Finite Group Databasebeta

← #31779926

Subgroups — #31779926

Order 17536.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#22578113C211yesmaximal, characteristic
2#22577783C211yesmaximal, characteristic, complemented
3#22577698C211yesmaximal, characteristic, complemented
4#22577694C211yesmaximal, characteristic
5#22577151C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic
7Q8⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×Dic548—22no—
9#17984088C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984080—21no—
12#17984066—22no—
13#17984064—21no—
14C4×D548—21no—
15C4×D548C2211yescharacteristic
16C4⋊C4×D137—21no—
17#17984021—21no—
18#17984020—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984012—21no—
21#17984010—21no—
22C4×Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×Dic274D412yes—
27C2×D548D411yescharacteristic
28C2×C4×D137—22no—
29C2×C4×D137—21no—
30Dic548—44no—
31C2×#11764040—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137D411yescharacteristic
37#14555489—22no—
38#14555489—22no—
39#14555487—22no—
40#14555487—21no—
41C2×Q8×C137D411yescharacteristic
42C2×C8×C137—21no—
43C2×C8×C137—21no—
44C4⋊C4×C137—21no—
45C4⋊C4×C137D411yescharacteristic
46C42×C137—21no—
47C42×C137C2311yescharacteristic
48C22×D137—21no—
49Dic274—42no—
50Dic274—42no—
51D548—21no—
52D548Q1612yescomplemented
53C2×Dic137—21no—
54C2×Dic137—22no—
55C2×Dic137—41no—
56C4×D137—41no—
57C4×D137—41no—
58C4×D137—82no—
59#11764040—41no—
60Q8×C137—21no—
61Q8×C137—21no—
62C2×C4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137C2×D411yescharacteristic
65C2×C4×C137C2×D411yescharacteristic
66C2×C4×C137C2×D411yescharacteristic
67C8×C137—22no—
68C8×C137—41no—
69C22×C137#12011yescharacteristic
70C2×D137—22no—
71C2×D137—41no—
72C2×D137—41no—
73C4×C137—21no—
74C4×C137—41no—
75C4×C137—41no—
76C4×C137C2×Q1611yescharacteristic
77C4×C137#13511yescharacteristic
78C4×C137#13711yescharacteristic
79C4×C137#13711yescharacteristic
80Dic137—41no—
81Dic137—42no—
82Dic137—81no—
83C2×C137#45111yescharacteristic
84C2×C137#45211yescharacteristic
85C2×C137#45311yescharacteristic
86D137—42no—
87C137#163211yescharacteristic, complemented, Sylow
88#1632—1371nomaximal, Sylow
89#554—1371no—
90#494—1371no—
91#477—1371no—
92#476—1371no—
93#333—1371no—
94Q8⋊C8D13711yescharacteristic, complemented
95D4⋊C8—1371no—
96C2×Q16—2742no—
97C4⋊Q8—1371no—
98C4⋊Q8—1371no—
99#125—2741no—
100C22⋊Q8—2742no—
101C22⋊Q8—2741no—
102C4×Q8C2×D13711yescharacteristic
103C4×D4—2741no—
104C4×D4—1371no—
105C2×C4⋊C4—2741no—
106#107—2741no—
107#106—2741no—
108C4⋊C8—1371no—
109C4⋊C8C2×D13711yescharacteristic
110#103—2741no—
111#102—2741no—
112C4×C8C2×D13711yescharacteristic
113C2×Q8—2741no—
114C2×Q8—1372no—
115C2×Q8D54811yescharacteristic
116C2×D4—1371no—
117C22×C4—2742no—
118C22×C4—2741no—
119Q16—5484no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2741no—
123C4⋊C4—21no—
124C4⋊C4—2742no—
125C4⋊C4—2742no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—1371no—
131C4⋊C4D54811yescharacteristic
132C22⋊C4—2742no—
133C22⋊C4—2741no—
134C42—21no—
135C42C22×D13711yescharacteristic
136C23—2741no—
137Q8—21no—
138Q8—21no—
139Q8—5482no—
140Q8—5482no—
141D4—2741no—
142D4—1372no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—2741no—
146C2×C4—2742no—
147C2×C4—5481no—
148C2×C4—5481no—
149C2×C4—5481no—
150C2×C4—10962no—
151C2×C4D4×D13711yescharacteristic
152C2×C4D4×D13711yescharacteristic
153C2×C4C2×D54811yescharacteristic
154C8—22no—
155C8—41no—
156C8—5481no—
157C22—2742no—
158C22—5481no—
159C22—5481no—
160C22#1798405811yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—5481no—
165C4—5482no—
166C4—10961no—
167C4Q16×D13711yescharacteristic
168C4#1798411011yescharacteristic
169C4#1798411711yescharacteristic
170C4#1798412111yescharacteristic
171C2—5482no—
172C2#2257756011yescharacteristic
173C2#2257756711yescharacteristic
174C2#2257757211yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socleC22×C137#120
prime radicalC2×D548D4
hypercenterQ8⋊C8D137
hypocenterC137#1632
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #22577783C2semidirect
3. #22577698C2semidirect
5. #22577151C2semidirect
7. Q8⋊C8×C137C2semidirect
52. D548Q16semidirect
87. C137#1632semidirect
94. Q8⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225781132
#225777832
#225776982
#225776942
#225771512
#225771322
Q8⋊C8×C1372
#1632137