Finite Group Databasebeta

← #31797092

Subgroups — #31797092

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
1#22578390C212yesmaximal
2#22578314C212yesmaximal, complemented
3#22578313C212yesmaximal, complemented
4#22577973C211yesmaximal, characteristic
5C2×#14555519—22no—
6#17984149C2214yes—
7#17984148—22no—
8C2×#14555509C2211yescharacteristic
9C2×Q8×D137C411yescharacteristic
10#17984088C2211yescharacteristic
11#17984086C2211yescharacteristic
12#17984075C411yescharacteristic
13#17984070—22no—
14#17984065—22no—
15#17984045C411yescharacteristic
16#124×C137C411yescharacteristic
17#14555519—24no—
18#14555519—42no—
19C2×#11764047—42no—
20#14555509—42no—
21C2×Dic274—21no—
22C2×Dic274C2×C411yescharacteristic
23C2×Dic274C2311yescharacteristic
24Q8×D137—42no—
25C2×C22⋊D137D412yes—
26C22×Dic137D412yes—
27C2×C4×D137—21no—
28C2×C4×D137C2×C411yescharacteristic
29C4⋊Dic137—22no—
30#14555489—21no—
31#14555489—22no—
32C22⋊Dic137—22no—
33#14555487—21no—
34C4×Dic137C2×C411yescharacteristic
35C2×Q8×C137C2×C411yescharacteristic
36C2×D4×C137C2×C411yescharacteristic
37C22⋊C4×C137—22no—
38C42×C137C2×C411yescharacteristic
39#11764047—44no—
40C22×D137C22⋊C411yescharacteristic
41Dic274—41no—
42Dic274—42no—
43Dic274—41no—
44C22⋊D137—42no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47C2×Dic137—21no—
48C2×Dic137—21no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D412yes—
51C4×D137—22no—
52C4×D137—42no—
53C23×C137C22⋊C412yes—
54Q8×C137—41no—
55D4×C137—41no—
56C2×C4×C137—21no—
57C2×C4×C137—21no—
58C2×C4×C137C22×C411yescharacteristic
59C22×C137—22no—
60C22×C137—42no—
61C22×C137C2×C22⋊C411yescharacteristic
62C2×D137—21no—
63C2×D137—41no—
64C4×C137—21no—
65C4×C137—41no—
66C4×C137—41no—
67Dic137—22no—
68Dic137—41no—
69Dic137—41no—
70Dic137#9912yes—
71C2×C137—21no—
72C2×C137—42no—
73C2×C137C2×#9911yescharacteristic
74D137—41no—
75C137#213511yescharacteristic, complemented, Sylow
76#2135—1371nomaximal, Sylow
77#536—1371no—
78#413—1372no—
79#356—1372no—
80#356—1372no—
81C2×#41—1371no—
82C22×Q8—1371no—
83C2×M16—2742no—
84C4⋊Q8—1371no—
85C4⋊Q8—1371no—
86#124—1371no—
87#124#1058773911yescharacteristic
88#123—2742no—
89C22⋊Q8—2742no—
90#117—1371no—
91#101—1374no—
92#100—2742no—
93#41—5482no—
94C2×Q8—2741no—
95C2×Q8—5482no—
96C2×Q8—1371no—
97C2×Q8—1371no—
98C2×Q8C2×#1058773911yescharacteristic
99C2×D4—1372no—
100C2×D4C2×#1058773911yescharacteristic
101C22×C4—2741no—
102C22×C4—1372no—
103C22×C4—1371no—
104M16—2744no—
105M16—5482no—
106C2×C8—5482no—
107C4⋊C4—2741no—
108C4⋊C4—2742no—
109C4⋊C4—2742no—
110C22⋊C4—22no—
111C22⋊C4—2741no—
112C22⋊C4—2742no—
113C42—1371no—
114C42C2×#1058773911yescharacteristic
115C23—1371no—
116C23C22⋊#1058773912yes—
117Q8—41no—
118Q8—5481no—
119Q8—5481no—
120Q8—5482no—
121D4—41no—
122D4—5482no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—2742no—
126C2×C4—2742no—
127C2×C4—2742no—
128C2×C4—2741no—
129C2×C4—2741no—
130C2×C4—5482no—
131C2×C4—1372no—
132C2×C4—1371no—
133C2×C4C22×#1058773911yescharacteristic
134C8—5484no—
135C22—22no—
136C22—42no—
137C22—2741no—
138C22—5481no—
139C22C2×C22⋊#1058773911yescharacteristic
140C4—21no—
141C4—41no—
142C4—41no—
143C4—2742no—
144C4—5481no—
145C4—5481no—
146C4—1372no—
147C2—21no—
148C2—42no—
149C2—5481no—
150C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#124×C137C4
socleC2×C137C2×#99
prime radicalC2×#14555509C22
cosocleC2×Dic274C23
hypercenter#124#10587739
hypocenterC137#2135
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #22578314C2semidirect
3. #22578313C2semidirect
75. C137#2135semidirect

Maximal subgroups

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

subgroupindex
#225783902
#225783142
#225783132
#225779732
#2135137