Finite Group Databasebeta

← #31796065

Subgroups — #31796065

Order 17536.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#22578373C211yesmaximal, characteristic, complemented
2C2×C22⋊#11764047C211yesmaximal, characteristic, complemented
3#22578247C212yesmaximal, complemented
4#22578246C212yesmaximal, complemented
5C2×#17984060C211yesmaximal, characteristic
6C2×#14555519—21no—
7C22×#11764047—21no—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10C22⋊#11764047C2212yes—
11C22⋊#11764047C2212yes—
12C22×C22⋊D137C411yescharacteristic
13C22×C22⋊D137C411yescharacteristic
14C23×Dic137C2211yescharacteristic
15#17984060—21no—
16#17984060—21no—
17#17984060—21no—
18#17984060—21no—
19C2×#14555489C411yescharacteristic
20C2×C22⋊Dic137C2211yescharacteristic
21C2×#14555487C2211yescharacteristic
22C22×D4×C137C411yescharacteristic
23#14555519—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×#11764047—42no—
27C23×D137C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C22⋊D137—21no—
30C2×C22⋊D137—21no—
31C2×C22⋊D137—21no—
32C2×C22⋊D137—42no—
33C2×C22⋊D137—42no—
34C22×Dic137—42no—
35C22×Dic137C2×C411yescharacteristic
36C22×Dic137C2×C411yescharacteristic
37C22×Dic137D411yescharacteristic
38C22×Dic137D411yescharacteristic
39C22×Dic137D411yescharacteristic
40C22×Dic137D411yescharacteristic
41C22×Dic137C2311yescharacteristic
42#14555489—21no—
43#14555489—21no—
44C22⋊Dic137—21no—
45C22⋊Dic137—21no—
46#14555487—21no—
47#14555487—21no—
48C24×C137C2×C411yescharacteristic
49C24×C137C2×C411yescharacteristic
50C2×D4×C137—21no—
51C2×D4×C137—21no—
52C2×D4×C137—42no—
53C22×C4×C137C2×C411yescharacteristic
54#11764047—42no—
55#11764047—42no—
56C22×D137—21no—
57C22×D137—22no—
58C22×D137—41no—
59C22×D137—41no—
60C22⋊D137—42no—
61C22⋊D137—42no—
62C22⋊D137—42no—
63C22⋊D137—42no—
64C2×Dic137—22no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67C2×Dic137—21no—
68C2×Dic137—41no—
69C2×Dic137—41no—
70C2×Dic137—41no—
71C2×Dic137—41no—
72C2×Dic137C22⋊C411yescharacteristic
73C2×Dic137C22⋊C411yescharacteristic
74C2×Dic137C2×D411yescharacteristic
75C2×Dic137C2×D411yescharacteristic
76C23×C137—22no—
77C23×C137—22no—
78C23×C137—21no—
79C23×C137—22no—
80C23×C137—41no—
81C23×C137—41no—
82C23×C137—41no—
83C23×C137C22⋊C411yescharacteristic
84C23×C137C22⋊C411yescharacteristic
85C23×C137C22×C411yescharacteristic
86D4×C137—42no—
87D4×C137—42no—
88C2×C4×C137—21no—
89C2×C4×C137—41no—
90C22×C137—21no—
91C22×C137—21no—
92C22×C137—21no—
93C22×C137—21no—
94C22×C137—21no—
95C22×C137—21no—
96C22×C137—21no—
97C22×C137—44no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137—41no—
101C22×C137—82no—
102C22×C137#10012yes—
103C22×C137C2×C22⋊C411yescharacteristic
104C22×C137#13112yes—
105C2×D137—41no—
106C2×D137—41no—
107C2×D137—41no—
108C2×D137—82no—
109C4×C137—41no—
110Dic137—21no—
111Dic137—21no—
112Dic137—21no—
113Dic137—41no—
114Dic137#9912yes—
115C2×C137—22no—
116C2×C137—22no—
117C2×C137—41no—
118C2×C137—81no—
119C2×C137#40811yescharacteristic
120C2×C137C2×#9911yescharacteristic
121C2×C137C2×#10011yescharacteristic
122D137—81no—
123C137#147111yescharacteristic, complemented, Sylow
124#1471—1371nomaximal, Sylow
125C2×#121—1371no—
126#407—1371no—
127C2×C22⋊C8—1371no—
128#323—1372no—
129#323—1372no—
130C22×D4—1371no—
131C22×D4—1371no—
132C22×D4#1058773911yescharacteristic
133C23×C4—1371no—
134C2×M16—2741no—
135C22×C8—2741no—
136#121—2741no—
137#121—2741no—
138#121—2741no—
139#121—2741no—
140C2×C4⋊C4—1371no—
141C2×C22⋊C4—1371no—
142C2×C22⋊C4—1371no—
143C22⋊C8—2741no—
144C22⋊C8—2741no—
145C22⋊C8—1372no—
146C22⋊C8—1372no—
147C24—1371no—
148C24C2×#1058773911yescharacteristic
149C24C2×#1058773911yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—2741no—
154C2×D4—2741no—
155C2×D4—2741no—
156C2×D4—2741no—
157C2×D4—5482no—
158C2×D4—5482no—
159C22×C4—5482no—
160C22×C4—1371no—
161C22×C4—1371no—
162C22×C4—1371no—
163C22×C4—1371no—
164C22×C4—1371no—
165C22×C4—1371no—
166C22×C4—1371no—
167C22×C4C2×#1058773911yescharacteristic
168M16—5482no—
169C2×C8—2742no—
170C2×C8—2742no—
171C2×C8—5482no—
172C4⋊C4—2741no—
173C4⋊C4—2741no—
174C22⋊C4—2741no—
175C22⋊C4—2741no—
176C22⋊C4—2741no—
177C22⋊C4—2741no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—2741no—
186C23—2742no—
187C23—5481no—
188C23—5481no—
189C23C22⋊#1058773911yescharacteristic
190C23C22⋊#1058773911yescharacteristic
191C23C22×#1058773911yescharacteristic
192D4—42no—
193D4—42no—
194D4—5482no—
195D4—5482no—
196D4—5482no—
197D4—5482no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—2742no—
201C2×C4—2742no—
202C2×C4—2742no—
203C2×C4—2741no—
204C2×C4—5481no—
205C2×C4—5481no—
206C2×C4—5481no—
207C2×C4—5481no—
208C2×C4—1371no—
209C2×C4—1371no—
210C2×C4—1371no—
211C2×C4—1371no—
212C8—5482no—
213C8—5482no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—44no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—82no—
226C22—5481no—
227C22—5481no—
228C22—5481no—
229C22—10962no—
230C22#1798414812yes—
231C22C2×C22⋊#1058773911yescharacteristic
232C22#1798417512yes—
233C4—41no—
234C4—2741no—
235C4—2741no—
236C4—2741no—
237C4—5481no—
238C4—1372no—
239C2—22no—
240C2—22no—
241C2—41no—
242C2—81no—
243C2—10961no—
244C2#2257837611yescharacteristic
245C2D137⋊#9911yescharacteristic
246C2C2×#1798414811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC22×D4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984060C2
cosocleC22×Dic137C23
hypercenterC22×D4#10587739
hypocenterC137#1471
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578373C2semidirect
2. C2×C22⋊#11764047C2semidirect
3. #22578247C2semidirect
4. #22578246C2semidirect
123. C137#1471semidirect

Maximal subgroups

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

subgroupindex
#225783732
C2×C22⋊#117640472
#225782472
#225782462
C2×#179840602
#1471137