Finite Group Databasebeta

← #31779919

Subgroups — #31779919

Order 17536.

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

normal subgroups only

This group has 256 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#22578074C211yesmaximal, characteristic
2#22577776C211yesmaximal, characteristic, complemented
3#22577699C211yesmaximal, characteristic, complemented
4#22577693C211yesmaximal, characteristic, complemented
5#22577146C211yesmaximal, characteristic, complemented
6#22577135C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555510—21no—
9C2×#14555499—22no—
10#17984088C2211yescharacteristic
11#17984085C2211yescharacteristic
12#17984084—21no—
13#17984062—21no—
14#17984061—22no—
15C4×Dic274C2211yescharacteristic
16#17984050—21no—
17#17984022—21no—
18#17984019—21no—
19C4⋊#11764040C2211yescharacteristic
20#17984013—21no—
21#17984009—21no—
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25#14555510—42no—
26#14555510—42no—
27C2×Dic274—21no—
28C2×Dic274D411yescharacteristic
29C2×D548—21no—
30C2×D548—21no—
31C2×D548D412yes—
32C2×C22⋊D137—22no—
33C2×C22⋊D137—22no—
34C2×C4×D137—22no—
35#14555499—44no—
36C2×#11764040—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137—21no—
39C4⋊Dic137D411yescharacteristic
40#14555489—21no—
41#14555487—22no—
42C4×Dic137—21no—
43C2×D4×C137D411yescharacteristic
44C22×C4×C137—21no—
45C2×C8×C137—21no—
46C2×C8×C137—21no—
47C4⋊C4×C137D411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137C2311yescharacteristic
50C22×D137—22no—
51Dic274—21no—
52Dic274—42no—
53Dic274SD1612yescomplemented
54D548—42no—
55D548—41no—
56D548—41no—
57D548—41no—
58D548—41no—
59C22⋊D137—44no—
60C22⋊D137—82no—
61C2×Dic137—21no—
62C2×Dic137—21no—
63C2×Dic137—41no—
64C4×D137—44no—
65#11764040—41no—
66C23×C137—21no—
67D4×C137—21no—
68D4×C137—21no—
69C2×C4×C137—21no—
70C2×C4×C137—41no—
71C2×C4×C137—41no—
72C2×C4×C137C2×D411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C2×C4×C137C2×D411yescharacteristic
75C8×C137—22no—
76C8×C137—41no—
77C22×C137—41no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137#12011yescharacteristic
81C2×D137—42no—
82C2×D137—42no—
83C2×D137—42no—
84C4×C137—21no—
85C4×C137—41no—
86C4×C137C2×SD1611yescharacteristic
87C4×C137#13511yescharacteristic
88C4×C137AGL(1,ℤ/8)11yescharacteristic
89C4×C137#13711yescharacteristic
90Dic137—22no—
91Dic137—41no—
92Dic137—81no—
93C2×C137—81no—
94C2×C137#45011yescharacteristic
95C2×C137#45211yescharacteristic
96C2×C137#45311yescharacteristic
97D137—82no—
98C137#163111yescharacteristic, complemented, Sylow
99#1631—1371nomaximal, Sylow
100#550—1371no—
101#492—1371no—
102#477—1371no—
103#476—1371no—
104#331—1371no—
105Q8⋊C8—1371no—
106D4⋊C8D13711yescharacteristic, complemented
107C2×#41—2741no—
108C2×SD16—2742no—
109C4⋊Q8—1371no—
110#127—2741no—
111#127—1371no—
112#121—2742no—
113#121—2741no—
114C4×Q8—1371no—
115C4×D4—2741no—
116C4×D4C2×D13711yescharacteristic
117#107—2741no—
118#106—2741no—
119C4⋊C8—1371no—
120C4⋊C8C2×D13711yescharacteristic
121#103—2741no—
122#102—2741no—
123C4×C8C2×D13711yescharacteristic
124#41—5482no—
125#41—5482no—
126C2×Q8—2741no—
127C2×Q8—1371no—
128C2×D4—2742no—
129C2×D4—2742no—
130C2×D4—2741no—
131C2×D4—2741no—
132C2×D4—1372no—
133C2×D4D54811yescharacteristic
134C22×C4—21no—
135C22×C4—2742no—
136SD16—5484no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—2741no—
140C4⋊C4—2741no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—1371no—
144C4⋊C4D54811yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—2742no—
147C42—2741no—
148C42C22×D13711yescharacteristic
149C23—21no—
150C23—2742no—
151Q8—2741no—
152Q8—5482no—
153Q8—1372no—
154D4—21no—
155D4—21no—
156D4—5482no—
157D4—5484no—
158D4—5481no—
159D4—5481no—
160D4—5481no—
161D4—5481no—
162D4—10962no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—2741no—
167C2×C4—2741no—
168C2×C4—5484no—
169C2×C4—5481no—
170C2×C4D4×D13711yescharacteristic
171C2×C4D4×D13711yescharacteristic
172C2×C4C2×D54811yescharacteristic
173C8—22no—
174C8—41no—
175C8—5481no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—5482no—
180C22—5482no—
181C22—5482no—
182C22#1798405811yescharacteristic
183C4—21no—
184C4—41no—
185C4—2742no—
186C4—5481no—
187C4—10961no—
188C4SD16×D13711yescharacteristic
189C4#1798411011yescharacteristic
190C4#1798411211yescharacteristic
191C4#1798412111yescharacteristic
192C2—81no—
193C2—10962no—
194C2#2257755511yescharacteristic
195C2#2257756711yescharacteristic
196C2#2257757211yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137#120
prime radical#22578074C2
hypercenterD4⋊C8D137
hypocenterC137#1631
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. #22577776C2semidirect
3. #22577699C2semidirect
4. #22577693C2semidirect
5. #22577146C2semidirect
6. #22577135C2semidirect
7. D4⋊C8×C137C2semidirect
53. Dic274SD16semidirect
98. C137#1631semidirect
106. D4⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225780742
#225777762
#225776992
#225776932
#225771462
#225771352
D4⋊C8×C1372
#1631137