Finite Group Databasebeta

← #31779943

Subgroups — #31779943

Order 17536.

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

normal subgroups only

This group has 234 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#22578058C211yesmaximal, characteristic
2#22577754C211yesmaximal, characteristic, complemented
3#22577694C211yesmaximal, characteristic, complemented
4#22577693C211yesmaximal, characteristic, complemented
5#22577150C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555510—21no—
9#17984088C2211yescharacteristic
10#17984088C2211yescharacteristic
11#17984070—22no—
12#17984067—21no—
13#17984059—21no—
14C4×D548C2211yescharacteristic
15C4⋊C4×D137—21no—
16#17984020—21no—
17#17984019—21no—
18C4⋊#11764040C2211yescharacteristic
19#17984013—22no—
20#17984010—21no—
21#17984009—21no—
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25#14555510—42no—
26#14555510—42no—
27C2×Dic274—21no—
28C2×Dic274—21no—
29C2×D548D411yescharacteristic
30C2×C22⋊D137—22no—
31C2×C4×D137—22no—
32C2×C4×D137—21no—
33C2×#11764040—21no—
34C4⋊Dic137—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137D412yes—
38C4⋊Dic137D411yescharacteristic
39#14555489—22no—
40C22⋊Dic137—22no—
41#14555487—21no—
42C2×D4×C137D411yescharacteristic
43C22×C4×C137—21no—
44C2×C8×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137D411yescharacteristic
47C22⋊C4×C137—21no—
48C42×C137C2311yescharacteristic
49C22×D137—21no—
50Dic274—41no—
51Dic274—41no—
52Dic274—42no—
53D548—21no—
54D548SD1612yes—
55C22⋊D137—44no—
56C2×Dic137—21no—
57C2×Dic137—22no—
58C2×Dic137—41no—
59C4×D137—44no—
60C4×D137—41no—
61C4×D137—41no—
62#11764040—41no—
63C23×C137—21no—
64D4×C137—21no—
65D4×C137—21no—
66C2×C4×C137—21no—
67C2×C4×C137—41no—
68C2×C4×C137—41no—
69C2×C4×C137C2×D411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137—41no—
74C22×C137—41no—
75C22×C137—41no—
76C22×C137—41no—
77C22×C137#12011yescharacteristic
78C2×D137—22no—
79C2×D137—41no—
80C2×D137—41no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137C2×SD1611yescharacteristic
84C4×C137#13511yescharacteristic
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137#13711yescharacteristic
87Dic137—42no—
88Dic137—41no—
89Dic137—81no—
90C2×C137—81no—
91C2×C137#45011yescharacteristic
92C2×C137#45211yescharacteristic
93C2×C137#45411yescharacteristic
94D137—42no—
95C137#163511yescharacteristic, complemented, Sylow
96#1635—1371nomaximal, Sylow
97#547—1371no—
98#487—1371no—
99#476—1371no—
100#476—1371no—
101#332—1371no—
102D4⋊C8—1371no—
103D4⋊C8D13711yescharacteristic, complemented
104C2×#41—2741no—
105C4⋊Q8—1371no—
106C4⋊Q8—1371no—
107#123—2742no—
108C22⋊Q8—2741no—
109#121—2741no—
110C4×D4—1371no—
111C4×D4C2×D13711yescharacteristic
112C2×C4⋊C4—2741no—
113#106—2741no—
114#106—2741no—
115C4⋊C8—1371no—
116C4⋊C8C2×D13711yescharacteristic
117#103—2742no—
118#102—2741no—
119#102—2741no—
120C4×C8C2×D13711yescharacteristic
121#41—5482no—
122#41—5482no—
123C2×Q8—2741no—
124C2×Q8—2741no—
125C2×D4—2742no—
126C2×D4—1371no—
127C2×D4D54811yescharacteristic
128C22×C4—21no—
129C22×C4—2742no—
130C22×C4—2741no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2741no—
134C4⋊C4—2742no—
135C4⋊C4—2741no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C4⋊C4—1372no—
139C4⋊C4—1371no—
140C4⋊C4D54811yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—2742no—
143C22⋊C4—2741no—
144C42C22×D13711yescharacteristic
145C23—21no—
146C23—2741no—
147Q8—5482no—
148Q8—5481no—
149Q8—5481no—
150D4—21no—
151D4—21no—
152D4—2741no—
153D4—5484no—
154D4—1372no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—2741no—
159C2×C4—2742no—
160C2×C4—5484no—
161C2×C4—5481no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C2×C4C2×D54811yescharacteristic
167C8—41no—
168C8—41no—
169C8—5481no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—2742no—
174C22—5481no—
175C22—5481no—
176C22#1798405811yescharacteristic
177C4—21no—
178C4—41no—
179C4—5482no—
180C4—5481no—
181C4—10961no—
182C4SD16×D13711yescharacteristic
183C4#1798411011yescharacteristic
184C4#1798411211yescharacteristic
185C4#1798412111yescharacteristic
186C2—81no—
187C2—5482no—
188C2#2257755511yescharacteristic
189C2#2257756711yescharacteristic
190C2#2257757711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137#120
prime radical#22578058C2
hypercenterD4⋊C8D137
hypocenterC137#1635
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. #22577754C2semidirect
3. #22577694C2semidirect
4. #22577693C2semidirect
5. #22577150C2semidirect
6. #22577132C2semidirect
7. D4⋊C8×C137C2semidirect
95. C137#1635semidirect
103. D4⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225780582
#225777542
#225776942
#225776932
#225771502
#225771322
D4⋊C8×C1372
#1635137