Finite Group Databasebeta

← #31780107

Subgroups — #31780107

Order 17536.

208 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 45 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#22578147C211yesmaximal, characteristic
2#22577812C211yesmaximal, characteristic, complemented
3#22577798C211yesmaximal, characteristic, complemented
4#22577763C211yesmaximal, characteristic, complemented
5#22577144C211yesmaximal, characteristic, complemented
6#22577142C211yesmaximal, characteristic, complemented
7#329×C137C211yesmaximal, characteristic, complemented
8C2×#14555500—21no—
9C2×#14555498—21no—
10#17984088C2211yescharacteristic
11#17984080C2211yescharacteristic
12#17984076C2211yescharacteristic
13#17984072—21no—
14#17984070—21no—
15#17984065—21no—
16#17984064—21no—
17#17984061—21no—
18#17984060—21no—
19#17984022—21no—
20#17984020—21no—
21#17984013—21no—
22#17984010—21no—
23#17983999C2211yescharacteristic
24#124×C137C2211yescharacteristic
25#97×C137C2211yescharacteristic
26#97×C137C2211yescharacteristic
27C2×Dic274—21no—
28C2×Dic274D411yescharacteristic
29C2×D548D411yescharacteristic
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C22×Dic137—21no—
33C2×C4×D137—21no—
34#14555500—42no—
35#14555498—42no—
36C2×#11764040—21no—
37C4⋊Dic137—21no—
38C4⋊Dic137—21no—
39C4⋊Dic137—21no—
40C4⋊Dic137D411yescharacteristic
41C4⋊Dic137D411yescharacteristic
42#14555489—21no—
43#14555489—21no—
44#14555489—21no—
45C22⋊Dic137—21no—
46#14555487—21no—
47#14555487—21no—
48#14555487—21no—
49C2×Q8×C137D411yescharacteristic
50C2×D4×C137D411yescharacteristic
51C2×C8×C137—21no—
52C2×C8×C137—21no—
53C22⋊C4×C137—21no—
54C22⋊C4×C137—21no—
55C42×C137C2311yescharacteristic
56C22×D137—21no—
57Dic274—41no—
58Dic274—41no—
59Dic274—41no—
60D548—41no—
61C22⋊D137—81no—
62C22⋊D137—81no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—41no—
67C2×Dic137—81no—
68C4×D137—81no—
69#11764040—22no—
70C23×C137—21no—
71Q8×C137—41no—
72D4×C137—41no—
73C2×C4×C137—21no—
74C2×C4×C137C2×D411yescharacteristic
75C2×C4×C137C2×D411yescharacteristic
76C2×C4×C137C2×D411yescharacteristic
77C8×C137—22no—
78C8×C137—41no—
79C22×C137—41no—
80C22×C137—41no—
81C22×C137—41no—
82C22×C137#12011yescharacteristic
83C2×D137—41no—
84C2×D137—41no—
85C2×D137—41no—
86C4×C137—21no—
87C4×C137—21no—
88C4×C137—21no—
89C4×C137—41no—
90Dic137—41no—
91Dic137—41no—
92Dic137—41no—
93Dic137—81no—
94C2×C137—81no—
95C2×C137#45311yescharacteristic
96C2×C137#45511yescharacteristic
97C2×C137#45611yescharacteristic
98D137—81no—
99C137#166111yescharacteristic, complemented, Sylow
100#1661—1371nomaximal, Sylow
101#562—1371no—
102#501—1371no—
103#497—1371no—
104#489—1371no—
105#330—1371no—
106#329—1371no—
107#329D13711yescharacteristic, complemented
108C2×Q16—2741no—
109C2×SD16—2741no—
110C4⋊Q8—1371no—
111#125—1371no—
112#124—1371no—
113#124C2×D13711yescharacteristic
114#123—2741no—
115#123—2741no—
116C22⋊Q8—2741no—
117C22⋊Q8—2741no—
118#121—2741no—
119#121—2741no—
120#107—2741no—
121#106—2741no—
122#103—2741no—
123#102—2741no—
124#97—1371no—
125#97C2×D13711yescharacteristic
126#97C2×D13711yescharacteristic
127C2×Q8—2741no—
128C2×Q8—1371no—
129C2×Q8D54811yescharacteristic
130C2×D4—2741no—
131C2×D4—2741no—
132C2×D4—1371no—
133C2×D4D54811yescharacteristic
134C22×C4—2741no—
135C22×C4—2741no—
136Q16—5482no—
137SD16—5482no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—2741no—
141C4⋊C4—2741no—
142C4⋊C4—2741no—
143C4⋊C4—2741no—
144C4⋊C4—2741no—
145C4⋊C4—2741no—
146C4⋊C4—2741no—
147C4⋊C4—1371no—
148C4⋊C4—1371no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—2741no—
152C22⋊C4—2741no—
153C22⋊C4—2741no—
154C22⋊C4—2741no—
155C42C22×D13711yescharacteristic
156C23—21no—
157C23—2741no—
158Q8—41no—
159Q8—5481no—
160Q8—5481no—
161Q8—5481no—
162D4—41no—
163D4—5481no—
164D4—10961no—
165D4—10961no—
166C2×C4—21no—
167C2×C4—2741no—
168C2×C4—2741no—
169C2×C4—2741no—
170C2×C4—5481no—
171C2×C4—10961no—
172C2×C4—10961no—
173C2×C4D4×D13711yescharacteristic
174C2×C4D4×D13711yescharacteristic
175C2×C4C2×D54811yescharacteristic
176C8—22no—
177C8—41no—
178C8—2742no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—5481no—
183C22—5481no—
184C22—5481no—
185C22#1798405811yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—5481no—
191C4—5481no—
192C4—5481no—
193C4—10961no—
194C2—81no—
195C2—10961no—
196C2#2257757311yescharacteristic
197C2#2257758211yescharacteristic
198C2#2257758711yescharacteristic
subgroupG/H
centerC22#17984058
commutator, cosocleC42×C137C23
FrattiniC42C22×D137
Fitting#329×C137C2
socleC22×C137#120
prime radical#22578147C2
hypercenter#329D137
hypocenterC137#1661
LCSG ▷ C42×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22577812C2semidirect
3. #22577798C2semidirect
4. #22577763C2semidirect
5. #22577144C2semidirect
6. #22577142C2semidirect
7. #329×C137C2semidirect
99. C137#1661semidirect
107. #329D137semidirect

Maximal subgroups

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

subgroupindex
#225781472
#225778122
#225777982
#225777632
#225771442
#225771422
#329×C1372
#1661137