemented
11#14556551C211yesmaximal, characteristic, complemented
12#14556548C211yesmaximal, characteristic, complemented
13C2×#11924005C211yesmaximal, characteristic, complemented
14C22⋊C4×D73C211yesmaximal, characteristic, complemented
15#117×C73C211yesmaximal, characteristic, complemented
16C23×D73C2211yescharacteristic
17C2×Dic146C2211yescharacteristic, complemented
18C2×D292C2211yescharacteristic
19C2×D292C2211yescharacteristic
20C2×C22⋊D73C2211yescharacteristic
21C22×Dic73C2211yescharacteristic
22C2×C4×D73C2211yescharacteristic
23C2×C4×D73C2211yescharacteristic
24C2×C4×D73C2211yescharacteristic, complemented
25C4⋊Dic73C2211yescharacteristic, complemented
26#11924007C2211yescharacteristic, complemented
27#11924007C2211yescharacteristic, complemented
28#11924007C2211yescharacteristic, complemented
29#11924007C2211yescharacteristic, complemented
30#11924007C2211yescharacteristic, complemented
31C22⋊Dic73C2211yescharacteristic
32#11924005C2211yescharacteristic
33#11924005C2211yescharacteristic
34#11924005C2211yescharacteristic
35#11924005C2211yescharacteristic
36#11924005C2211yescharacteristic
37#11924005C2211yescharacteristic
38#11924005C2211yescharacteristic
39#11924005C2211yescharacteristic
40#11924005C2211yescharacteristic, complemented
41#11924005C2211yescharacteristic, complemented
42#11924005C2211yescharacteristic, complemented
43C4×Dic73C2211yescharacteristic, complemented
44C22×C4×C73C2211yescharacteristic
45C4⋊C4×C73C2211yescharacteristic, complemented
46C4⋊C4×C73C2211yescharacteristic, complemented
47C22⋊C4×C73C2211yescharacteristic
48C22⋊C4×C73C2211yescharacteristic
49C42×C73C2211yescharacteristic, complemented
50C42×C73C2211yescharacteristic, complemented
51C23×C73C2311yescharacteristic
52C22×D73—22no—
53C22×D73—22no—
54C22×D73—41no—
55C22×D73C2311yescharacteristic
56C22×D73C2311yescharacteristic
57C22×D73C2311yescharacteristic
58C2×C4×C73—22no—
59C2×C4×C73C2311yescharacteristic
60C2×C4×C73C2311yescharacteristic
61C2×C4×C73C2311yescharacteristic
62C2×C4×C73C2311yescharacteristic
63C2×C4×C73C2311yescharacteristic
64C2×C4×C73C2311yescharacteristic
65C22⋊D73—22no—
66C2×Dic73—41no—
67C2×Dic73C2311yescharacteristic
68C2×Dic73C2311yescharacteristic
69C2×Dic73C2311yescharacteristic
70C2×Dic73C2311yescharacteristic
71C2×Dic73C2311yescharacteristic
72D292—22no—
73D292—41no—
74C4×D73—22no—
75C4×D73—41no—
76C4×D73—41no—
77Dic146—41no—
78C22×C73—21no—
79C22×C73—21no—
80C22×C73#4112yes—
81C22×C73C2411yescharacteristic
82C2×D73—21no—
83C2×D73—21no—
84C2×D73—21no—
85C2×D73—21no—
86C2×D73—21no—
87C2×D73—21no—
88C2×D73—21no—
89C2×D73—21no—
90C2×D73—44no—
91C2×D73#4112yes—
92C4×C73—21no—
93C4×C73—21no—
94C4×C73—21no—
95C4×C73—21no—
96C4×C73—21no—
97C4×C73—21no—
98Dic73—21no—
99Dic73—21no—
100Dic73—21no—
101Dic73—21no—
102Dic73—21no—
103C2×C73—22no—
104C2×C73C2×#4111yescharacteristic
105C2×C73C2×#4111yescharacteristic
106C2×C73H(5,𝔽2)11yescharacteristic
107D73—22no—
108D73—41no—
109D73—41no—
110C73#55111yescharacteristic, complemented, Sylow
111#551—731nomaximal, Sylow
112#126—731no—
113#126—731no—
114#124—731no—
115#123—731no—
116#123—731no—
117#123—731no—
118C22⋊Q8—731no—
119C22⋊Q8—731no—
120#120—731no—
121C4×D4—731no—
122C4×D4—731no—
123#117—731no—
124#117D7311yescharacteristic, complemented
