Finite Group Databasebeta

← #31779880

Subgroups — #31779880

Order 17536.

256 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 57 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#22578057C211yesmaximal, characteristic, complemented
2#22577750C211yesmaximal, characteristic, complemented
3#22577599C211yesmaximal, characteristic, complemented
4#22577598C211yesmaximal, characteristic, complemented
5#22577148C211yesmaximal, characteristic, complemented
6#22577132C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555509—21no—
9C2×D1096—21no—
10C2×#14555495—21no—
11#17984088C2211yescharacteristic
12#17984085C2211yescharacteristic
13#17984072—22no—
14#17984068—21no—
15#17984062—21no—
16C4×D548C2211yescharacteristic
17C2×C4⋊Dic137—21no—
18C4⋊#11764040C2211yescharacteristic
19#17984010—22no—
20#17984010—21no—
21#17984008—21no—
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4×C8×C137C2211yescharacteristic
25#14555509—42no—
26#14555509—42no—
27C2×Dic274—21no—
28C2×D548—21no—
29C2×D548—21no—
30C2×D548—21no—
31C2×D548D411yescharacteristic
32C2×C22⋊D137—22no—
33C22×Dic137—22no—
34C2×C4×D137—21no—
35D1096—42no—
36#14555495—42no—
37C2×#11764040—21no—
38C4⋊Dic137—22no—
39C4⋊Dic137—21no—
40C4⋊Dic137D411yescharacteristic
41C4⋊Dic137D412yes—
42#14555487—22no—
43#14555487—21no—
44C2×D4×C137D411yescharacteristic
45C22×C4×C137—21no—
46C2×C8×C137—21no—
47C2×C8×C137—21no—
48C4⋊C4×C137D411yescharacteristic
49C22⋊C4×C137—21no—
50C42×C137C2311yescharacteristic
51C22×D137—21no—
52C22×D137—41no—
53Dic274—41no—
54Dic274—41no—
55D548—21no—
56D548—21no—
57D548—42no—
58D548—41no—
59D548—41no—
60D548—41no—
61D548—41no—
62C22⋊D137—44no—
63C2×Dic137—21no—
64C2×Dic137—22no—
65C2×Dic137—44no—
66C4×D137—41no—
67C4×D137—41no—
68#11764040—41no—
69C23×C137—21no—
70D4×C137—21no—
71D4×C137D812yescomplemented
72C2×C4×C137—21no—
73C2×C4×C137—41no—
74C2×C4×C137—41no—
75C2×C4×C137C2×D411yescharacteristic
76C2×C4×C137C2×D411yescharacteristic
77C2×C4×C137C2×D411yescharacteristic
78C8×C137—41no—
79C8×C137—41no—
80C22×C137—22no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137#12011yescharacteristic
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—41no—
87C2×D137—81no—
88C2×D137—81no—
89C2×D137—81no—
90C4×C137—21no—
91C4×C137—41no—
92C4×C137C2×D811yescharacteristic
93C4×C137#13511yescharacteristic
94C4×C137AGL(1,ℤ/8)11yescharacteristic
95C4×C137AGL(1,ℤ/8)11yescharacteristic
96Dic137—41no—
97Dic137—42no—
98C2×C137—42no—
99C2×C137#44711yescharacteristic
100C2×C137#44911yescharacteristic
101C2×C137#45411yescharacteristic
102D137—81no—
103D137—161no—
104C137#162511yescharacteristic, complemented, Sylow
105#1625—1371nomaximal, Sylow
106#547—1371no—
107#486—1371no—
108#459—1371no—
109#459—1371no—
110#332—1371no—
111D4⋊C8—1371no—
112D4⋊C8D13711yescharacteristic, complemented
113C2×#41—2741no—
114C2×D8—2741no—
115C2×D8—2741no—
116C4⋊Q8—1371no—
117#127—1371no—
118#123—2742no—
119C22⋊Q8—2741no—
120#121—2741no—
121C4×D4—1371no—
122C4×D4C2×D13711yescharacteristic
123C2×C4⋊C4—2741no—
124C4⋊C8—1371no—
125C4⋊C8C2×D13711yescharacteristic
126#102—2741no—
127#102—2742no—
128#102—2741no—
129C4×C8C2×D13711yescharacteristic
130#41—5482no—
131#41—5482no—
132C2×Q8—2741no—
133C2×D4—2742no—
134C2×D4—2741no—
135C2×D4—2741no—
136C2×D4—2741no—
137C2×D4—1371no—
138C2×D4D54811yescharacteristic
139C22×C4—21no—
140C22×C4—2742no—
141C22×C4—2741no—
142D8—5482no—
143D8—5482no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—2741no—
147C4⋊C4—2742no—
148C4⋊C4—2741no—
149C4⋊C4—1371no—
150C4⋊C4—1372no—
151C4⋊C4D54811yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2742no—
154C22⋊C4—2741no—
155C42C22×D13711yescharacteristic
156C23—21no—
157C23—2741no—
158C23—5481no—
159Q8—5481no—
160Q8—5481no—
161D4—21no—
162D4—2741no—
163D4—2741no—
164D4—5482no—
165D4—5484no—
166D4—5481no—
167D4—5481no—
168D4—5481no—
169D4—5481no—
170D4D109612yescomplemented
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—2741no—
175C2×C4—2742no—
176C2×C4—5484no—
177C2×C4—5481no—
178C2×C4—5481no—
179C2×C4D4×D13711yescharacteristic
180C2×C4D4×D13711yescharacteristic
181C2×C4C2×D54811yescharacteristic
182C8—41no—
183C8—41no—
184C8—5481no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—5481no—
189C22—5481no—
190C22—5481no—
191C22—10961no—
192C22—10961no—
193C22—10961no—
194C22#1798405811yescharacteristic
195C4—21no—
196C4—41no—
197C4—5481no—
198C4—5482no—
199C4C2×D109611yescharacteristic
200C4#1798410911yescharacteristic
201C4#1798411211yescharacteristic
202C4#1798411611yescharacteristic
203C2—42no—
204C2—10961no—
205C2—21921no—
206C2#2257753911yescharacteristic
207C2#2257754911yescharacteristic
208C2#2257757711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137#120
hypercenterD4⋊C8D137
hypocenterC137#1625
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
1. #22578057C2semidirect
2. #22577750C2semidirect
3. #22577599C2semidirect
4. #22577598C2semidirect
5. #22577148C2semidirect
6. #22577132C2semidirect
7. D4⋊C8×C137C2semidirect
71. D4×C137D8semidirect
104. C137#1625semidirect
112. D4⋊C8D137semidirect
170. D4D1096semidirect

Maximal subgroups

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

subgroupindex
#225780572
#225777502
#225775992
#225775982
#225771482
#225771322
D4⋊C8×C1372
#1625137