Finite Group Databasebeta

← #31796260

Subgroups — #31796260

Order 17536.

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

normal subgroups only

This group has 256 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#22578362C211yesmaximal, characteristic, complemented
3#22578259C214yesmaximal, complemented
4#22577954C211yesmaximal, characteristic
5C2×#14555518—21no—
6C4⋊#11764047—21no—
7#17984145—21no—
8#17984143—21no—
9#17984143C2212yes—
10#17984143C2212yes—
11C2×#14555511C412yes—
12#17984085C411yescharacteristic
13#17984077C2212yes—
14#17984063—21no—
15#17984063—21no—
16#17984050—21no—
17#17984050—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic
20#14555518—24no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23#14555511—44no—
24C2×D548—22no—
25C2×D548—21no—
26C2×D548—21no—
27C2×D548C2×C412yes—
28C2×C4×D137—22no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32#14555487—22no—
33#14555487—22no—
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137C2311yescharacteristic
37C2×Q8×C137C2×C412yes—
38C4⋊C4×C137—21no—
39C4⋊C4×C137—21no—
40C42×C137C2×C411yescharacteristic
41#11764047—24no—
42#11764047—42no—
43C22×D137—22no—
44C22×D137C22⋊C411yescharacteristic
45D548—42no—
46D548—42no—
47D548—44no—
48D548—42no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D411yescharacteristic
51C2×Dic137C2×D411yescharacteristic
52C4×D137—22no—
53C4×D137—22no—
54C4×D137—41no—
55C4×D137—44no—
56Q8×C137—42no—
57C2×C4×C137—22no—
58C2×C4×C137C22⋊C411yescharacteristic
59C2×C4×C137C22⋊C411yescharacteristic
60C2×C4×C137C22×C411yescharacteristic
61C22×C137C2×C22⋊C411yescharacteristic
62C2×D137—21no—
63C2×D137—21no—
64C2×D137—42no—
65C2×D137—42no—
66C2×D137—42no—
67C2×D137#10012yes—
68C4×C137—21no—
69C4×C137—21no—
70C4×C137—42no—
71C4×C137#10412yes—
72Dic137—21no—
73Dic137—21no—
74Dic137—21no—
75C2×C137C2×#10011yescharacteristic
76C2×C137C2×#10411yescharacteristic
77C2×C137#42111yescharacteristic
78D137—22no—
79D137—82no—
80C137#153011yescharacteristic, complemented, Sylow
81#1530—1371nomaximal, Sylow
82#532—1371no—
83#432—1371no—
84C2×#97—1371no—
85#329—1374no—
86C2×#41—1372no—
87C22×C8—2741no—
88C4⋊Q8#1058773911yescharacteristic
89#127—1371no—
90#124—1372no—
91#121—2741no—
92#121—2741no—
93C4×D4—2741no—
94C4×D4—2741no—
95C2×C42—1371no—
96C4⋊C8—2741no—
97C22⋊C8—2741no—
98#97—2741no—
99#97—1372no—
100#97—1372no—
101#41—5484no—
102C2×Q8C2×#1058773912yes—
103C2×D4—2742no—
104C2×D4—2741no—
105C2×D4—2741no—
106C2×D4—1372no—
107C22×C4—2742no—
108C22×C4—1371no—
109C22×C4—1371no—
110C22×C4—1371no—
111C2×C8—2744no—
112C2×C8—2742no—
113C2×C8—2742no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C22⋊C4—2742no—
117C22⋊C4—2742no—
118C42—1371no—
119C42—1371no—
120C42—1371no—
121C42C2×#1058773911yescharacteristic
122C23—2742no—
123C23—1371no—
124Q8—42no—
125D4—5482no—
126D4—5482no—
127D4—5484no—
128D4—5482no—
129C2×C4—22no—
130C2×C4—2742no—
131C2×C4—2742no—
132C2×C4—5481no—
133C2×C4—5484no—
134C2×C4—1371no—
135C2×C4—1371no—
136C2×C4—1371no—
137C2×C4C22⋊#1058773911yescharacteristic
138C2×C4C22⋊#1058773911yescharacteristic
139C2×C4C22×#1058773911yescharacteristic
140C8—2744no—
141C8—5482no—
142C22—2741no—
143C22—2741no—
144C22—5482no—
145C22—5482no—
146C22—5482no—
147C22—1372no—
148C22C2×C22⋊#1058773911yescharacteristic
149C4—21no—
150C4—21no—
151C4—42no—
152C4—2741no—
153C4—2741no—
154C4—2741no—
155C4#1798415312yes—
156C2—2742no—
157C2—10962no—
158C2D137⋊#10011yescharacteristic
159C2C2×#1798415311yescharacteristic
160C2#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#1530
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578425C2semidirect
2. #22578362C2semidirect
3. #22578259C2semidirect
80. C137#1530semidirect

Maximal subgroups

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

subgroupindex
#225784252
#225783622
#225782592
#225779542
#1530137