Finite Group Databasebeta

← #31797047

Subgroups — #31797047

Order 17536.

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 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⋊#100C211yesmaximal, characteristic, complemented
2D137⋊#99C211yesmaximal, characteristic, complemented
3#22578305C212yesmaximal, complemented
4#22578304C212yesmaximal, complemented
5#22577964C211yesmaximal, characteristic
6D137⋊M16—21no—
7C2×C22⋊#10587739—21no—
8#17984148—21no—
9#17984148C2212yes—
10#17984147—21no—
11#17984146C2212yes—
12C22×D548C411yescharacteristic, complemented
13C2×D4×D137C2211yescharacteristic
14#17984073—21no—
15#17984071C2211yescharacteristic
16#17984063—21no—
17#17984061—21no—
18#17984058—21no—
19#17984058C2211yescharacteristic
20#17984056C411yescharacteristic, complemented
21C22⋊C4×D137C411yescharacteristic, complemented
22#123×C137C411yescharacteristic, complemented
23C22×#10587739—41no—
24#14555520—41no—
25#14555519—22no—
26#14555518—41no—
27C22⋊#10587739—22no—
28C22⋊#10587739—41no—
29C23×D137C2×C411yescharacteristic
30C23×D137C2×C411yescharacteristic, complemented
31C23×D137D411yescharacteristic
32C2×D548—21no—
33C2×D548—21no—
34C2×D548—21no—
35C2×D548—41no—
36C2×D548—41no—
37C2×D548D411yescharacteristic
38C2×C22⋊D137—21no—
39C2×C22⋊D137D411yescharacteristic
40C2×C22⋊D137C2311yescharacteristic
41D4×D137—41no—
42D4×D137—41no—
43C2×C4×D137—21no—
44C2×C4×D137D411yescharacteristic
45#14555489—21no—
46C22⋊Dic137C2×C411yescharacteristic, complemented
47#14555487—21no—
48#14555487—21no—
49#14555487—21no—
50#14555487—21no—
51C2×D4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic, complemented
53C4⋊C4×C137—21no—
54C22⋊C4×C137—21no—
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—21no—
63C22×D137—21no—
64C22×D137—41no—
65C22×D137—81no—
66C22×D137—81no—
67C22×D137—81no—
68C22×D137C22⋊C411yescharacteristic
69C22×D137C22⋊C411yescharacteristic
70C22×D137C2×D411yescharacteristic
71D548—41no—
72D548—42no—
73D548—41no—
74D548—41no—
75D548—81no—
76C22⋊D137—41no—
77C22⋊D137—41no—
78C22⋊D137—81no—
79C2×Dic137—21no—
80C2×Dic137C2×D411yescharacteristic
81C4×D137—21no—
82C4×D137—21no—
83C4×D137—81no—
84C23×C137C22⋊C411yescharacteristic
85C23×C137C22×C411yescharacteristic
86D4×C137—41no—
87C2×C4×C137—21no—
88C2×C4×C137—21no—
89C2×C4×C137—41no—
90C2×C4×C137C22⋊C411yescharacteristic
91C22×C137—21no—
92C22×C137—21no—
93C22×C137—41no—
94C22×C137—41no—
95C22×C137C2×C22⋊C411yescharacteristic
96C2×D137—21no—
97C2×D137—21no—
98C2×D137—41no—
99C2×D137—42no—
100C2×D137—41no—
101C2×D137—41no—
102C2×D137—41no—
103C2×D137—41no—
104C2×D137—41no—
105C2×D137—81no—
106C2×D137—81no—
107C2×D137—81no—
108C2×D137—82no—
109C2×D137#9911yescharacteristic
110C2×D137#9911yescharacteristic
111#10587739—82no—
112C4×C137—21no—
113C4×C137—41no—
114C4×C137—41no—
115Dic137—21no—
116Dic137—41no—
117C2×C137—21no—
118C2×C137—41no—
119C2×C137—41no—
120C2×C137C2×#9911yescharacteristic
121D137—21no—
122D137—21no—
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#1058773911yescharacteristic, complemented
139#121—2741no—
140#121—2741no—
141#120—2741no—
142#120—1371no—
143#120—1371no—
144C2×C22⋊C4—2741no—
145C2×C22⋊C4—1371no—
146#100—2741no—
147#100—1372no—
148#99—2741no—
149#99—1372no—
150C24—1371no—
151C24—1371no—
152C24—1371no—
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×D4C2×#1058773911yescharacteristic
165C22×C4—2741no—
166C22×C4—5481no—
167C22×C4—1371no—
168C22×C4C2×#1058773911yescharacteristic, complemented
169M16—2742no—
170M16—5481no—
171C2×C8—5481no—
172C4⋊C4—21no—
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⋊C4—1371no—
182C22⋊C4C2×#1058773911yescharacteristic, complemented
183C23—2741no—
184C23—2741no—
185C23—2741no—
186C23—2741no—
187C23—2741no—
188C23—5481no—
189C23—10961no—
190C23—10961no—
191C23—10961no—
192C23—1371no—
193C23—1371no—
194C23—1371no—
195C23C22⋊#1058773911yescharacteristic
196C23C22×#1058773911yescharacteristic
197D4—41no—
198D4—5481no—
199D4—5482no—
200D4—5481no—
201D4—5481no—
202D4—5481no—
203D4—5481no—
204D4—10961no—
205D4—10961no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—2741no—
210C2×C4—2741no—
211C2×C4—2741no—
212C2×C4—5482no—
213C2×C4—10961no—
214C2×C4—10962no—
215C2×C4—1371no—
216C2×C4C22⋊#1058773911yescharacteristic
217C8—5482no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—2741no—
223C22—2741no—
224C22—5481no—
225C22—5481no—
226C22—5481no—
227C22—5481no—
228C22—5481no—
229C22—5482no—
230C22—5481no—
231C22—10961no—
232C22—10961no—
233C22—10961no—
234C22—10962no—
235C22—1371no—
236C22—1371no—
237C22C2×C22⋊#1058773911yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—2741no—
242C4—5481no—
243C4—10962no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—2741no—
248C2—2741no—
249C2—5481no—
250C2—5481no—
251C2—10961no—
252C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#123×C137C4
socleC2×C137C2×#99
prime radical#22577964C2
cosocleC2×C22⋊D137C23
hypercenter#123#10587739
hypocenterC137#2121
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. D137⋊#100C2semidirect
2. D137⋊#99C2semidirect
3. #22578305C2semidirect
4. #22578304C2semidirect
12. C22×D548C4semidirect
20. #17984056C4semidirect
21. C22⋊C4×D137C4semidirect
22. #123×C137C4semidirect
30. C23×D137C2×C4semidirect
46. C22⋊Dic137C2×C4semidirect
52. C22×C4×C137C2×C4semidirect
55. C22⋊C4×C137C2×C4semidirect
126. C137#2121semidirect
138. #123#10587739semidirect
168. 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
D137⋊#1002
D137⋊#992
#225783052
#225783042
#225779642
#2121137