Finite Group Databasebeta

← #16838043

Subgroups — #16838043

Order 3776.

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×#11745689C211yesmaximal, characteristic, complemented
2C22×D236C211yesmaximal, characteristic, complemented
3#14273716C211yesmaximal, characteristic, complemented
4#14273712C212yesmaximal, complemented
5#14273702C211yesmaximal, characteristic, complemented
6#14273702C212yesmaximal, complemented
7#14273697C212yesmaximal, complemented
8#14273689C211yesmaximal, characteristic, complemented
9C4×C22⋊D59C211yesmaximal, characteristic, complemented
10C22⋊C4×D59C212yesmaximal, complemented
11C22⋊Q8×C59C211yesmaximal, characteristic, complemented
12C23×D59C2212yes—
13C2×Q8×C59C2211yescharacteristic, complemented
14C22×C4×C59C2211yescharacteristic
15C2×C22⋊D59C2211yescharacteristic
16#11745689—24no—
17C2×D236—24no—
18C2×D236C2212yes—
19C2×D236C2211yescharacteristic
20C2×D236C2211yescharacteristic
21C2×D236C2212yescomplemented
22C2×D236C2212yescomplemented
23C2×C4×D59C2212yescomplemented
24C2×C4×D59C2211yescharacteristic, complemented
25C2×C4×D59C2212yescomplemented
26C4⋊C4×C59C2211yescharacteristic, complemented
27C4⋊C4×C59C2212yescomplemented
28C22⋊C4×C59C2212yes—
29C22⋊Dic59C2211yescharacteristic
30(C2×D59)⋊C4C2212yescomplemented
31(C2×D59)⋊C4C2212yescomplemented
32(C2×D59)⋊C4C2211yescharacteristic, complemented
33(C2×D59)⋊C4C2212yescomplemented
34(C2×D59)⋊C4C2212yescomplemented
35Dic59⋊C4C2211yescharacteristic, complemented
36C4×Dic59C2211yescharacteristic, complemented
37C23×C59C2311yescharacteristic
38C22×D59—24no—
39C22×D59—24no—
40C22×D59—42no—
41C22×D59C2312yes—
42C22×D59C2311yescharacteristic
43C22×D59C2312yes—
44Q8×C59—22no—
45C2×C4×C59—22no—
46C2×C4×C59C2311yescharacteristic
47C2×C4×C59C2312yes—
48C2×C4×C59C2312yes—
49C2×C4×C59C2311yescharacteristic
50C22⋊D59—22no—
51C2×Dic59C2311yescharacteristic
52C2×Dic59C2311yescharacteristic
53C2×Dic59C2311yescharacteristic
54D236—24no—
55D236—24no—
56D236—22no—
57D236—42no—
58D236D414yescomplemented
59C4×D59—24no—
60C4×D59—22no—
61C4×D59—42no—
62C22×C59—21no—
63C22×C59—21no—
64C22×C59#4112yescomplemented
65C22×C59C2411yescharacteristic
66C2×D59—22no—
67C2×D59—22no—
68C2×D59—21no—
69C2×D59—22no—
70C2×D59—21no—
71C2×D59—22no—
72C2×D59—21no—
73C2×D59—22no—
74C2×D59—48no—
75C2×D59C2×D414yes—
76C4×C59—22no—
77C4×C59—21no—
78C4×C59—22no—
79C4×C59C2×D412yes—
80Dic59—21no—
81Dic59—21no—
82Dic59—21no—
83C2×C59—22no—
84C2×C59C22×D411yescharacteristic
85C2×C59C2×#4111yescharacteristic
86C2×C59H(5,𝔽2)11yescharacteristic
87D59—24no—
88D59—41no—
89D59—42no—
90#546—591nomaximal, Sylow
91C59#54611yescharacteristic, complemented, Sylow
92C2×#41—591no—
93C22×D4—591no—
94#124—591no—
95#123—592no—
96C22⋊Q8D5911yescharacteristic, complemented
97#121—591no—
98#121—592no—
99#120—592no—