125C2×C22⋊C4—731no—
126C2×C22⋊C4—731no—
127C24—731no—
128C2×Q8—731no—
129C2×D4—731no—
130C2×D4—731no—
131C2×D4—731no—
132C22×C4—731no—
133C22×C4—731no—
134C22×C4—731no—
135C22×C4—731no—
136C22×C4C2×D7311yescharacteristic
137C4⋊C4—731no—
138C4⋊C4—731no—
139C4⋊C4—731no—
140C4⋊C4—731no—
141C4⋊C4—731no—
142C4⋊C4—731no—
143C4⋊C4C2×D7311yescharacteristic, complemented
144C4⋊C4C2×D7311yescharacteristic, complemented
145C22⋊C4—731no—
146C22⋊C4—731no—
147C22⋊C4—731no—
148C22⋊C4—731no—
149C22⋊C4—731no—
150C22⋊C4—731no—
151C22⋊C4—731no—
152C22⋊C4—731no—
153C22⋊C4—731no—
154C22⋊C4—731no—
155C22⋊C4—731no—
156C22⋊C4—731no—
157C22⋊C4C2×D7311yescharacteristic
158C22⋊C4C2×D7311yescharacteristic
159C42—731no—
160C42C2×D7311yescharacteristic, complemented
161C42C2×D7311yescharacteristic, complemented
162C23—2921no—
163C23—731no—
164C23—731no—
165C23—731no—
166C23—1462no—
167C23—1462no—
168C23C22×D7311yescharacteristic
169Q8—2921no—
170D4—2921no—
171D4—1462no—
172D4—1462no—
173C2×C4—22no—
174C2×C4—2921no—
175C2×C4—2921no—
176C2×C4—2921no—
177C2×C4—731no—
178C2×C4—731no—
179C2×C4—731no—
180C2×C4—731no—
181C2×C4—731no—
182C2×C4—1462no—
183C2×C4C22×D7311yescharacteristic
184C2×C4C22×D7311yescharacteristic
185C2×C4C22×D7311yescharacteristic
186C2×C4C22×D7311yescharacteristic
187C2×C4C22×D7311yescharacteristic
188C2×C4C22×D7311yescharacteristic
189C22—21no—
190C22—21no—
191C22—2924no—
192C22—732no—
193C22—1461no—
194C22—1461no—
195C22—1461no—
196C22—1461no—
197C22—1461no—
198C22—1461no—
199C22—1461no—
200C22—1461no—
201C22#1192402812yes—
202C22C23×D7311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—1461no—
210C4—1461no—
211C4—1461no—
212C4—1461no—
213C4—1461no—
214C2—22no—
215C2—2921no—
216C2—2921no—
217C2—1462no—
218C2#41×D7311yescharacteristic
219C2C2×#1192402811yescharacteristic
220C2#1455663711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D73
commutator, socle, cosocleC22×C73C24
Fitting#117×C73C2
prime radical#14556561C2
hypercenter#117D73
hypocenterC73#551
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #14556586C2semidirect
2. #14556585C2semidirect
3. #14556579C2semidirect
4. #14556576C2semidirect
5. #14556576C2semidirect
6. #14556574C2semidirect
7. #14556569C2semidirect
8. #14556568C2semidirect
10. C4×D292C2semidirect
11. #14556551C2semidirect
12. #14556548C2semidirect
13. C2×#11924005C2semidirect
14. C22⋊C4×D73C2semidirect
15. #117×C73C2semidirect
17. C2×Dic146C22semidirect
24. C2×C4×D73C22semidirect
25. C4⋊Dic73C22semidirect
26. #11924007C22semidirect
27. #11924007C22semidirect
28. #11924007C22semidirect
29. #11924007C22semidirect
30. #11924007C22semidirect
40. #11924005C22semidirect
41. #11924005C22semidirect
42. #11924005C22semidirect
43. C4×Dic73C22semidirect
45. C4⋊C4×C73C22semidirect
46. C4⋊C4×C73C22semidirect
49. C42×C73C22semidirect
50. C42×C73C22semidirect
110. C73#551semidirect
124. #117D73semidirect
143. C4⋊C4C2×D73semidirect
144. C4⋊C4C2×D73semidirect
160. C42C2×D73semidirect
161. C42C2×D73semidirect

Maximal subgroups

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

subgroupindex
#145565862
#145565852
#145565792
#145565762
#145565742
#145565692
#145565682
#145565612
C4×D2922
#145565512
#145565482
C2×#119240052
C22⋊C4×D732
#117×C732
#55173