Finite Group Databasebeta

← #31797046

Subgroups — #31797046

Order 17536.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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×#17984148C211yesmaximal, characteristic, complemented
2C2×#17984146C211yesmaximal, characteristic, complemented
3#22578306C212yesmaximal, complemented
4#22578304C212yesmaximal, complemented
5#22577965C211yesmaximal, characteristic
6C2×#14555519—21no—
7C2×C22⋊#10587739—21no—
8#17984148—21no—
9#17984148C2212yes—
10#17984146—21no—
11#17984146C2212yes—
12C22×C22⋊D137C411yescharacteristic, complemented
13C22×C22⋊D137C2211yescharacteristic
14#17984072—21no—
15#17984072C411yescharacteristic, complemented
16#17984072C2211yescharacteristic
17#17984062—21no—
18#17984062—21no—
19#17984058—21no—
20#17984058C411yescharacteristic, complemented
21#17984058C2211yescharacteristic
22C2×C22⋊C4×C137C411yescharacteristic, complemented
23C22×#10587739—41no—
24#14555519—22no—
25#14555519—41no—
26C2×#11764047—41no—
27C22⋊#10587739—22no—
28C22⋊#10587739—41no—
29C23×D137C2×C411yescharacteristic, complemented
30C23×D137D411yescharacteristic
31C2×D548—21no—
32C2×D548—21no—
33C2×C22⋊D137—21no—
34C2×C22⋊D137—21no—
35C2×C22⋊D137—41no—
36C2×C22⋊D137—41no—
37C2×C22⋊D137—41no—
38C2×C22⋊D137—41no—
39C2×C22⋊D137C2×C411yescharacteristic
40C2×C22⋊D137D411yescharacteristic
41C2×C22⋊D137D411yescharacteristic
42C2×C22⋊D137C2311yescharacteristic
43C22×Dic137C2×C411yescharacteristic, complemented
44C22×Dic137D411yescharacteristic
45C4⋊Dic137—21no—
46C4⋊Dic137—21no—
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50#14555487—21no—
51C24×C137C2×C411yescharacteristic
52C22×C4×C137—21no—
53C22⋊C4×C137—21no—
54C22⋊C4×C137C2×C411yescharacteristic, complemented
55C22⋊C4×C137C2×C411yescharacteristic, complemented
56C2×#10587739—42no—
57C2×#10587739—82no—
58#11764047—42no—
59C22×D137—21no—
60C22×D137—21no—
61C22×D137—21no—
62C22×D137—41no—
63C22×D137—81no—
64C22×D137—81no—
65C22×D137C22⋊C411yescharacteristic
66C22×D137C2×D411yescharacteristic
67D548—81no—
68D548—81no—
69C22⋊D137—41no—
70C22⋊D137—41no—
71C22⋊D137—41no—
72C22⋊D137—41no—
73C22⋊D137—42no—
74C22⋊D137—41no—
75C22⋊D137—41no—
76C2×Dic137—21no—
77C2×Dic137—21no—
78C2×Dic137—21no—
79C2×Dic137—41no—
80C2×Dic137C22⋊C411yescharacteristic
81C2×Dic137C2×D411yescharacteristic
82C23×C137—21no—
83C23×C137—21no—
84C23×C137—81no—
85C23×C137C22⋊C411yescharacteristic
86C23×C137C22⋊C411yescharacteristic
87C23×C137C22×C411yescharacteristic
88C2×C4×C137—21no—
89C2×C4×C137—21no—
90C2×C4×C137—81no—
91C22×C137—21no—
92C22×C137—21no—
93C22×C137—41no—
94C22×C137—41no—
95C22×C137—41no—
96C22×C137—81no—
97C22×C137—81no—
98C22×C137#9911yescharacteristic
99C22×C137#9911yescharacteristic
100C22×C137C2×C22⋊C411yescharacteristic
101C2×D137—21no—
102C2×D137—21no—
103C2×D137—42no—
104C2×D137—41no—
105C2×D137—41no—
106C2×D137—41no—
107C2×D137—41no—
108C2×D137—41no—
109C2×D137—81no—
110C2×D137—82no—
111#10587739—82no—
112C4×C137—41no—
113C4×C137—41no—
114Dic137—21no—
115Dic137—21no—
116Dic137—41no—
117C2×C137—21no—
118C2×C137—21no—
119C2×C137—21no—
120C2×C137—41no—
121C2×C137—41no—
122C2×C137C2×#9911yescharacteristic