100C4×D4—591no—
101C4×D4—591no—
102C2×C22⋊C4—592no—
103C24—592no—
104#41—1184no—
105C2×Q8C2×D5911yescharacteristic, complemented
106C2×D4—591no—
107C2×D4—591no—
108C2×D4—592no—
109C2×D4—592no—
110C2×D4—591no—
111C2×D4—592no—
112C2×D4—1184no—
113C22×C4—592no—
114C22×C4—591no—
115C22×C4—592no—
116C22×C4C2×D5911yescharacteristic
117C4⋊C4—591no—
118C4⋊C4C2×D5911yescharacteristic, complemented
119C4⋊C4C2×D5912yescomplemented
120C22⋊C4—591no—
121C22⋊C4—592no—
122C22⋊C4—592no—
123C22⋊C4—592no—
124C22⋊C4—591no—
125C22⋊C4—592no—
126C22⋊C4C2×D5912yes—
127C42—591no—
128C23—592no—
129C23—592no—
130C23—591no—
131C23—1184no—
132C23—1184no—
133C23—2362no—
134C23C22×D5911yescharacteristic
135Q8—22no—
136D4—594no—
137D4—1184no—
138D4—1184no—
139D4—1182no—
140D4—1182no—
141D4—2362no—
142C2×C4—22no—
143C2×C4—591no—
144C2×C4—591no—
145C2×C4—591no—
146C2×C4—1184no—
147C2×C4—1182no—
148C2×C4—2362no—
149C2×C4C22×D5912yes—
150C2×C4C22×D5912yes—
151C2×C4C22×D5911yescharacteristic
152C2×C4C22×D5911yescharacteristic
153C22—21no—
154C22—21no—
155C22—594no—
156C22—1182no—
157C22—1182no—
158C22—1181no—
159C22—1182no—
160C22—1181no—
161C22—1182no—
162C22—1181no—
163C22—1182no—
164C22—2368no—
165C22#1174568912yescomplemented
166C22C23×D5911yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—1181no—
171C4—1181no—
172C4—1181no—
173C4D4×D5912yes—
174C2—22no—
175C2—1184no—
176C2—2362no—
177C2—2361no—
178C2C2×D4×D5911yescharacteristic
179C2C2×#1174568911yescharacteristic
180C2#1427377311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D59
commutator, socle, cosocleC22×C59C24
FittingC22⋊Q8×C59C2
hypercenterC22⋊Q8D59
hypocenterC59#546
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11745689C2semidirect
2. C22×D236C2semidirect
3. #14273716C2semidirect
4. #14273712C2semidirect
5. #14273702C2semidirect
6. #14273702C2semidirect
7. #14273697C2semidirect
8. #14273689C2semidirect
9. C4×C22⋊D59C2semidirect
10. C22⋊C4×D59C2semidirect
11. C22⋊Q8×C59C2semidirect
13. C2×Q8×C59C22semidirect
21. C2×D236C22semidirect
22. C2×D236C22semidirect
23. C2×C4×D59C22semidirect
24. C2×C4×D59C22semidirect
25. C2×C4×D59C22semidirect
26. C4⋊C4×C59C22semidirect
27. C4⋊C4×C59C22semidirect
30. (C2×D59)⋊C4C22semidirect
31. (C2×D59)⋊C4C22semidirect
32. (C2×D59)⋊C4C22semidirect
33. (C2×D59)⋊C4C22semidirect
34. (C2×D59)⋊C4C22semidirect
35. Dic59⋊C4C22semidirect
36. C4×Dic59C22semidirect
58. D236D4semidirect
64. C22×C59#41semidirect
91. C59#546semidirect
96. C22⋊Q8D59semidirect
105. C2×Q8C2×D59semidirect
118. C4⋊C4C2×D59semidirect
119. C4⋊C4C2×D59semidirect
165. C22#11745689semidirect

Maximal subgroups

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

subgroupindex
C2×#117456892
C22×D2362
#142737162
#142737122
#142737022
#142736972
#142736892
C4×C22⋊D592
C22⋊C4×D592
C22⋊Q8×C592
#54659