Finite Group Databasebeta

← #31796299

Subgroups — #31796299

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#22578421C211yesmaximal, characteristic
2#22578408C211yesmaximal, characteristic
3#22578262C214yesmaximal, complemented
4C4⋊Q8×D137C211yesmaximal, characteristic
5C2×#14555520—21no—
6#17984155—21no—
7#17984155C2212yes—
8#17984155C2212yes—
9#17984145—21no—
10C4×#11764047—21no—
11C2×Q8×D137—21no—
12#17984088C411yescharacteristic
13#17984087—21no—
14#17984087C2212yes—
15#17984086—21no—
16C4⋊C4×D137—21no—
17C4⋊C4×D137C412yes—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic
20#14555520—42no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C2×Dic274—22no—
24C2×Dic274—21no—
25Q8×D137—42no—
26Q8×D137—42no—
27C2×C4×D137—22no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137C2×C412yes—
33#14555489—22no—
34#14555489—22no—
35C4×Dic137D411yescharacteristic
36C4×Dic137D411yescharacteristic
37C4×Dic137C2311yescharacteristic
38C2×Q8×C137—21no—
39C4⋊C4×C137—21no—
40C4⋊C4×C137C2×C412yes—
41C42×C137C2×C411yescharacteristic
42#11764047—42no—
43#11764047—42no—
44C22×D137C22⋊C411yescharacteristic
45Dic274—42no—
46Dic274—42no—
47Dic274—42no—
48C2×Dic137—22no—
49C2×Dic137C22⋊C411yescharacteristic
50C2×Dic137C2×D411yescharacteristic
51C2×Dic137C2×D411yescharacteristic
52C4×D137—22no—
53C4×D137—22no—
54C4×D137—44no—
55C4×D137D812yes—
56C4×D137SD1612yes—
57Q8×C137—42no—
58C2×C4×C137—22no—
59C2×C4×C137C22⋊C411yescharacteristic
60C2×C4×C137C22⋊C411yescharacteristic
61C2×C4×C137C22×C411yescharacteristic
62C22×C137C2×C22⋊C411yescharacteristic
63C2×D137—21no—
64C2×D137—21no—
65C2×D137#10212yes—
66C4×C137—21no—
67C4×C137—21no—
68C4×C137—42no—
69C4×C137#10212yes—
70Dic137—42no—
71Dic137#10112yes—
72Dic137C2×D811yescharacteristic
73Dic137C2×SD1611yescharacteristic
74Dic137#13712yes—
75C2×C137C2×#10111yescharacteristic
76C2×C137C2×#10211yescharacteristic
77C2×C137#41911yescharacteristic
78D137—22no—
79C137#154611yescharacteristic, complemented, Sylow
80#1546—1371nomaximal, Sylow
81C2×C4⋊Q8—1371no—
82#431—1371no—
83#423—1371no—
84#332—1374no—
85C22×Q8—2741no—
86C2×M16—2741no—
87C4⋊Q8—2741no—
88C4⋊Q8—2741no—
89C4⋊Q8—1372no—
90C4⋊Q8—1371no—
91C4⋊Q8#1058773911yescharacteristic
92C2×C4⋊C4—2741no—
93C2×C4⋊C4—1372no—
94C2×C42—1371no—
95C4⋊C8—2741no—
96C4⋊C8—1372no—
97C4⋊C8—1372no—
98C22⋊C8—2741no—
99C4×C8—2741no—
100C2×Q8—21no—
101C2×Q8—2742no—
102C2×Q8—2741no—
103C2×Q8—5482no—
104C2×Q8—5482no—
105C22×C4—2742no—
106C22×C4—1371no—
107C22×C4—1371no—
108C22×C4—1371no—
109M16—5482no—
110C2×C8—2742no—
111C2×C8—2742no—
112C4⋊C4—21no—
113C4⋊C4—2742no—
114C4⋊C4—2742no—
115C4⋊C4—2741no—
116C4⋊C4—1372no—
117C4⋊C4C2×#1058773912yes—
118C42—1371no—
119C42—1371no—
120C42—1371no—
121C42C2×#1058773911yescharacteristic
122C23—1371no—
123Q8—42no—
124Q8—5482no—
125Q8—5482no—
126Q8—5482no—
127C2×C4—22no—
128C2×C4—2742no—
129C2×C4—2742no—
130C2×C4—2742no—
131C2×C4—5484no—
132C2×C4—1371no—
133C2×C4—1371no—
134C2×C4—1371no—
135C2×C4—1372no—
136C2×C4—1372no—
137C2×C4C22⋊#1058773911yescharacteristic
138C2×C4C22⋊#1058773911yescharacteristic
139C2×C4C22×#1058773911yescharacteristic
140C8—5482no—
141C8—5482no—
142C22—2741no—
143C22—2741no—
144C22—1372no—
145C22C2×C22⋊#1058773911yescharacteristic
146C4—21no—
147C4—21no—
148C4—42no—
149C4—5482no—
150C4—1372no—
151C4—1371no—
152C4—1371no—
153C4—1372no—
154C4#1798415012yes—
155C2—2742no—
156C2D137⋊#10111yescharacteristic
157C2C2×#1798415011yescharacteristic
158C2#2257839611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC4×Dic137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1546
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
3. #22578262C2semidirect
79. C137#1546semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225784082
#225782622
C4⋊Q8×D1372
#1546137