Finite Group Databasebeta

← #31797086

Subgroups — #31797086

Order 17536.

230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 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
1C2×#17984149C212yesmaximal, complemented
2#22578314C212yesmaximal, complemented
3#22578312C212yesmaximal, complemented
4#22577979C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984149—22no—
7#17984149C2214yes—
8C2×#14555510C411yescharacteristic
9C22×Dic274C2211yescharacteristic
10#17984088C411yescharacteristic
11#17984088C2211yescharacteristic
12#17984076C411yescharacteristic
13#17984076C2211yescharacteristic
14#17984072—22no—
15#17984068—22no—
16#117×C137C411yescharacteristic
17#14555519—24no—
18#14555519—42no—
19C2×#11764047—42no—
20#14555510—42no—
21C2×Dic274—42no—
22C2×Dic274C2×C411yescharacteristic
23C2×Dic274C2×C411yescharacteristic
24C2×Dic274D412yes—
25C2×Dic274C2311yescharacteristic
26C2×D548C2×C411yescharacteristic
27C2×C22⋊D137—21no—
28C22×Dic137D412yes—
29C2×C4×D137—21no—
30C4⋊Dic137—22no—
31C4⋊Dic137—22no—
32#14555487—22no—
33#14555487—22no—
34C22×C4×C137C2×C411yescharacteristic
35C4⋊C4×C137—21no—
36C22⋊C4×C137—21no—
37C42×C137C2×C411yescharacteristic
38C42×C137C2×C411yescharacteristic
39#11764047—44no—
40C22×D137—21no—
41Dic274—41no—
42Dic274—42no—
43Dic274—41no—
44Dic274—41no—
45D548—41no—
46C22⋊D137—42no—
47C2×Dic137—22no—
48C2×Dic137—22no—
49C2×Dic137—21no—
50C2×Dic137C22⋊C412yes—
51C2×Dic137C2×D412yes—
52C4×D137—42no—
53C23×C137C22⋊C411yescharacteristic
54C2×C4×C137—22no—
55C2×C4×C137—21no—
56C2×C4×C137—21no—
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22×C411yescharacteristic
59C22×C137—41no—
60C22×C137#9912yes—
61C22×C137C2×C22⋊C411yescharacteristic
62C2×D137—42no—
63C2×D137—41no—
64C4×C137—21no—
65C4×C137—21no—
66C4×C137—41no—
67C4×C137—41no—
68Dic137—22no—
69Dic137—22no—
70Dic137—41no—
71C2×C137—21no—
72C2×C137—22no—
73C2×C137C2×#9911yescharacteristic
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—1371no—
86#124—1371no—
87#124—1371no—
88#123—2742no—
89C22⋊Q8—2742no—
90#117#1058773911yescharacteristic
91#101—2742no—
92#101—1374no—
93#41—5482no—
94C2×Q8—5482no—
95C2×Q8—1371no—
96C2×Q8—1372no—
97C2×Q8—1371no—
98C2×Q8—1371no—
99C2×D4—2741no—
100C2×D4—1371no—
101C22×C4—2741no—
102C22×C4—1372no—
103C22×C4C2×#1058773911yescharacteristic
104M16—2744no—
105M16—5482no—
106C2×C8—5482no—
107C4⋊C4—21no—
108C4⋊C4—2742no—
109C4⋊C4—2742no—
110C22⋊C4—21no—
111C22⋊C4—2742no—
112C22⋊C4—2742no—
113C42C2×#1058773911yescharacteristic
114C42C2×#1058773911yescharacteristic
115C23—2741no—
116C23C22⋊#1058773911yescharacteristic
117Q8—5482no—
118Q8—5481no—
119Q8—5481no—
120Q8—5481no—
121D4—5481no—
122D4—5482no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—2742no—
127C2×C4—2742no—
128C2×C4—2741no—
129C2×C4—5482no—
130C2×C4—1372no—
131C2×C4—1372no—
132C2×C4C22⋊#1058773911yescharacteristic
133C2×C4C22×#1058773911yescharacteristic
134C8—5484no—
135C22—41no—
136C22—5482no—
137C22—5481no—
138C22#1798414712yes—
139C22C2×C22⋊#1058773911yescharacteristic
140C4—21no—
141C4—21no—
142C4—41no—
143C4—41no—
144C4—2742no—
145C4—2742no—
146C4—5481no—
147C2—21no—
148C2—22no—
149C2—10961no—
150C2C2×#1798414711yescharacteristic
subgroupG/H
centerC2C2×#17984147
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555510C4
cosocleC2×Dic274C23
hypercenter#117#10587739
hypocenterC137#2133
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#17984149C2semidirect
2. #22578314C2semidirect
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
C2×#179841492
#225783142
#225783122
#225779792
#2133137