Finite Group Databasebeta

← #31779876

Subgroups — #31779876

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#22577749C211yesmaximal, characteristic, complemented
3#22577595C211yesmaximal, characteristic, complemented
4#22577150C211yesmaximal, characteristic, complemented
5D4⋊#11764040C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7#459×C137C211yesmaximal, characteristic, complemented
8C2×#14555509—21no—
9C2×#14555495—21no—
10#17984088C2211yescharacteristic
11#17984072—22no—
12#17984068—21no—
13#17984062—21no—
14C4×D548C2211yescharacteristic
15C2×C4⋊Dic137—21no—
16C4⋊#11764040C2211yescharacteristic
17#17984009—22no—
18#17984009—21no—
19C4×#11764040C2211yescharacteristic
20C2×D8×C137—21no—
21#127×C137C2211yescharacteristic
22C4×D4×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24#102×C137—21no—
25#14555509—42no—
26#14555509—42no—
27C2×Dic274—21no—
28C2×D548D411yescharacteristic
29C2×C22⋊D137—22no—
30C22×Dic137—22no—
31C2×C4×D137—21no—
32#14555495—42no—
33C2×#11764040—21no—
34C2×#11764040—21no—
35C4⋊Dic137—22no—
36C4⋊Dic137—21no—
37C4⋊Dic137D411yescharacteristic
38C4⋊Dic137D412yes—
39#14555487—22no—
40#14555487—21no—
41C2×D4×C137—21no—
42C2×D4×C137—21no—
43C2×D4×C137—21no—
44C2×D4×C137D411yescharacteristic
45C22×C4×C137—21no—
46D8×C137—42no—
47C2×C8×C137—21no—
48C4⋊C4×C137D411yescharacteristic
49C22⋊C4×C137—21no—
50C42×C137C2311yescharacteristic
51C22×D137—21no—
52Dic274—41no—
53Dic274—41no—
54D548—21no—
55D548—21no—
56C22⋊D137—44no—
57C2×Dic137—21no—
58C2×Dic137—22no—
59C2×Dic137—44no—
60C4×D137—41no—
61C4×D137—41no—
62#11764040—41no—
63#11764040—41no—
64C23×C137—21no—
65C23×C137—41no—
66D4×C137—21no—
67D4×C137—42no—
68D4×C137—41no—
69D4×C137—41no—
70D4×C137—41no—
71D4×C137—41no—
72D4×C137D812yescomplemented
73C2×C4×C137—21no—
74C2×C4×C137—41no—
75C2×C4×C137—41no—
76C2×C4×C137C2×D411yescharacteristic
77C2×C4×C137C2×D411yescharacteristic
78C2×C4×C137C2×D411yescharacteristic
79C8×C137—41no—
80C22×C137—22no—
81C22×C137—41no—
82C22×C137—41no—
83C22×C137—81no—
84C22×C137—81no—
85C22×C137—81no—
86C22×C137#12011yescharacteristic
87C2×D137—41no—
88C2×D137—41no—
89C2×D137—41no—
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—161no—
100C2×C137#44711yescharacteristic
101C2×C137#44911yescharacteristic
102C2×C137#45411yescharacteristic
103D137—81no—
104C137#162511yescharacteristic, complemented, Sylow
105#1625—1371nomaximal, Sylow
106#547—1371no—
107#486—1371no—
108#459—1371no—
109#459D13711yescharacteristic, complemented
110#332—1371no—
111D4⋊C8—1371no—
112D4⋊C8—1371no—
113C2×#41—2741no—
114C2×D8—21no—
115C2×D8—2741no—
116C4⋊Q8—1371no—
117#127C2×D13711yescharacteristic
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—21no—
127#102—2741no—
128#102—2742no—
129C4×C8—1371no—
130#41—5482no—
131#41—5482no—
132C2×Q8—2741no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—2742no—
137C2×D4—1371no—
138C2×D4C22⋊D13711yescharacteristic
139C22×C4—21no—
140C22×C4—2742no—
141C22×C4—2741no—
142D8—42no—
143D8—5482no—
144C2×C8—21no—
145C2×C8—2741no—
146C2×C8—2741no—
147C4⋊C4—2742no—
148C4⋊C4—2741no—
149C4⋊C4—1371no—
150C4⋊C4—1372no—
151C4⋊C4C22⋊D13711yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2742no—
154C22⋊C4—2741no—
155C42C22×D13711yescharacteristic
156C23—21no—
157C23—41no—
158C23—2741no—
159Q8—5481no—
160Q8—5481no—
161D4—21no—
162D4—42no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—2741no—
168D4—2741no—
169D4—5484no—
170D4#1455549512yescomplemented
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×C22⋊D13711yescharacteristic
182C8—41no—
183C8—5481no—
184C8—5481no—
185C22—22no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—5481no—
192C22—5481no—
193C22—5481no—
194C22#1798405611yescharacteristic
195C4—21no—
196C4—41no—
197C4—5481no—
198C4—5482no—
199C4C2×#1455549511yescharacteristic
200C4#1798410711yescharacteristic
201C4#1798411211yescharacteristic
202C4#1798411311yescharacteristic
203C2—42no—
204C2—161no—
205C2—10961no—
206C2#2257753611yescharacteristic
207C2#2257754511yescharacteristic
208C2#2257757411yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#459×C137C2
socleC22×C137#120
hypercenter#459D137
hypocenterC137#1625
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22578057C2semidirect
2. #22577749C2semidirect
3. #22577595C2semidirect
4. #22577150C2semidirect
5. D4⋊#11764040C2semidirect
6. #22577130C2semidirect
7. #459×C137C2semidirect
72. D4×C137D8semidirect
104. C137#1625semidirect
109. #459D137semidirect
170. D4#14555495semidirect

Maximal subgroups

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

subgroupindex
#225780572
#225777492
#225775952
#225771502
D4⋊#117640402
#225771302
#459×C1372
#1625137