Finite Group Databasebeta

← #31796172

Subgroups — #31796172

Order 17536.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578422C211yesmaximal, characteristic
2C4×C4⋊#10587739C211yesmaximal, characteristic
3#22578256C214yesmaximal
4#22577958C211yesmaximal, characteristic
5C2×C4⋊#10587739—21no—
6C2×C4×#10587739—21no—
7C4⋊#11764047—21no—
8#17984145C2212yes—
9C4×#11764047—21no—
10(C2×#10587739)⋊C4C2212yes—
11#17984088C411yescharacteristic, complemented
12#17984079—21no—
13#17984068C2212yes—
14#17984064—21no—
15Q8×Dic137—21no—
16#17984053C412yescomplemented
17#17984046—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4⋊#10587739—24no—
23C4×#10587739—42no—
24C2×Dic274C2×C412yes—
25C2×C4×D137D411yescharacteristic
26C2×C4×D137D411yescharacteristic
27C2×C4×D137C2311yescharacteristic
28C4⋊Dic137—22no—
29C4⋊Dic137—21no—
30C4⋊Dic137—21no—
31#14555489—22no—
32#14555487—22no—
33C4×Dic137—22no—
34C4×Dic137C2×C411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137D411yescharacteristic
37C2×Q8×C137—21no—
38C4⋊C4×C137—21no—
39C4⋊C4×C137C2×C412yes—
40C42×C137C2×C411yescharacteristic
41C2×#10587739—24no—
42C2×#10587739—42no—
43C2×#10587739—42no—
44#11764047—42no—
45C22×D137C2×D411yescharacteristic
46Dic274—42no—
47C2×Dic137—22no—
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D411yescharacteristic
51C4×D137—22no—
52C4×D137—41no—
53C4×D137SD1612yes—
54C4×D137Q1612yes—
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×D137C2×SD1611yescharacteristic
64C2×D137C2×Q1611yescharacteristic
65#10587739—44no—
66C4×C137—21no—
67C4×C137—42no—
68C4×C137#10312yes—
69C4×C137#10412yes—
70Dic137—21no—
71Dic137—21no—
72Dic137—42no—
73Dic137#10312yes—
74C2×C137#41011yescharacteristic
75C2×C137C2×#10311yescharacteristic
76C2×C137C2×#10411yescharacteristic
77D137—21no—
78D137—21no—
79C137#150311yescharacteristic, complemented, Sylow
80#1503—1371nomaximal, Sylow
81#533—1371no—
82#431—1371no—
83C4×C4⋊C4—1371no—
84#328—1374no—
85C4⋊Q8—1371no—
86C4⋊Q8#1058773911yescharacteristic, complemented
87#125—2741no—
88C22⋊Q8—2741no—
89C22⋊Q8—1372no—
90C4×Q8—2741no—
91C4×Q8—1372no—
92#117—2741no—
93C2×C4⋊C4—2741no—
94C2×C42—2741no—
95C2×C42—1371no—
96C4⋊C8—2741no—
97C22⋊C8—1372no—
98C4×C8—2741no—
99#95—1372no—
100C2×Q8—21no—
101C2×Q8—1372no—
102C22×C4—2742no—
103C22×C4—1371no—
104C22×C4—1371no—
105C22×C4—1371no—
106C2×C8—2742no—
107C4⋊C4—21no—
108C4⋊C4—2742no—
109C4⋊C4—2742no—
110C4⋊C4—2744no—
111C4⋊C4—2741no—
112C4⋊C4—2741no—
113C4⋊C4C2×#1058773912yes—
114C22⋊C4—2742no—
115C42—2742no—
116C42—5482no—
117C42—1371no—
118C42—1371no—
119C42—1371no—
120C42C2×#1058773911yescharacteristic
121C23—1371no—
122Q8—42no—
123Q8—5482no—
124C2×C4—22no—
125C2×C4—2742no—
126C2×C4—2744no—
127C2×C4—2742no—
128C2×C4—5481no—
129C2×C4—5482no—
130C2×C4—5482no—
131C2×C4—1371no—
132C2×C4—1371no—
133C2×C4—1371no—
134C2×C4—1372no—
135C2×C4—1372no—
136C2×C4C22⋊#1058773911yescharacteristic
137C2×C4C22⋊#1058773911yescharacteristic
138C2×C4C22×#1058773911yescharacteristic
139C8—5482no—
140C22—2741no—
141C22—2741no—
142C22—1371no—
143C22—1371no—
144C22C2×C22⋊#1058773911yescharacteristic
145C4—21no—
146C4—42no—
147C4—2741no—
148C4—2741no—
149C4—5484no—
150C4—5482no—
151C4—1372no—
152C4#1798415112yes—
153C4#1798415212yes—
154C2—2741no—
155C2—2741no—
156C2#2257838411yescharacteristic
157C2C2×#1798415111yescharacteristic
158C2C2×#1798415211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C2×D4
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1503
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
11. #17984088C4semidirect
16. #17984053C4semidirect
19. C4⋊Q8×C137C4semidirect
79. C137#1503semidirect
86. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784222
C4×C4⋊#105877392
#225782562
#225779582
#1503137