Finite Group Databasebeta

← #31779912

Subgroups — #31779912

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#22578138C211yesmaximal, characteristic
2#22577776C211yesmaximal, characteristic
3#22577699C211yesmaximal, characteristic, complemented
4#22577698C211yesmaximal, characteristic, complemented
5#22577146C211yesmaximal, characteristic
6#22577135C211yesmaximal, characteristic, complemented
7Q8⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555499—22no—
9#17984088C2211yescharacteristic
10#17984085C2211yescharacteristic
11#17984077—21no—
12#17984076—21no—
13#17984073—22no—
14C4×Dic274C2211yescharacteristic
15C4×D548—21no—
16#17984050—21no—
17#17984022—21no—
18#17984021—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984013—21no—
21#17984012—21no—
22C4×Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274—21no—
26C2×Dic274D411yescharacteristic
27C2×D548—21no—
28C2×D548—21no—
29C2×D548D412yes—
30C2×C4×D137—22no—
31#14555499—44no—
32C2×#11764040—21no—
33C4⋊Dic137—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137D411yescharacteristic
36#14555489—21no—
37#14555487—22no—
38#14555487—22no—
39#14555487—22no—
40C4×Dic137—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—22no—
49Dic274—21no—
50Dic274—21no—
51Dic274—42no—
52D548—42no—
53D548—41no—
54D548—41no—
55D548—41no—
56D548—41no—
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137—41no—
60C4×D137—82no—
61#11764040—41no—
62Q8×C137—21no—
63Q8×C137—21no—
64C2×C4×C137—21no—
65C2×C4×C137—21no—
66C2×C4×C137C2×D411yescharacteristic
67C2×C4×C137C2×D411yescharacteristic
68C2×C4×C137C2×D411yescharacteristic
69C8×C137—22no—
70C8×C137—41no—
71C22×C137#12011yescharacteristic
72C2×D137—42no—
73C2×D137—42no—
74C2×D137—42no—
75C4×C137—21no—
76C4×C137—41no—
77C4×C137—41no—
78C4×C137#13511yescharacteristic
79C4×C137#13511yescharacteristic
80C4×C137#13711yescharacteristic
81C4×C137#13711yescharacteristic
82Dic137—41no—
83Dic137—41no—
84Dic137—81no—
85C2×C137#45211yescharacteristic
86C2×C137#45211yescharacteristic
87C2×C137#45311yescharacteristic
88D137—82no—
89C137#163011yescharacteristic, complemented, Sylow
90#1630—1371nomaximal, Sylow
91#559—1371no—
92#492—1371no—
93#477—1371no—
94#477—1371no—
95#331—1371no—
96Q8⋊C8—1371no—
97Q8⋊C8D13711yescharacteristic, complemented
98C2×SD16—2742no—
99C4⋊Q8—1371no—
100#127—1371no—
101#124—2741no—
102#124—2741no—
103#123—2742no—
104C4×Q8—1371no—
105C4×Q8C2×D13711yescharacteristic
106C4×D4—2741no—
107C4×D4—2741no—
108#107—2741no—
109#107—2741no—
110C4⋊C8—1371no—
111C4⋊C8C2×D13711yescharacteristic
112#103—2741no—
113#103—2741no—
114C4×C8C2×D13711yescharacteristic
115C2×Q8—2741no—
116C2×Q8—1371no—
117C2×Q8D54811yescharacteristic
118C2×D4—2741no—
119C2×D4—2741no—
120C2×D4—1372no—
121C22×C4—2742no—
122SD16—5484no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—2741no—
126C4⋊C4—21no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—1371no—
131C4⋊C4D54811yescharacteristic
132C22⋊C4—2742no—
133C22⋊C4—2742no—
134C22⋊C4—2742no—
135C42—21no—
136C42—2741no—
137C42C22×D13711yescharacteristic
138C23—2742no—
139Q8—21no—
140Q8—21no—
141Q8—2741no—
142Q8—2741no—
143Q8—5482no—
144D4—5482no—
145D4—5481no—
146D4—5481no—
147D4—5481no—
148D4—5481no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—2741no—
152C2×C4—2741no—
153C2×C4—5481no—
154C2×C4—10962no—
155C2×C4D4×D13711yescharacteristic
156C2×C4D4×D13711yescharacteristic
157C2×C4C2×D54811yescharacteristic
158C8—22no—
159C8—41no—
160C8—5481no—
161C22—5482no—
162C22—5482no—
163C22—5482no—
164C22#1798405811yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—5481no—
169C4—5481no—
170C4—10961no—
171C4#1798410911yescharacteristic
172C4#1798411011yescharacteristic
173C4#1798412011yescharacteristic
174C4#1798412111yescharacteristic
175C2—10962no—
176C2#2257756711yescharacteristic
177C2#2257756811yescharacteristic
178C2#2257757211yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socleC22×C137#120
prime radical#17984085C22
hypercenterQ8⋊C8D137
hypocenterC137#1630
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
3. #22577699C2semidirect
4. #22577698C2semidirect
6. #22577135C2semidirect
7. Q8⋊C8×C137C2semidirect
89. C137#1630semidirect
97. Q8⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225781382
#225777762
#225776992
#225776982
#225771462
#225771352
Q8⋊C8×C1372
#1630137