Finite Group Databasebeta

← #31796150

Subgroups — #31796150

Order 17536.

246 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578425C211yesmaximal, characteristic, complemented
2#22578421C211yesmaximal, characteristic, complemented
3#22578251C214yesmaximal, complemented
4#22577954C211yesmaximal, characteristic
5#17984155—21no—
6C4⋊#11764047C2212yes—
7#17984145—21no—
8#17984145—21no—
9#17984143—21no—
10C4×#11764047C2212yes—
11C2×#14555511—21no—
12#17984085C411yescharacteristic
13#17984077—21no—
14#17984063—21no—
15#17984063C412yes—
16#17984050—21no—
17#17984050C2212yes—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic
20C2×#11764047—22no—
21C2×#11764047—22no—
22#14555511—44no—
23C2×D548—22no—
24C2×D548—21no—
25C2×D548—21no—
26C2×D548C2×C412yes—
27C2×C4×D137—22no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31#14555487—22no—
32#14555487—22no—
33C4×Dic137D411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137C2311yescharacteristic
36C2×Q8×C137—21no—
37C4⋊C4×C137—21no—
38C4⋊C4×C137C2×C412yes—
39C42×C137C2×C411yescharacteristic
40#11764047—42no—
41#11764047—42no—
42C22×D137—22no—
43C22×D137C22⋊C411yescharacteristic
44D548—24no—
45D548—42no—
46D548—42no—
47D548—44no—
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C2×D411yescharacteristic
50C2×Dic137C2×D411yescharacteristic
51C4×D137—22no—
52C4×D137—22no—
53C4×D137—41no—
54C4×D137—44no—
55Q8×C137—42no—
56C2×C4×C137—22no—
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C22×C411yescharacteristic
60C22×C137C2×C22⋊C411yescharacteristic
61C2×D137—21no—
62C2×D137—21no—
63C2×D137—21no—
64C2×D137—42no—
65C2×D137—42no—
66C2×D137—42no—
67C4×C137—21no—
68C4×C137—21no—
69C4×C137—42no—
70C4×C137#13112yes—
71Dic137—21no—
72Dic137—21no—
73Dic137AGL(1,ℤ/8)12yes—
74C2×C137#40811yescharacteristic
75C2×C137#41811yescharacteristic
76C2×C137#42111yescharacteristic
77D137—41no—
78D137—82no—
79C137#149611yescharacteristic, complemented, Sylow
80#1496—1371nomaximal, Sylow
81#532—1371no—
82#432—1371no—
83#431—1371no—
84D4⋊C8—1374no—
85C2×#41—2741no—
86C4⋊Q8#1058773911yescharacteristic
87#127—1371no—
88#124—2741no—
89#121—2741no—
90#121—1372no—
91C4×D4—2741no—
92C4×D4—1372no—
93C2×C42—1371no—
94C4⋊C8—2741no—
95C4⋊C8—1372no—
96C22⋊C8—2741no—
97C22⋊C8—2741no—
98#97—2741no—
99C4×C8—1372no—
100#41—5484no—
101C2×Q8—21no—
102C2×D4—2742no—
103C2×D4—2741no—
104C2×D4—2741no—
105C2×D4—1372no—
106C22×C4—2742no—
107C22×C4—1371no—
108C22×C4—1371no—
109C22×C4—1371no—
110C2×C8—2742no—
111C2×C8—2742no—
112C4⋊C4—21no—
113C4⋊C4C2×#1058773912yes—
114C22⋊C4—2742no—
115C22⋊C4—2742no—
116C42—1371no—
117C42—1371no—
118C42—1371no—
119C42C2×#1058773911yescharacteristic
120C23—2742no—
121C23—1371no—
122Q8—42no—
123D4—2744no—
124D4—5482no—
125D4—5482no—
126D4—5484no—
127C2×C4—22no—
128C2×C4—2742no—
129C2×C4—2742no—
130C2×C4—5481no—
131C2×C4—5484no—
132C2×C4—1371no—
133C2×C4—1371no—
134C2×C4—1371no—
135C2×C4C22⋊#1058773911yescharacteristic
136C2×C4C22⋊#1058773911yescharacteristic
137C2×C4C22×#1058773911yescharacteristic
138C8—5482no—
139C8—5482no—
140C22—2741no—
141C22—2741no—
142C22—2741no—
143C22—5482no—
144C22—5482no—
145C22—5482no—
146C22C2×C22⋊#1058773911yescharacteristic
147C4—21no—
148C4—21no—
149C4—42no—
150C4—2741no—
151C4—2741no—
152C4—1372no—
153C4#1798417612yes—
154C2—5481no—
155C2—10962no—
156C2#2257837811yescharacteristic
157C2#2257839511yescharacteristic
158C2#2257840211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC4×Dic137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1496
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578425C2semidirect
2. #22578421C2semidirect
3. #22578251C2semidirect
79. C137#1496semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225784212
#225782512
#225779542
#1496137