123D137—41no—
124D137—41no—
125D137—81no—
126C137#212111yescharacteristic, complemented, Sylow
127#2121—1371nomaximal, Sylow
128#534—1371no—
129C2×#100—1371no—
130C2×#99—1371no—
131#352—1372no—
132AΣL(1,𝔽16)—1372no—
133C22×D4—1371no—
134C22×D4—1371no—
135C2×M16—2741no—
136#123—2741no—
137#123—1371no—
138#123—1371no—
139#121—2741no—
140#121—2741no—
141#120—2741no—
142#120—1371no—
143#120—1371no—
144C2×C22⋊C4—2741no—
145C2×C22⋊C4#1058773911yescharacteristic, complemented
146#100—2741no—
147#100—1372no—
148#99—2741no—
149#99—1372no—
150C24—1371no—
151C24—1371no—
152C24C2×#1058773911yescharacteristic
153C2×D4—2741no—
154C2×D4—2741no—
155C2×D4—2741no—
156C2×D4—2741no—
157C2×D4—5481no—
158C2×D4—5481no—
159C2×D4—5481no—
160C2×D4—5481no—
161C2×D4—1371no—
162C2×D4—1371no—
163C2×D4—1371no—
164C2×D4—1371no—
165C22×C4—21no—
166C22×C4—5481no—
167C22×C4—1371no—
168C22×C4—1371no—
169M16—2742no—
170M16—5481no—
171C2×C8—5481no—
172C4⋊C4—2741no—
173C4⋊C4—2741no—
174C22⋊C4—21no—
175C22⋊C4—2741no—
176C22⋊C4—2741no—
177C22⋊C4—2741no—
178C22⋊C4—2741no—
179C22⋊C4—2742no—
180C22⋊C4—5481no—
181C22⋊C4C2×#1058773911yescharacteristic, complemented
182C22⋊C4C2×#1058773911yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—81no—
186C23—2741no—
187C23—2741no—
188C23—2741no—
189C23—5481no—
190C23—10961no—
191C23—10961no—
192C23—1371no—
193C23—1371no—
194C23C22⋊#1058773911yescharacteristic
195C23C22⋊#1058773911yescharacteristic
196C23C22×#1058773911yescharacteristic
197D4—5481no—
198D4—5481no—
199D4—5481no—
200D4—5481no—
201D4—5482no—
202D4—5481no—
203D4—5481no—
204D4—10961no—
205D4—10961no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—81no—
209C2×C4—2741no—
210C2×C4—2741no—
211C2×C4—2741no—
212C2×C4—5481no—
213C2×C4—5482no—
214C2×C4—10962no—
215C2×C4—1371no—
216C2×C4—1371no—
217C8—5482no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—2741no—
226C22—2741no—
227C22—5481no—
228C22—5481no—
229C22—5482no—
230C22—5481no—
231C22—5481no—
232C22—5481no—
233C22—10961no—
234C22—10962no—
235C22#1798414711yescharacteristic
236C22#1798414711yescharacteristic
237C22C2×C22⋊#1058773911yescharacteristic
238C4—41no—
239C4—41no—
240C4—2741no—
241C4—2741no—
242C4—5481no—
243C4—10962no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—5481no—
250C2—5481no—
251C2—10961no—
252C2C2×#1798414711yescharacteristic
subgroupG/H
centerC2C2×#17984147
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC2×C137C2×#99
prime radical#22577965C2
cosocleC2×C22⋊D137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#2121
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17984148C2semidirect
2. C2×#17984146C2semidirect
3. #22578306C2semidirect
4. #22578304C2semidirect
12. C22×C22⋊D137C4semidirect
15. #17984072C4semidirect
20. #17984058C4semidirect
22. C2×C22⋊C4×C137C4semidirect
29. C23×D137C2×C4semidirect
43. C22×Dic137C2×C4semidirect
54. C22⋊C4×C137C2×C4semidirect
55. C22⋊C4×C137C2×C4semidirect
126. C137#2121semidirect
145. C2×C22⋊C4#10587739semidirect
181. C22⋊C4C2×#10587739semidirect
182. C22⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841482
C2×#179841462
#225783062
#225783042
#225779652
#2121137