Finite Group Databasebeta

← #31796223

Subgroups — #31796223

Order 17536.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578347C211yesmaximal, characteristic
3#22578257C214yesmaximal
4#22577958C211yesmaximal, characteristic
5C2×C4⋊#10587739—21no—
6#17984155—21no—
7#17984145C2212yes—
8C4×#11764047—21no—
9(C2×#10587739)⋊C4—21no—
10(C2×#10587739)⋊C4C2212yes—
11#17984088C411yescharacteristic, complemented
12#17984079—21no—
13#17984068—21no—
14#17984064C2212yes—
15Q8×Dic137C412yescomplemented
16#17984053—21no—
17#17984046—21no—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21C2×#11764047—22no—
22C4⋊#10587739—42no—
23C2×Dic274—21no—
24C2×C4×D137D411yescharacteristic
25C2×C4×D137D411yescharacteristic
26C2×C4×D137C2311yescharacteristic
27C4⋊Dic137—22no—
28C4⋊Dic137—21no—
29C4⋊Dic137C2×C412yes—
30#14555489—22no—
31#14555487—22no—
32C4×Dic137—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C2×Q8×C137C2×C412yes—
37C4⋊C4×C137—21no—
38C4⋊C4×C137—21no—
39C42×C137C2×C411yescharacteristic
40C2×#10587739—42no—
41C2×#10587739—42no—
42C2×#10587739—42no—
43#11764047—42no—
44C22×D137C2×D411yescharacteristic
45Dic274—42no—
46C2×Dic137—22no—
47C2×Dic137C22⋊C411yescharacteristic
48C2×Dic137C22⋊C411yescharacteristic
49C2×Dic137C2×D411yescharacteristic
50C4×D137—41no—
51C4×D137—41no—
52C4×D137SD1612yes—
53C4×D137Q1612yes—
54Q8×C137—42no—
55C2×C4×C137—22no—
56C2×C4×C137C22⋊C411yescharacteristic
57C2×C4×C137C22⋊C411yescharacteristic
58C2×C4×C137C22×C411yescharacteristic
59C22×C137C2×C22⋊C411yescharacteristic
60C2×D137—21no—
61C2×D137—21no—
62C2×D137C2×SD1611yescharacteristic
63C2×D137C2×Q1611yescharacteristic
64#10587739—82no—
65C4×C137—21no—
66C4×C137—21no—
67C4×C137—42no—
68C4×C137#10312yes—
69Dic137—21no—
70Dic137—21no—
71Dic137—42no—
72Dic137#10312yes—
73C2×C137#41011yescharacteristic
74C2×C137C2×#10311yescharacteristic
75C2×C137#42111yescharacteristic
76D137—21no—
77D137—21no—
78C137#151811yescharacteristic, complemented, Sylow
79#1518—1371nomaximal, Sylow
80#533—1371no—
81#431—1371no—
82#382—1371no—
83#328—1374no—
84C4⋊Q8—1371no—
85C4⋊Q8#1058773911yescharacteristic, complemented
86#125—2741no—
87C22⋊Q8—2741no—
88C22⋊Q8—1372no—
89C4×Q8—2741no—
90C4×Q8—1372no—
91#117—2741no—
92C2×C4⋊C4—2741no—
93C2×C42—1371no—
94C4⋊C8—2741no—
95C22⋊C8—1372no—
96C4×C8—2741no—
97#95—2741no—
98#95—1372no—
99C2×Q8—2741no—
100C2×Q8C2×#1058773912yes—
101C22×C4—2742no—
102C22×C4—1371no—
103C22×C4—1371no—
104C22×C4—1371no—
105C2×C8—2742no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—2742no—
109C4⋊C4—2742no—
110C4⋊C4—2741no—
111C4⋊C4—5482no—
112C4⋊C4—1372no—
113C22⋊C4—2742no—
114C42—2742no—
115C42—1371no—
116C42—1371no—
117C42—1371no—
118C42C2×#1058773911yescharacteristic
119C23—1371no—
120Q8—42no—
121Q8—5482no—
122C2×C4—22no—
123C2×C4—2742no—
124C2×C4—5481no—
125C2×C4—5481no—
126C2×C4—5482no—
127C2×C4—5482no—
128C2×C4—5482no—
129C2×C4—1371no—
130C2×C4—1371no—
131C2×C4—1371no—
132C2×C4—1372no—
133C2×C4—1372no—
134C2×C4C22⋊#1058773911yescharacteristic
135C2×C4C22⋊#1058773911yescharacteristic
136C2×C4C22×#1058773911yescharacteristic
137C8—5482no—
138C22—2741no—
139C22—2741no—
140C22—1371no—
141C22—1371no—
142C22C2×C22⋊#1058773911yescharacteristic
143C4—21no—
144C4—21no—
145C4—42no—
146C4—2741no—
147C4—2741no—
148C4—5482no—
149C4—10962no—
150C4—1372no—
151C4#1798415112yes—
152C2—2741no—
153C2—2741no—
154C2#2257838311yescharacteristic
155C2C2×#1798415111yescharacteristic
156C2#2257840211yescharacteristic
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#1518
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
15. Q8×Dic137C4semidirect
19. C4⋊Q8×C137C4semidirect
78. C137#1518semidirect
85. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784212
#225783472
#225782572
#225779582
#1518137