Finite Group Databasebeta

← #31796225

Subgroups — #31796225

Order 17536.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578371C211yesmaximal, characteristic, complemented
2#22578345C211yesmaximal, characteristic, complemented
3#22578254C214yesmaximal, complemented
4C2×#17984059C211yesmaximal, characteristic
5C2×#14555518—21no—
6C2×C4⋊#10587739—21no—
7C2×C22⋊#10587739—21no—
8#17984145C2212yescomplemented
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4C2212yescomplemented
11C22×C22⋊D137—21no—
12C22×C4×D137C2211yescharacteristic
13#17984059—21no—
14#17984059—21no—
15#17984059C412yescomplemented
16#17984059C2212yes—
17C2×C4⋊Dic137C411yescharacteristic, complemented
18C2×C22⋊Dic137—21no—
19C22×D4×C137C411yescharacteristic, complemented
20C22×#10587739—22no—
21#14555518—42no—
22C2×#11764047—22no—
23C4⋊#10587739—42no—
24C22⋊#10587739—42no—
25C23×D137D411yescharacteristic
26C2×C22⋊D137—22no—
27C2×C22⋊D137—22no—
28C2×C22⋊D137—42no—
29C2×C22⋊D137—42no—
30C22×Dic137—21no—
31C22×Dic137D411yescharacteristic
32C2×C4×D137—42no—
33C2×C4×D137C2×C411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×C4×D137D411yescharacteristic
36C2×C4×D137C2311yescharacteristic
37C4⋊Dic137—21no—
38C4⋊Dic137C2×C412yes—
39C22⋊Dic137—22no—
40C22⋊Dic137—22no—
41C24×C137—21no—
42C2×D4×C137—21no—
43C2×D4×C137—42no—
44C2×D4×C137C2×C412yes—
45C22×C4×C137C2×C411yescharacteristic
46C2×#10587739—42no—
47C2×#10587739—42no—
48C2×#10587739—42no—
49#11764047—42no—
50C22×D137—41no—
51C22×D137—41no—
52C22×D137C22⋊C411yescharacteristic
53C22×D137D812yes—
54C22×D137SD1612yes—
55C22×D137C2×D411yescharacteristic
56C22⋊D137—44no—
57C22⋊D137—42no—
58C22⋊D137—42no—
59C2×Dic137—22no—
60C2×Dic137—41no—
61C2×Dic137—42no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C2×D411yescharacteristic
64C4×D137—22no—
65C4×D137—22no—
66C4×D137—41no—
67C4×D137—41no—
68C23×C137—22no—
69C23×C137—42no—
70C23×C137—42no—
71C23×C137—42no—
72C23×C137C22⋊C411yescharacteristic
73D4×C137—42no—
74D4×C137—42no—
75C2×C4×C137—22no—
76C2×C4×C137C22⋊C411yescharacteristic
77C2×C4×C137C22×C411yescharacteristic
78C22×C137—21no—
79C22×C137—21no—
80C22×C137—42no—
81C22×C137—42no—
82C22×C137—42no—
83C22×C137—84no—
84C22×C137#10212yes—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—42no—
91C2×D137—42no—
92C2×D137#10212yes—
93C2×D137C2×D811yescharacteristic
94C2×D137C2×SD1611yescharacteristic
95#10587739—82no—
96C4×C137—21no—
97C4×C137#9912yes—
98Dic137—21no—
99Dic137—21no—
100Dic137—42no—
101C2×C137—22no—
102C2×C137—82no—
103C2×C137C2×#9911yescharacteristic
104C2×C137C2×#10211yescharacteristic
105C2×C137#42111yescharacteristic
106D137—21no—
107D137—22no—
108D137—21no—
109C137#151911yescharacteristic, complemented, Sylow
110#1519—1371nomaximal, Sylow
111C2×#121—1371no—
112C2×C22⋊C8—1371no—
113#380—1371no—
114#327—1374no—
115C22×D4—2741no—
116C22×D4#1058773911yescharacteristic, complemented
117C23×C4—1371no—
118C22×C8—2741no—
119#121—2741no—
120#121—2741no—
121#121—1372no—
122#121—1372no—
123C2×C4⋊C4—2741no—
124C2×C4⋊C4—1371no—
125C2×C22⋊C4—2741no—
126C2×C22⋊C4—2741no—
127C22⋊C8—2741no—
128C22⋊C8—1372no—
129#95—1372no—
130C24—21no—
131C24—1371no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—2742no—
135C2×D4—2742no—
136C2×D4—5482no—
137C2×D4—5482no—
138C2×D4C2×#1058773912yes—
139C22×C4—2741no—
140C22×C4—2742no—
141C22×C4—5482no—
142C22×C4—1371no—
143C22×C4—1371no—
144C22×C4—1371no—
145C22×C4—1371no—
146C22×C4—1371no—
147C22×C4C2×#1058773911yescharacteristic
148C2×C8—2742no—
149C2×C8—5482no—
150C4⋊C4—2741no—
151C4⋊C4—5482no—
152C4⋊C4—1372no—
153C22⋊C4—2742no—
154C22⋊C4—2742no—
155C22⋊C4—5482no—
156C23—22no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—5481no—
161C23—5481no—
162C23—1371no—
163C23—1371no—
164C23—1372no—
165C23—1372no—
166C23C22⋊#1058773911yescharacteristic
167D4—42no—
168D4—42no—
169D4—5482no—
170D4—5484no—
171D4—5482no—
172C2×C4—22no—
173C2×C4—2742no—
174C2×C4—2742no—
175C2×C4—2742no—
176C2×C4—5481no—
177C2×C4—5481no—
178C2×C4—5481no—
179C2×C4—5482no—
180C2×C4—5482no—
181C2×C4—5482no—
182C2×C4—5482no—
183C2×C4—1371no—
184C2×C4—1371no—
185C2×C4C22⋊#1058773911yescharacteristic
186C2×C4C22×#1058773911yescharacteristic
187C8—5482no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—84no—
194C22—2741no—
195C22—2741no—
196C22—2741no—
197C22—2741no—
198C22—5482no—
199C22—5482no—
200C22—1371no—
201C22—1372no—
202C22—1371no—
203C22#1798415012yes—
204C22C2×C22⋊#1058773911yescharacteristic
205C4—21no—
206C4—2741no—
207C4—2741no—
208C4—5482no—
209C4—10962no—
210C4#1798414612yes—
211C2—22no—
212C2—82no—
213C2—2741no—
214C2—2741no—
215C2—2742no—
216C2C2×#1798414611yescharacteristic
217C2C2×#1798415011yescharacteristic
218C2#2257840111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984059C2
cosocleC2×C4×D137C23
hypercenterC22×D4#10587739
hypocenterC137#1519
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578371C2semidirect
2. #22578345C2semidirect
3. #22578254C2semidirect
8. #17984145C22semidirect
10. (C2×#10587739)⋊C4C22semidirect
15. #17984059C4semidirect
17. C2×C4⋊Dic137C4semidirect
19. C22×D4×C137C4semidirect
109. C137#1519semidirect
116. C22×D4#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783712
#225783452
#225782542
C2×#179840592
#1519137