Finite Group Databasebeta

← #16838079

Subgroups — #16838079

Order 3776.

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#14273722C211yesmaximal, characteristic, complemented
2#14273722C211yesmaximal, characteristic, complemented
3#14273716C211yesmaximal, characteristic, complemented
4#14273712C211yesmaximal, characteristic, complemented
5#14273712C211yesmaximal, characteristic, complemented
6#14273711C211yesmaximal, characteristic, complemented
7#14273703C211yesmaximal, characteristic, complemented
8#14273697C211yesmaximal, characteristic
9#14273689C211yesmaximal, characteristic, complemented
10C4×C22⋊D59C211yesmaximal, characteristic, complemented
11#14273685C211yesmaximal, characteristic, complemented
12#14273683C211yesmaximal, characteristic, complemented
13C2×(C2×D59)⋊C4C211yesmaximal, characteristic, complemented
14C22⋊C4×D59C211yesmaximal, characteristic, complemented
15C22⋊Q8×C59C211yesmaximal, characteristic, complemented
16C23×D59C2211yescharacteristic
17C2×Q8×C59C2211yescharacteristic, complemented
18C22×C4×C59C2211yescharacteristic
19C2×C22⋊D59C2211yescharacteristic
20C22×Dic59C2211yescharacteristic
21C2×D236C2211yescharacteristic
22C2×D236C2211yescharacteristic
23C2×C4×D59C2211yescharacteristic
24C2×C4×D59C2211yescharacteristic
25C2×C4×D59C2211yescharacteristic, complemented
26C4⋊C4×C59C2211yescharacteristic, complemented
27C4⋊C4×C59C2211yescharacteristic, complemented
28C4⋊C4×C59C2211yescharacteristic, complemented
29C22⋊C4×C59C2211yescharacteristic
30C22⋊C4×C59C2211yescharacteristic
31C22⋊Dic59C2211yescharacteristic
32(C2×D59)⋊C4C2211yescharacteristic
33(C2×D59)⋊C4C2211yescharacteristic
34(C2×D59)⋊C4C2211yescharacteristic
35(C2×D59)⋊C4C2211yescharacteristic
36(C2×D59)⋊C4C2211yescharacteristic
37(C2×D59)⋊C4C2211yescharacteristic
38(C2×D59)⋊C4C2211yescharacteristic
39(C2×D59)⋊C4C2211yescharacteristic
40(C2×D59)⋊C4C2211yescharacteristic, complemented
41(C2×D59)⋊C4C2211yescharacteristic, complemented
42(C2×D59)⋊C4C2211yescharacteristic, complemented
43C4⋊Dic59C2211yescharacteristic, complemented
44C4⋊Dic59C2211yescharacteristic, complemented
45Dic59⋊C4C2211yescharacteristic, complemented
46Dic59⋊C4C2211yescharacteristic, complemented
47Dic59⋊C4C2211yescharacteristic, complemented
48C4×Dic59C2211yescharacteristic, complemented
49C4×Dic59C2211yescharacteristic, complemented
50C4×Dic59C2211yescharacteristic, complemented
51C23×C59C2311yescharacteristic
52C22×D59—22no—
53C22×D59—22no—
54C22×D59—41no—
55C22×D59C2311yescharacteristic
56C22×D59C2311yescharacteristic
57C22×D59C2311yescharacteristic
58Q8×C59—41no—
59C2×C4×C59—41no—
60C2×C4×C59C2311yescharacteristic
61C2×C4×C59C2311yescharacteristic
62C2×C4×C59C2311yescharacteristic
63C2×C4×C59C2311yescharacteristic
64C2×C4×C59C2311yescharacteristic
65C2×C4×C59C2311yescharacteristic
66C22⋊D59—22no—
67C2×Dic59—22no—
68C2×Dic59C2311yescharacteristic
69C2×Dic59C2311yescharacteristic
70C2×Dic59C2311yescharacteristic
71C2×Dic59C2311yescharacteristic
72C2×Dic59C2311yescharacteristic
73D236—22no—
74D236—41no—
75C4×D59—22no—
76C4×D59—41no—
77C4×D59—41no—
78C22×C59—21no—
79C22×C59—21no—
80C22×C59#4112yes—
81C22×C59C2411yescharacteristic
82C2×D59—21no—
83C2×D59—21no—
84C2×D59—21no—
85C2×D59—21no—
86C2×D59—21no—
87C2×D59—21no—
88C2×D59—21no—
89C2×D59—21no—
90C2×D59—44no—
91C2×D59#4112yes—
92C4×C59—21no—
93C4×C59—21no—
94C4×C59—21no—
95C4×C59—21no—
96C4×C59—21no—
97C4×C59—21no—
98Dic59—21no—
99Dic59—21no—
100Dic59—21no—
101Dic59—21no—
102Dic59—21no—
103C2×C59—22no—
104C2×C59C2×#4111yescharacteristic
105C2×C59C2×#4111yescharacteristic
106C2×C59H(5,𝔽2)11yescharacteristic
