Finite Group Databasebeta

← #31779864

Subgroups — #31779864

Order 17536.

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

normal subgroups only

This group has 236 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, complemented
2#22577776C211yesmaximal, characteristic, complemented
3#22577603C211yesmaximal, characteristic, complemented
4#22577599C211yesmaximal, characteristic
5#22577151C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7Q8⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555499—22no—
9C2×#14555497—21no—
10C2×D1096—21no—
11#17984088C2211yescharacteristic
12#17984085C2211yescharacteristic
13#17984080—21no—
14#17984066—22no—
15#17984064—21no—
16C4×D548—21no—
17C4×D548C2211yescharacteristic
18C4⋊C4×D137—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984010—21no—
21#17984008—21no—
22C4×Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25C2×Dic274D412yes—
26C2×D548—21no—
27C2×D548—21no—
28C2×D548—21no—
29C2×D548D411yescharacteristic
30C2×C4×D137—22no—
31C2×C4×D137—21no—
32#14555499—44no—
33#14555497—42no—
34D1096—42no—
35C2×#11764040—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137D411yescharacteristic
39#14555489—22no—
40#14555489—22no—
41#14555487—22no—
42#14555487—21no—
43C2×Q8×C137D411yescharacteristic
44C2×C8×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137D411yescharacteristic
48C42×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—21no—
51C22×D137—41no—
52Dic274—42no—
53D548—21no—
54D548—42no—
55D548—41no—
56D548—41no—
57D548—41no—
58D548—41no—
59D548SD1612yescomplemented
60C2×Dic137—21no—
61C2×Dic137—22no—
62C4×D137—41no—
63C4×D137—41no—
64C4×D137—82no—
65#11764040—41no—
66Q8×C137—21no—
67Q8×C137—21no—
68C2×C4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—22no—
74C8×C137—41no—
75C22×C137#12011yescharacteristic
76C2×D137—22no—
77C2×D137—41no—
78C2×D137—41no—
79C2×D137—81no—
80C2×D137—81no—
81C2×D137—81no—
82C4×C137—21no—
83C4×C137—41no—
84C4×C137—41no—
85C4×C137C2×SD1611yescharacteristic
86C4×C137#13511yescharacteristic
87C4×C137AGL(1,ℤ/8)11yescharacteristic
88C4×C137#13711yescharacteristic
89Dic137—41no—
90Dic137—42no—
91C2×C137#44811yescharacteristic
92C2×C137#44911yescharacteristic
93C2×C137#45311yescharacteristic
94D137—42no—
95D137—161no—
96C137#162211yescharacteristic, complemented, Sylow
97#1622—1371nomaximal, Sylow
98#554—1371no—
99#492—1371no—
100#460—1371no—
101#459—1371no—
102#333—1371no—
103Q8⋊C8D13711yescharacteristic, complemented
104D4⋊C8—1371no—
105C2×SD16—2742no—
106C2×SD16—2741no—
107C2×D8—2741no—
108C4⋊Q8—1371no—
109#127—1371no—
110#125—2741no—
111C22⋊Q8—2742no—
112C22⋊Q8—2741no—
113C4×Q8C2×D13711yescharacteristic
114C4×D4—2741no—
115C4×D4—1371no—
116C2×C4⋊C4—2741no—
117C4⋊C8—1371no—
118C4⋊C8C2×D13711yescharacteristic
119#102—2741no—
120#102—2741no—
121C4×C8C2×D13711yescharacteristic
122C2×Q8—1372no—
123C2×Q8D54811yescharacteristic
124C2×D4—2741no—
125C2×D4—2741no—
126C2×D4—2741no—
127C2×D4—1371no—
128C22×C4—2742no—
129C22×C4—2741no—
130SD16—5484no—
131SD16—5482no—
132D8—5482no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2741no—
136C4⋊C4—21no—
137C4⋊C4—2742no—
138C4⋊C4—2742no—
139C4⋊C4—2741no—
140C4⋊C4—2741no—
141C4⋊C4—1371no—
142C4⋊C4D54811yescharacteristic
143C22⋊C4—2742no—
144C22⋊C4—2741no—
145C42—21no—
146C42C22×D13711yescharacteristic
147C23—2741no—
148C23—5481no—
149Q8—21no—
150Q8—21no—
151Q8—5482no—
152D4—2741no—
153D4—5482no—
154D4—5481no—
155D4—5481no—
156D4—5481no—
157D4—5481no—
158D4—1372no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—2741no—
162C2×C4—2742no—
163C2×C4—5481no—
164C2×C4—5481no—
165C2×C4—10962no—
166C2×C4D4×D13711yescharacteristic
167C2×C4D4×D13711yescharacteristic
168C2×C4C2×D54811yescharacteristic
169C8—22no—
170C8—41no—
171C8—5481no—
172C22—2742no—
173C22—5481no—
174C22—5481no—
175C22—10961no—
176C22—10961no—
177C22—10961no—
178C22#1798405811yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—5481no—
183C4—5482no—
184C4SD16×D13711yescharacteristic
185C4#1798411011yescharacteristic
186C4#1798411611yescharacteristic
187C4#1798412011yescharacteristic
188C2—5482no—
189C2—21921no—
190C2#2257754311yescharacteristic
191C2#2257755011yescharacteristic
192C2#2257757211yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socleC22×C137#120
prime radical#22577599C2
hypercenterQ8⋊C8D137
hypocenterC137#1622
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
1. #22578113C2semidirect
2. #22577776C2semidirect
3. #22577603C2semidirect
5. #22577151C2semidirect
6. #22577132C2semidirect
7. Q8⋊C8×C137C2semidirect
59. D548SD16semidirect
96. C137#1622semidirect
103. Q8⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225781132
#225777762
#225776032
#225775992
#225771512
#225771322
Q8⋊C8×C1372
#1622137