Finite Group Databasebeta

← #31796968

Subgroups — #31796968

Order 17536.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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#22578404C211yesmaximal, characteristic
2C4×C4⋊#10587739C211yesmaximal, characteristic
3#22578276C214yesmaximal
4C4⋊Q8×D137C211yesmaximal, characteristic
5C2×#14555518C2212yes—
6C2×C4⋊#10587739C2212yes—
7C2×C4×#10587739—21no—
8#17984155—21no—
9C4⋊#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11C2×Q8×D137—21no—
12#17984088C411yescharacteristic, complemented
13#17984087—21no—
14#17984087C412yescomplemented
15#17984086—21no—
16C4⋊C4×D137—21no—
17C4⋊C4×D137C2212yes—
18C42×D137C2211yescharacteristic
19C4⋊Q8×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555518—24no—
22C2×#11764047—22no—
23C4⋊#10587739—22no—
24C4⋊#10587739Q814yescomplemented
25C4×#10587739—42no—
26C2×Dic274—22no—
27C2×Dic274—21no—
28Q8×D137—42no—
29Q8×D137—42no—
30C2×C4×D137—22no—
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137C2311yescharacteristic
34C4⋊Dic137—21no—
35C4⋊Dic137C2×C412yes—
36#14555489—22no—
37#14555489—22no—
38C4×Dic137C2×C411yescharacteristic
39C4×Dic137D411yescharacteristic
40C4×Dic137D411yescharacteristic
41C2×Q8×C137—21no—
42C4⋊C4×C137—21no—
43C4⋊C4×C137C2×C412yes—
44C42×C137C2×C411yescharacteristic
45C2×#10587739—24no—
46C2×#10587739—42no—
47C2×#10587739—42no—
48#11764047—42no—
49C22×D137C2×D411yescharacteristic
50Dic274—42no—
51Dic274—44no—
52C2×Dic137—22no—
53C2×Dic137C22⋊C411yescharacteristic
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C4×D137—22no—
57C4×D137—42no—
58C4×D137—42no—
59C4×D137D812yes—
60C4×D137SD1612yes—
61C4×D137C2×Q812yes—
62C4×D137#4112yes—
63Q8×C137—42no—
64C2×C4×C137—22no—
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22×C411yescharacteristic
68C22×C137C2×C22⋊C411yescharacteristic
69C2×D137C22⋊Q812yes—
70C2×D137C2×D811yescharacteristic
71C2×D137C2×SD1611yescharacteristic
72C2×D137#13712yes—
73#10587739—44no—
74C4×C137—21no—
75C4×C137—42no—
76C4×C137#10212yes—
77C4×C137C4×Q812yes—
78Dic137—21no—
79Dic137—42no—
80Dic137#10212yes—
81Dic137#12411yescharacteristic
82Dic137C4⋊Q811yescharacteristic
83C2×C137#39111yescharacteristic
84C2×C137C2×#10211yescharacteristic
85C2×C137#41911yescharacteristic
86D137#47412yes—
87D137#47512yes—
88C137#198811yescharacteristic, complemented, Sylow
89#1988—1371nomaximal, Sylow
90C2×C4⋊Q8—1371no—
91C2×C4⋊C8—1371no—
92C4×C4⋊C4—1371no—
93#340—1374no—
94C22×Q8—2741no—
95C22×C8—1372no—
96C4⋊Q8—2741no—
97C4⋊Q8—2741no—
98C4⋊Q8—1372no—
99C4⋊Q8—1371no—
100C4⋊Q8#1058773911yescharacteristic, complemented
101C2×C4⋊C4—2741no—
102C2×C4⋊C4—1372no—
103C2×C4⋊C4—1372no—
104C2×C42—2741no—
105C2×C42—1371no—
106C4⋊C8—2741no—
107C4⋊C8—2741no—
108#95—2741no—
109C2×Q8—21no—
110C2×Q8—2742no—
111C2×Q8—2741no—
112C2×Q8—5482no—
113C2×Q8—5482no—
114C22×C4—2742no—
115C22×C4—2742no—
116C22×C4—1371no—
117C22×C4—1371no—
118C22×C4—1371no—
119C2×C8—2742no—
120C2×C8—2744no—
121C4⋊C4—21no—
122C4⋊C4—2742no—
123C4⋊C4—2742no—
124C4⋊C4—2742no—
125C4⋊C4—2741no—
126C4⋊C4—1374no—
127C4⋊C4—1372no—
128C4⋊C4C2×#1058773912yes—
129C42—5482no—
130C42—1371no—
131C42—1371no—
132C42—1371no—
133C42C2×#1058773911yescharacteristic
134C23—1371no—
135Q8—42no—
136Q8—5484no—
137Q8—5482no—
138C2×C4—22no—
139C2×C4—2742no—
140C2×C4—2744no—
141C2×C4—2742no—
142C2×C4—5482no—
143C2×C4—5482no—
144C2×C4—5482no—
145C2×C4—5482no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1372no—
150C2×C4—1372no—
151C2×C4—1372no—
152C2×C4—1372no—
153C2×C4C22⋊#1058773911yescharacteristic
154C2×C4C22⋊#1058773911yescharacteristic
155C2×C4C22×#1058773911yescharacteristic
156C8—5482no—
157C22—1372no—
158C22—1371no—
159C22—1371no—
160C22—1372no—
161C22C2×C22⋊#1058773911yescharacteristic
162C4—21no—
163C4—42no—
164C4—2741no—
165C4—5484no—
166C4—5482no—
167C4—1371no—
168C4—1371no—
169C4—1372no—
170C4#1798415012yes—
171C4Q8×#1058773912yes—
172C2—1372no—
173C2—1372no—
174C2(C2×#10587739)⋊Q811yescharacteristic
175C2C2×#1798415011yescharacteristic
176C2#2257839611yescharacteristic
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#1988
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. #17984088C4semidirect
14. #17984087C4semidirect
19. C4⋊Q8×C137C4semidirect
24. C4⋊#10587739Q8semidirect
88. C137#1988semidirect
100. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784042
C4×C4⋊#105877392
#225782762
C4⋊Q8×D1372
#1988137