Finite Group Databasebeta

← #31797087

Subgroups — #31797087

Order 17536.

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

normal subgroups only

This group has 230 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⋊#101C212yesmaximal, complemented
2#22578313C212yesmaximal, complemented
3#22578312C212yesmaximal, complemented
4#22577978C211yesmaximal, characteristic
5D137⋊M16—22no—
6#17984149—22no—
7#17984149C2214yes—
8C2×#14555511C411yescharacteristic
9C2×Q8×D137C2211yescharacteristic
10#17984086C2211yescharacteristic
11#17984077C411yescharacteristic
12#17984076C2211yescharacteristic
13#17984073—22no—
14#17984066—22no—
15#17984045C411yescharacteristic
16C4⋊Q8×C137C411yescharacteristic
17#14555520—42no—
18#14555519—24no—
19#14555518—42no—
20#14555511—42no—
21C2×Dic274D412yes—
22C2×Dic274C2311yescharacteristic
23Q8×D137—42no—
24C2×D548—21no—
25C2×D548C2×C411yescharacteristic
26C2×C4×D137—21no—
27C2×C4×D137C2×C411yescharacteristic
28C2×C4×D137D412yes—
29#14555489—21no—
30#14555489—22no—
31#14555487—21no—
32#14555487—22no—
33#14555487—22no—
34C4×Dic137C2×C411yescharacteristic
35C2×Q8×C137C2×C411yescharacteristic
36C2×Q8×C137C2×C411yescharacteristic
37C4⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic
39#11764047—44no—
40C22×D137—21no—
41C22×D137C22⋊C411yescharacteristic
42Dic274—42no—
43Dic274—41no—
44D548—41no—
45D548—42no—
46C2×Dic137—21no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C2×D412yes—
49C4×D137—22no—
50C4×D137—22no—
51C4×D137—22no—
52C4×D137—42no—
53Q8×C137—41no—
54Q8×C137—41no—
55C2×C4×C137—21no—
56C2×C4×C137—21no—
57C2×C4×C137C22⋊C412yes—
58C2×C4×C137C22×C411yescharacteristic
59C22×C137C2×C22⋊C411yescharacteristic
60C2×D137—41no—
61C2×D137—42no—
62C2×D137—41no—
63C2×D137#9912yes—
64C4×C137—22no—
65C4×C137—21no—
66C4×C137—41no—
67C4×C137—41no—
68Dic137—21no—
69Dic137—22no—
70Dic137—41no—
71C2×C137—21no—
72C2×C137C2×#9911yescharacteristic
73D137—22no—
74D137—81no—
75C137#213311yescharacteristic, complemented, Sylow
76#2133—1371nomaximal, Sylow
77#536—1371no—
78C2×#101—1372no—
79#356—1372no—
80#355—1372no—
81C2×#41—1371no—
82C22×Q8—1371no—
83C2×M16—2742no—
84C4⋊Q8—1371no—
85C4⋊Q8#1058773911yescharacteristic
86#124—1371no—
87#124—1371no—
88#123—2742no—
89C22⋊Q8—2742no—
90#117—1371no—
91#101—2742no—
92#101—1374no—
93#41—5482no—
94C2×Q8—5482no—
95C2×Q8—1372no—
96C2×Q8—1371no—
97C2×Q8C2×#1058773911yescharacteristic
98C2×Q8C2×#1058773911yescharacteristic
99C2×D4—2741no—
100C2×D4—1371no—
101C22×C4—2741no—
102C22×C4—1372no—
103C22×C4—1371no—
104M16—2744no—
105M16—5482no—
106C2×C8—5482no—
107C4⋊C4—22no—
108C4⋊C4—2741no—
109C4⋊C4—2742no—
110C22⋊C4—2741no—
111C22⋊C4—2742no—
112C22⋊C4—2742no—
113C42—1371no—
114C42C2×#1058773911yescharacteristic
115C23—2741no—
116C23—1371no—
117Q8—41no—
118Q8—41no—
119Q8—5482no—
120Q8—5481no—
121D4—5481no—
122D4—5482no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—2742no—
126C2×C4—2742no—
127C2×C4—2742no—
128C2×C4—2741no—
129C2×C4—5482no—
130C2×C4—1372no—
131C2×C4—1371no—
132C2×C4C22⋊#1058773912yes—
133C2×C4C22×#1058773911yescharacteristic
134C8—5484no—
135C22—5481no—
136C22—5482no—
137C22—5481no—
138C22—1372no—
139C22C2×C22⋊#1058773911yescharacteristic
140C4—22no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—2742no—
145C4—2741no—
146C4—5481no—
147C2—21no—
148C2—2742no—
149C2—10961no—
150C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555511C4
cosocleC2×Dic274C23
hypercenterC4⋊Q8#10587739
hypocenterC137#2133
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. D137⋊#101C2semidirect
2. #22578313C2semidirect
3. #22578312C2semidirect
75. C137#2133semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1012
#225783132
#225783122
#225779782
#2133137