107D59—22no—
108D59—41no—
109D59—41no—
110#551—591nomaximal, Sylow
111C59#55111yescharacteristic, complemented, Sylow
112#126—591no—
113#126—591no—
114#124—591no—
115#123—591no—
116#123—591no—
117#123—591no—
118C22⋊Q8—591no—
119C22⋊Q8D5911yescharacteristic, complemented
120#120—591no—
121C4×D4—591no—
122C4×D4—591no—
123#117—591no—
124#117—591no—
125C2×C22⋊C4—591no—
126C2×C22⋊C4—591no—
127C24—591no—
128C2×Q8C2×D5911yescharacteristic, complemented
129C2×D4—591no—
130C2×D4—591no—
131C2×D4—591no—
132C22×C4—591no—
133C22×C4—591no—
134C22×C4—591no—
135C22×C4—591no—
136C22×C4C2×D5911yescharacteristic
137C4⋊C4—591no—
138C4⋊C4—591no—
139C4⋊C4—591no—
140C4⋊C4—591no—
141C4⋊C4—591no—
142C4⋊C4C2×D5911yescharacteristic, complemented
143C4⋊C4C2×D5911yescharacteristic, complemented
144C4⋊C4C2×D5911yescharacteristic, complemented
145C22⋊C4—591no—
146C22⋊C4—591no—
147C22⋊C4—591no—
148C22⋊C4—591no—
149C22⋊C4—591no—
150C22⋊C4—591no—
151C22⋊C4—591no—
152C22⋊C4—591no—
153C22⋊C4—591no—
154C22⋊C4—591no—
155C22⋊C4—591no—
156C22⋊C4—591no—
157C22⋊C4C2×D5911yescharacteristic
158C22⋊C4C2×D5911yescharacteristic
159C42—591no—
160C42—591no—
161C42—591no—
162C23—591no—
163C23—591no—
164C23—591no—
165C23—1182no—
166C23—1182no—
167C23—2361no—
168C23C22×D5911yescharacteristic
169Q8—41no—
170D4—1182no—
171D4—1182no—
172D4—2361no—
173C2×C4—41no—
174C2×C4—591no—
175C2×C4—591no—
176C2×C4—591no—
177C2×C4—591no—
178C2×C4—591no—
179C2×C4—1182no—
180C2×C4—1182no—
181C2×C4—2361no—
182C2×C4—2361no—
183C2×C4C22×D5911yescharacteristic
184C2×C4C22×D5911yescharacteristic
185C2×C4C22×D5911yescharacteristic
186C2×C4C22×D5911yescharacteristic
187C2×C4C22×D5911yescharacteristic
188C2×C4C22×D5911yescharacteristic
189C22—21no—
190C22—21no—
191C22—592no—
192C22—1181no—
193C22—1181no—
194C22—1181no—
195C22—1181no—
196C22—1181no—
197C22—1181no—
198C22—1181no—
199C22—1181no—
200C22—2364no—
201C22#1174568912yes—
202C22C23×D5911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—1181no—
210C4—1181no—
211C4—1181no—
212C4—1181no—
213C4—1181no—
214C2—22no—
215C2—1182no—
216C2—2361no—
217C2—2361no—
218C2#41×D5911yescharacteristic
219C2C2×#1174568911yescharacteristic
220C2#1427377211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D59
commutator, socle, cosocleC22×C59C24
FittingC22⋊Q8×C59C2
prime radical#14273697C2
hypercenterC22⋊Q8D59
hypocenterC59#551
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #14273722C2semidirect
2. #14273722C2semidirect
3. #14273716C2semidirect
4. #14273712C2semidirect
5. #14273712C2semidirect
6. #14273711C2semidirect
7. #14273703C2semidirect
9. #14273689C2semidirect
10. C4×C22⋊D59C2semidirect
11. #14273685C2semidirect
12. #14273683C2semidirect
13. C2×(C2×D59)⋊C4C2semidirect
14. C22⋊C4×D59C2semidirect
15. C22⋊Q8×C59C2semidirect
17. C2×Q8×C59C22semidirect
25. C2×C4×D59C22semidirect
26. C4⋊C4×C59C22semidirect
27. C4⋊C4×C59C22semidirect
28. C4⋊C4×C59C22semidirect
40. (C2×D59)⋊C4C22semidirect
41. (C2×D59)⋊C4C22semidirect
42. (C2×D59)⋊C4C22semidirect
43. C4⋊Dic59C22semidirect
44. C4⋊Dic59C22semidirect
45. Dic59⋊C4C22semidirect
46. Dic59⋊C4C22semidirect
47. Dic59⋊C4C22semidirect
48. C4×Dic59C22semidirect
49. C4×Dic59C22semidirect
50. C4×Dic59C22semidirect
111. C59#551semidirect
119. C22⋊Q8D59semidirect
128. C2×Q8C2×D59semidirect
142. C4⋊C4C2×D59semidirect
143. C4⋊C4C2×D59semidirect
144. C4⋊C4C2×D59semidirect

Maximal subgroups

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

subgroupindex
#142737222
#142737162
#142737122
#142737112
#142737032
#142736972
#142736892
C4×C22⋊D592
#142736852
#142736832
C2×(C2×D59)⋊C42
C22⋊C4×D592
C22⋊Q8×C592
#55159