Finite Group Databasebeta

← #31795875

Subgroups — #31795875

Order 17536.

234 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 40 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
1D137⋊#99C211yesmaximal, characteristic
2#22578351C211yesmaximal, characteristic
3#100×D137C211yesmaximal, characteristic
4C23×#10587739C411yescharacteristic, complemented
5C2×C22⋊#10587739—21no—
6C2×C22⋊#10587739—21no—
7C2×C22⋊#10587739C411yescharacteristic, complemented
8#17984147—21no—
9#17984146C412yescomplemented
10(C2×#10587739)⋊C4—21no—
11C2×D4×D137C2211yescharacteristic
12M16×D137—21no—
13#17984005C411yescharacteristic, complemented
14#17984004—21no—
15#100×C137C411yescharacteristic, complemented
16C22×#10587739—21no—
17C22×#10587739—21no—
18C22×#10587739—41no—
19C22×#10587739—41no—
20C22×#10587739—42no—
21C22⋊#10587739—22no—
22C22⋊#10587739—41no—
23C22⋊#10587739—41no—
24C22⋊#10587739—42no—
25C23×D137C2×C411yescharacteristic
26C23×D137D411yescharacteristic
27C2×D548D411yescharacteristic
28C2×C22⋊D137C2×C411yescharacteristic
29C2×C22⋊D137Q811yescharacteristic
30D4×D137—41no—
31D4×D137—41no—
32C2×C4×D137D411yescharacteristic
33#14555494—21no—
34#14555493—41no—
35C8×D137—41no—
36C2×D4×C137C2×C411yescharacteristic
37M16×C137—21no—
38C2×#10587739—22no—
39C2×#10587739—41no—
40C2×#10587739—42no—
41C2×#10587739—41no—
42C2×#10587739—42no—
43C2×#10587739—42no—
44C2×#10587739—41no—
45C2×#10587739—41no—
46C2×#10587739—41no—
47C2×#10587739—81no—
48C2×#10587739—81no—
49C22×D137—21no—
50C22×D137—21no—
51C22×D137—21no—
52C22×D137—21no—
53C22×D137—41no—
54C22×D137—41no—
55C22×D137—41no—
56C22×D137—41no—
57C22×D137C22⋊C411yescharacteristic
58C22×D137C22⋊C411yescharacteristic
59C22×D137C4⋊C411yescharacteristic
60D548—41no—
61C22⋊D137—41no—
62C22⋊D137—41no—
63C2×Dic137C4⋊C411yescharacteristic
64C4×D137—21no—
65C4×D137—21no—
66#11764040—41no—
67C23×C137C4211yescharacteristic
68C23×C137C4⋊C411yescharacteristic
69D4×C137—41no—
70C2×C4×C137C22⋊C411yescharacteristic
71C8×C137—41no—
72C22×C137—21no—
73C22×C137—21no—
74C22×C137—41no—
75C22×C137—41no—
76C22×C137#9511yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—21no—
80C2×D137—21no—
81C2×D137—42no—
82C2×D137—41no—
83C2×D137—41no—
84C2×D137—41no—
85C2×D137—41no—
86C2×D137—41no—
87C2×D137—41no—
88C2×D137—81no—
89C2×D137#9911yescharacteristic
90C2×D137#9911yescharacteristic
91#10587739—42no—
92#10587739—81no—
93#10587739—82no—
94C4×C137—21no—
95Dic137—21no—
96C2×C137—21no—
97C2×C137—41no—
98C2×C137—41no—
99C2×C137#34211yescharacteristic
100D137—21no—
101D137—41no—
102D137—41no—
103D137AΣL(1,𝔽16)11yescharacteristic
104D137#35211yescharacteristic
105C137#139011yescharacteristic, complemented, Sylow
106#1390—1371nomaximal, Sylow
107C2×#100—1371no—
108C2×#99—1371no—
109#386—1371no—
110C22×D4—1371no—
111C23×C4—1371no—
112C2×M16—2741no—
113C2×C22⋊C4—2741no—
114C2×C22⋊C4—2741no—
115C2×C22⋊C4—1371no—
116#100—2741no—
117#100—1371no—
118#100#1058773911yescharacteristic, complemented
119#99—2741no—
120#99—1372no—
121#95—2741no—
122C24—1371no—
123C24—1371no—
124C2×D4—5481no—
125C2×D4—5481no—
126C2×D4—1371no—
127C2×D4—1371no—
128C2×D4—1371no—
129C2×D4C2×#1058773911yescharacteristic
130C22×C4—2741no—
131C22×C4—2741no—
132C22×C4—5482no—
133C22×C4—5481no—
134C22×C4—5481no—
135C22×C4—1371no—
136M16—21no—
137M16—2741no—
138M16—5481no—
139C2×C8—5481no—
140C22⋊C4—2742no—
141C22⋊C4—5482no—
142C22⋊C4—5481no—
143C22⋊C4—5481no—
144C23—2741no—
145C23—2741no—
146C23—2741no—
147C23—2741no—
148C23—5481no—
149C23—5481no—
150C23—5481no—
151C23—5481no—
152C23—1371no—
153C23—1371no—
154C23—1371no—
155C23C4×#1058773911yescharacteristic
156C23C4⋊#1058773911yescharacteristic
157D4—41no—
158D4—5481no—
159D4—5481no—
160D4—5481no—
161C2×C4—2741no—
162C2×C4—2741no—
163C2×C4—2742no—
164C2×C4—5481no—
165C2×C4—5482no—
166C2×C4—5481no—
167C2×C4—5482no—
168C2×C4—5482no—
169C2×C4—5481no—
170C2×C4—5481no—
171C2×C4—5481no—
172C2×C4—10961no—
173C2×C4—10961no—
174C2×C4—1371no—
175C2×C4C22⋊#1058773911yescharacteristic
176C8—41no—
177C8—5481no—
178C22—21no—
179C22—21no—
180C22—41no—
181C22—41no—
182C22—2741no—
183C22—2741no—
184C22—2741no—
185C22—2741no—
186C22—5481no—
187C22—5481no—
188C22—5482no—
189C22—5481no—
190C22—5481no—
191C22—5481no—
192C22—5481no—
193C22—10961no—
194C22—1371no—
195C22—1371no—
196C22(C2×#10587739)⋊C411yescharacteristic
197C4—21no—
198C4—2741no—
199C4—5482no—
200C4—10961no—
201C4—10962no—
202C2—21no—
203C2—41no—
204C2—41no—
205C2—2741no—
206C2—5481no—
207C2—5481no—
208C2—1371no—
209C2—1371no—
210C2#2257828011yescharacteristic
subgroupG/H
centerC2#22578280
commutatorC23×C137C42
FrattiniC2×D4C2×#10587739
Fitting#100×C137C4
socleC2×C137#342
prime radical, cosocleC2×D4×D137C22
hypercenter#100#10587739
hypocenterC137#1390
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
4. C23×#10587739C4semidirect
7. C2×C22⋊#10587739C4semidirect
9. #17984146C4semidirect
13. #17984005C4semidirect
15. #100×C137C4semidirect
105. C137#1390semidirect
118. #100#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#992
#225783512
#100×D1372
#1390137