Finite Group Databasebeta

← #31796799

Subgroups — #31796799

Order 17536.

386 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 68 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#22578470C211yesmaximal, characteristic
2D137⋊#99C211yesmaximal, characteristic
3#22578346C212yesmaximal, complemented
4#22578280C212yesmaximal, complemented
5#123×D137C211yesmaximal, characteristic
6D4×#10587739—24no—
7#17984167—21no—
8C2×C22⋊#10587739—21no—
9C2×C22⋊#10587739C2212yes—
10C2×C22⋊#10587739C2212yes—
11#17984147—21no—
12#17984146—21no—
13(C2×#10587739)⋊C4—24no—
14C2×D4×D137C2211yescharacteristic
15C22×C4×D137C2211yescharacteristic
16#17984073—21no—
17#17984072C411yescharacteristic, complemented
18#17984071C411yescharacteristic, complemented
19#17984070—21no—
20#17984069C411yescharacteristic, complemented
21C4⋊C4×D137—21no—
22C22⋊C4×D137—21no—
23C22⋊C4×D137C2211yescharacteristic
24#123×C137C411yescharacteristic, complemented
25C22×#10587739—22no—
26C22×#10587739—22no—
27C22×#10587739D414yes—
28C4⋊#10587739—22no—
29C22⋊#10587739—22no—
30C22⋊#10587739—22no—
31C22⋊#10587739—22no—
32C22⋊#10587739—22no—
33C22⋊#10587739—22no—
34C4×#10587739—22no—
35C23×D137D411yescharacteristic
36C23×D137C2311yescharacteristic
37C2×D548D411yescharacteristic
38C2×C22⋊D137C2×C411yescharacteristic
39C2×C22⋊D137D411yescharacteristic
40D4×D137—22no—
41D4×D137—22no—
42C22×Dic137C2×C411yescharacteristic, complemented
43C2×C4×D137—21no—
44C2×C4×D137—21no—
45C2×C4×D137—21no—
46C2×C4×D137—22no—
47C2×C4×D137—22no—
48C2×C4×D137D411yescharacteristic
49C4⋊Dic137—21no—
50#14555489—21no—
51#14555489—21no—
52C22⋊Dic137—21no—
53C22⋊Dic137C2×C411yescharacteristic, complemented
54#14555487—21no—
55#14555487—21no—
56#14555487—21no—
57C2×D4×C137C2×C411yescharacteristic
58C22×C4×C137C2×C411yescharacteristic, complemented
59C4⋊C4×C137—21no—
60C22⋊C4×C137—21no—
61C22⋊C4×C137C2×C411yescharacteristic, complemented
62C2×#10587739—24no—
63C2×#10587739—44no—
64C2×#10587739—44no—
65C2×#10587739—42no—
66C2×#10587739—82no—
67C22×D137—21no—
68C22×D137—21no—
69C22×D137—42no—
70C22×D137—42no—
71C22×D137C22⋊C411yescharacteristic
72C22×D137C2×D412yes—
73C22×D137C2×D411yescharacteristic
74C22×D137C2×D411yescharacteristic
75C22×D137#4112yes—
76D548—22no—
77C22⋊D137—22no—
78C22⋊D137—22no—
79C2×Dic137—21no—
80C2×Dic137—21no—
81C2×Dic137—22no—
82C2×Dic137C22⋊C411yescharacteristic
83C4×D137—21no—
84C4×D137—21no—
85C4×D137—42no—
86C4×D137—41no—
87C4×D137—42no—
88C4×D137—41no—
89C23×C137C22⋊C411yescharacteristic
90C23×C137C22×C411yescharacteristic
91D4×C137—22no—
92C2×C4×C137—21no—
93C2×C4×C137—21no—
94C2×C4×C137—22no—
95C2×C4×C137C22⋊C411yescharacteristic
96C22×C137—21no—
97C22×C137—41no—
98C22×C137—41no—
99C22×C137C2×C22⋊C411yescharacteristic
100C22×C137C4×D412yes—
101C2×D137—22no—
102C2×D137—24no—
103C2×D137—21no—
104C2×D137—41no—
105C2×D137—42no—
106C2×D137—42no—
107C2×D137—41no—
108C2×D137—81no—
109C2×D137#12011yescharacteristic
110C2×D137#12112yes—
111C2×D137#12311yescharacteristic
112#10587739—44no—
113#10587739—82no—
114C4×C137—21no—
115C4×C137—41no—
116C4×C137—41no—
117Dic137—21no—
118Dic137—41no—
119Dic137—41no—
120C2×C137—21no—
121C2×C137—22no—
122C2×C137—41no—
123C2×C137#38611yescharacteristic
124D137—22no—
125D137—21no—
126D137—41no—
127D137#45812yes—
128C137#189011yescharacteristic, complemented, Sylow
129#1890—1371nomaximal, