Finite Group Databasebeta

← #31779936

Subgroups — #31779936

Order 17536.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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#22578118C211yesmaximal, characteristic
2#22577783C211yesmaximal, characteristic, complemented
3#22577694C211yesmaximal, characteristic, complemented
4#22577693C211yesmaximal, characteristic, complemented
5#22577151C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×Dic548—22no—
9#17984088C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984074—22no—
12#17984062—21no—
13#17984059—21no—
14C4×D548C2211yescharacteristic
15#17984047—21no—
16#17984046—21no—
17#17984020—21no—
18#17984019—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984010—21no—
21#17984009—21no—
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×Dic274D412yes—
27C2×D548D411yescharacteristic
28C2×C22⋊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
37C22⋊Dic137—22no—
38#14555487—22no—
39#14555487—21no—
40C4×Dic137—22no—
41C2×D4×C137D411yescharacteristic
42C22×C4×C137—21no—
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137D411yescharacteristic
46C22⋊C4×C137—21no—
47C42×C137C2311yescharacteristic
48C22×D137—21no—
49Dic274—42no—
50Dic274—42no—
51D548—21no—
52D548—21no—
53C22⋊D137—82no—
54C2×Dic137—21no—
55C2×Dic137—22no—
56C2×Dic137—41no—
57C4×D137—41no—
58C4×D137—41no—
59#11764040—41no—
60C23×C137—21no—
61D4×C137—21no—
62D4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137—41no—
65C2×C4×C137—41no—
66C2×C4×C137C2×D411yescharacteristic
67C2×C4×C137C2×D411yescharacteristic
68C2×C4×C137C2×D411yescharacteristic
69C8×C137—22no—
70C8×C137—41no—
71C22×C137—41no—
72C22×C137—41no—
73C22×C137—41no—
74C22×C137#12011yescharacteristic
75C2×D137—41no—
76C2×D137—41no—
77C2×D137—41no—
78C4×C137—21no—
79C4×C137—41no—
80C4×C137#13511yescharacteristic
81C4×C137#13511yescharacteristic
82C4×C137#13711yescharacteristic
83C4×C137#13711yescharacteristic
84Dic137—41no—
85Dic137—42no—
86Dic137—81no—
87C2×C137—81no—
88C2×C137#45211yescharacteristic
89C2×C137#45211yescharacteristic
90C2×C137#45311yescharacteristic
91D137—81no—
92C137#163411yescharacteristic, complemented, Sylow
93#1634—1371nomaximal, Sylow
94#555—1371no—
95#494—1371no—
96#476—1371no—
97#476—1371no—
98#333—1371no—
99D4⋊C8—1371no—
100D4⋊C8D13711yescharacteristic, complemented
101C2×Q16—2742no—
102C4⋊Q8—1371no—
103C4⋊Q8—1371no—
104#124—2742no—
105#121—2741no—
106#121—2741no—
107C4×D4—1371no—
108C4×D4C2×D13711yescharacteristic
109#117—2741no—
110#117—2741no—
111#106—2741no—
112#106—2741no—
113C4⋊C8—1371no—
114C4⋊C8C2×D13711yescharacteristic
115#102—2741no—
116#102—2741no—
117C4×C8C2×D13711yescharacteristic
118C2×Q8—2741no—
119C2×Q8—1372no—
120C2×D4—2742no—
121C2×D4—1371no—
122C2×D4D54811yescharacteristic
123C22×C4—21no—
124C22×C4—2741no—
125Q16—5484no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—2741no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C4⋊C4—2741no—
132C4⋊C4—2741no—
133C4⋊C4—1371no—
134C4⋊C4D54811yescharacteristic
135C22⋊C4—21no—
136C22⋊C4—2742no—
137C22⋊C4—2742no—
138C22⋊C4—2741no—
139C42—2742no—
140C42C22×D13711yescharacteristic
141C23—21no—
142C23—2741no—
143Q8—5482no—
144Q8—5482no—
145D4—21no—
146D4—21no—
147D4—2741no—
148D4—2741no—
149D4—10962no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—2741no—
154C2×C4—2742no—
155C2×C4—5481no—
156C2×C4—5481no—
157C2×C4—5481no—
158C2×C4D4×D13711yescharacteristic
159C2×C4D4×D13711yescharacteristic
160C2×C4C2×D54811yescharacteristic
161C8—22no—
162C8—41no—
163C8—5481no—
164C22—41no—
165C22—41no—
166C22—41no—
167C22—5481no—
168C22—5481no—
169C22—5481no—
170C22#1798405811yescharacteristic
171C4—21no—
172C4—41no—
173C4—5481no—
174C4—5482no—
175C4—10961no—
176C4#1798410711yescharacteristic
177C4#1798411011yescharacteristic
178C4#1798411711yescharacteristic
179C4#1798412111yescharacteristic
180C2—81no—
181C2—10961no—
182C2#2257756611yescharacteristic
183C2#2257756711yescharacteristic
184C2#2257757211yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137#120
prime radical#22578118C2
hypercenterD4⋊C8D137
hypocenterC137#1634
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22577783C2semidirect
3. #22577694C2semidirect
4. #22577693C2semidirect
5. #22577151C2semidirect
6. #22577132C2semidirect
7. D4⋊C8×C137C2semidirect
92. C137#1634semidirect
100. D4⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225781182
#225777832
#225776942
#225776932
#225771512
#225771322
D4⋊C8×C1372
#1634137