Finite Group Databasebeta

← #31797067

Subgroups — #31797067

Order 17536.

286 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D137⋊#99C212yesmaximal, complemented
2#22578309C212yesmaximal, complemented
3#22578308C212yesmaximal, complemented
4#22577971C211yesmaximal, characteristic
5C2×C22⋊#10587739—22no—
6#17984147C2214yes—
7#17984146—22no—
8C2×#14555511C411yescharacteristic, complemented
9C2×D4×D137C2211yescharacteristic
10#17984085C2211yescharacteristic
11#17984084C411yescharacteristic, complemented
12#17984077C2211yescharacteristic
13#17984061—22no—
14#17984058—22no—
15#17984045C411yescharacteristic, complemented
16#124×C137C411yescharacteristic, complemented
17C22×#10587739—42no—
18C22⋊#10587739—24no—
19C22⋊#10587739—42no—
20C23×D137D412yes—
21#14555511—42no—
22C2×D548—21no—
23C2×D548—22no—
24C2×D548C2×C411yescharacteristic, complemented
25C2×D548C2311yescharacteristic
26C2×C22⋊D137—22no—
27C2×C22⋊D137D412yes—
28D4×D137—42no—
29C2×C4×D137—21no—
30C2×C4×D137C2×C411yescharacteristic
31#14555489—21no—
32#14555487—21no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic, complemented
35C2×Q8×C137C2×C411yescharacteristic, complemented
36C2×D4×C137C2×C411yescharacteristic
37C22⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic, complemented
39C2×#10587739—44no—
40C2×#10587739—84no—
41C22×D137—22no—
42C22×D137—22no—
43C22×D137—21no—
44C22×D137—82no—
45C22×D137C22⋊C411yescharacteristic
46C22×D137C2×D412yes—
47D548—41no—
48D548—41no—
49D548—42no—
50D548—82no—
51C22⋊D137—42no—
52C22⋊D137—82no—
53C2×Dic137—21no—
54C2×Dic137C22⋊C411yescharacteristic
55C4×D137—22no—
56C4×D137—42no—
57C23×C137C22⋊C412yes—
58Q8×C137—41no—
59D4×C137—41no—
60C2×C4×C137—21no—
61C2×C4×C137—21no—
62C2×C4×C137C22×C411yescharacteristic
63C22×C137—22no—
64C22×C137—42no—
65C22×C137C2×C22⋊C411yescharacteristic
66C2×D137—22no—
67C2×D137—41no—
68C2×D137—44no—
69C2×D137—42no—
70C2×D137—42no—
71C2×D137—41no—
72C2×D137—82no—
73C2×D137#9912yes—
74#10587739—84no—
75C4×C137—21no—
76C4×C137—41no—
77C4×C137—41no—
78Dic137—21no—
79Dic137—41no—
80C2×C137—21no—
81C2×C137—42no—
82C2×C137C2×#9911yescharacteristic
83D137—22no—
84D137—42no—
85D137—81no—
86C137#212711yescharacteristic, complemented, Sylow
87#2127—1371nomaximal, Sylow
88#535—1371no—
89C2×#99—1372no—
90#354—1372no—
91#353—1372no—
92C2×#41—1371no—
93C22×D4—1371no—
94#127—1371no—
95#127—1371no—
96#124—1371no—
97#124#1058773911yescharacteristic, complemented
98#121—2742no—
99#120—2742no—
100#117—1371no—
101C2×C22⋊C4—2742no—
102#99—2742no—
103#99—1374no—
104C24—1372no—
105#41—5482no—
106C2×Q8C2×#1058773911yescharacteristic, complemented
107C2×D4—2741no—
108C2×D4—2742no—
109C2×D4—2742no—
110C2×D4—5482no—
111C2×D4—1372no—
112C2×D4—1371no—
113C2×D4—1371no—
114C2×D4C2×#1058773911yescharacteristic
115C22×C4—2741no—
116C22×C4—5482no—
117C22×C4—1371no—
118C4⋊C4—2741no—
119C22⋊C4—22no—
120C22⋊C4—2741no—
121C22⋊C4—2742no—
122C22⋊C4—2744no—
123C22⋊C4—5482no—
124C42—1371no—
125C42C2×#1058773911yescharacteristic, complemented
126C23—2742no—
127C23—2742no—
128C23—2741no—
129C23—10962no—
130C23—1372no—
131C23—1371no—
132C23C22⋊#1058773912yes—
133Q8—41no—
134D4—41no—
135D4—5481no—
136D4—5481no—
137D4—5482no—
138D4—5482no—
139D4—10962no—
140D4—10962no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—2742no—
144C2×C4—2741no—
145C2×C4—5482no—
146C2×C4—5484no—
147C2×C4—10964no—
148C2×C4—1371no—
149C2×C4C22×#1058773911yescharacteristic
150C22—22no—
151C22—42no—
152C22—2742no—
153C22—5481no—
154C22—5482no—
155C22—5482no—
156C22—5484no—
157C22—5481no—
158C22—10962no—
159C22—1372no—
160C22C2×C22⋊#1058773911yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—2741no—
165C4—5481no—
166C4—10964no—
167C2—21no—
168C2—42no—
169C2—2742no—
170C2—5482no—
171C2—10961no—
172C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137C2×#99
prime radical#22577971C2
cosocleC2×D548C23
hypercenter#124#10587739
hypocenterC137#2127
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D137⋊#99C2semidirect
2. #22578309C2semidirect
3. #22578308C2semidirect
8. C2×#14555511C4semidirect
11. #17984084C4semidirect
15. #17984045C4semidirect
16. #124×C137C4semidirect
24. C2×D548C2×C4semidirect
34. C4×Dic137C2×C4semidirect
35. C2×Q8×C137C2×C4semidirect
38. C42×C137C2×C4semidirect
86. C137#2127semidirect
97. #124#10587739semidirect
106. C2×Q8C2×#10587739semidirect
125. C42C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#992
#225783092
#225783082
#225779712
#2127137