Finite Group Databasebeta

← #31796552

Subgroups — #31796552

Order 17536.

320 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×#17984152C212yesmaximal
2C4×#14555520C211yesmaximal, characteristic, complemented
3#22578339C211yesmaximal, characteristic, complemented
4#22578273C212yesmaximal, complemented
5C4×#14555509C211yesmaximal, characteristic
6C2×#14555520C2212yes—
7C2×C4×#10587739C2212yes—
8#17984152C418yescomplemented
9#17984143—21no—
10C4×#11764047—21no—
11(C2×#10587739)⋊C4—21no—
12(C2×#10587739)⋊C4—21no—
13C2×#14555509C2211yescharacteristic
14C4×Dic274C411yescharacteristic, complemented
15#17984053C411yescharacteristic, complemented
16D4×Dic137C411yescharacteristic, complemented
17C4×C22⋊D137—21no—
18#17984048—21no—
19#17984046C2211yescharacteristic
20#17984044—21no—
21C2×C4×Dic137—21no—
22C42×D137C2211yescharacteristic
23C4×D4×C137C411yescharacteristic, complemented
24C22×#10587739—22no—
25#14555520—22no—
26#14555520C2×C414yescomplemented
27C2×#11764047—22no—
28C4×#10587739—22no—
29C4×#10587739C2×C414yescomplemented
30#14555509—22no—
31#14555509C2×C414yes—
32C2×Dic274C2×C411yescharacteristic
33C2×C22⋊D137—21no—
34C22×Dic137—21no—
35C22×Dic137—21no—
36C2×C4×D137D411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137C2311yescharacteristic
39C4⋊Dic137C2×C411yescharacteristic
40#14555489—21no—
41#14555489—21no—
42C22⋊Dic137—21no—
43#14555487—21no—
44C4×Dic137—21no—
45C4×Dic137—21no—
46C4×Dic137—21no—
47C4×Dic137C2×C411yescharacteristic
48C4×Dic137D411yescharacteristic
49C4×Dic137D411yescharacteristic
50C2×D4×C137C2×C411yescharacteristic
51C22×C4×C137—21no—
52C4⋊C4×C137C2×C411yescharacteristic
53C22⋊C4×C137—21no—
54C42×C137C2×C411yescharacteristic
55C2×#10587739—24no—
56C2×#10587739—42no—
57C2×#10587739—42no—
58#11764047—24no—
59C22×D137C2×D411yescharacteristic
60Dic274—21no—
61Dic274C4212yes—
62C22⋊D137—24no—
63C2×Dic137—21no—
64C2×Dic137—24no—
65C2×Dic137—21no—
66C2×Dic137—42no—
67C2×Dic137C22⋊C411yescharacteristic
68C2×Dic137C22⋊C411yescharacteristic
69C2×Dic137C2×D411yescharacteristic
70C4×D137—41no—
71C4×D137—41no—
72C4×D137C22×C412yes—
73C4×D137#4112yes—
74C23×C137—21no—
75D4×C137—21no—
76D4×C137C4212yes—
77C2×C4×C137—21no—
78C2×C4×C137—42no—
79C2×C4×C137C22⋊C411yescharacteristic
80C2×C4×C137C22⋊C411yescharacteristic
81C2×C4×C137C22×C411yescharacteristic
82C22×C137—22no—
83C22×C137—41no—
84C22×C137—41no—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—21no—
87C2×D137—21no—
88C2×D137C4×D412yes—
89#10587739—44no—
90C4×C137—21no—
91C4×C137—21no—
92C4×C137—41no—
93C4×C137C2×C4211yescharacteristic
94C4×C137#11711yescharacteristic
95Dic137—22no—
96Dic137—21no—
97Dic137—21no—
98Dic137—41no—
99Dic137C4×D412yes—
100C2×C137—42no—
101C2×C137C4×C22⋊C411yescharacteristic
102C2×C137#42111yescharacteristic
103C2×C137#42111yescharacteristic
104D137—22no—
105C137#175911yescharacteristic, complemented, Sylow
106#1759—1371nomaximal, Sylow
107C4×#41—1371no—
108C2×#104—1372no—
109C4×M16—1371no—
110#376—1371no—
111#339—1372no—
112C2×#41—1371no—
113C2×M16—1372no—
114C4×Q8—1371no—
115C4×Q8—1371no—
116C4×D4—2741no—
117C4×D4—2741no—
118C4×D4—1371no—
119C4×D4#1058773911yescharacteristic, complemented
120#117—2741no—
121#117—1371no—
122C2×C42—2741no—
123C2×C42—1372no—
124C2×C42—1371no—
125#104—1378no—
126#97—2741no—
127C4×C8—2741no—
128#95—2741no—
129#95—2741no—
130#41—2742no—
131#41—1374no—
132C2×Q8—1371no—
133C2×D4—2741no—
134C2×D4C2×#1058773911yescharacteristic
135C22×C4—21no—
136C22×C4—2741no—
137C22×C4—2742no—
138C22×C4—2741no—
139C22×C4—1371no—
140C22×C4—1371no—
141C22×C4—1371no—
142M16—2742no—
143M16—1374no—
144C2×C8—2742no—
145C4⋊C4—2741no—
146C4⋊C4—2741no—
147C4⋊C4—1371no—
148C4⋊C4C2×#1058773911yescharacteristic
149C22⋊C4—21no—
150C22⋊C4—2741no—
151C22⋊C4—2741no—
152C42—2741no—
153C42—2741no—
154C42—2742no—
155C42—2741no—
156C42—1374no—
157C42—1371no—
158C42—1371no—
159C42—1371no—
160C42C2×#1058773911yescharacteristic
161C23—21no—
162C23—1371no—
163Q8—2741no—
164Q8—1372no—
165D4—21no—
166D4—2744no—
167D4C4×#1058773912yes—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—2741no—
171C2×C4—2744no—
172C2×C4—2744no—
173C2×C4—2741no—
174C2×C4—5481no—
175C2×C4—5481no—
176C2×C4—5482no—
177C2×C4—5482no—
178C2×C4—5482no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1372no—
183C2×C4—1372no—
184C2×C4C22⋊#1058773911yescharacteristic
185C2×C4C22⋊#1058773911yescharacteristic
186C2×C4C22×#1058773911yescharacteristic
187C8—2744no—
188C22—22no—
189C22—41no—
190C22—41no—
191C22—2741no—
192C22—2741no—
193C22—1372no—
194C22C2×C22⋊#1058773911yescharacteristic
195C4—21no—
196C4—21no—
197C4—41no—
198C4—2742no—
199C4—2741no—
200C4—2741no—
201C4—5484no—
202C4—5481no—
203C4—1372no—
204C4C2×C4×#1058773911yescharacteristic
205C4#1798416711yescharacteristic
206C2—42no—
207C2—2742no—
208C2C4×C22⋊#1058773911yescharacteristic
209C2#2257840011yescharacteristic
210C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC4×C137C2×C42
FrattiniC2×C4C22×#10587739
FittingC4×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#14555509C22
cosocleC2×C4×D137C23
hypercenterC4×D4#10587739
hypocenterC137#1759
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. C4×#14555520C2semidirect
3. #22578339C2semidirect
4. #22578273C2semidirect
8. #17984152C4semidirect
14. C4×Dic274C4semidirect
15. #17984053C4semidirect
16. D4×Dic137C4semidirect
23. C4×D4×C137C4semidirect
26. #14555520C2×C4semidirect
29. C4×#10587739C2×C4semidirect
105. C137#1759semidirect
119. C4×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841522
C4×#145555202
#225783392
#225782732
C4×#145555092
#1759137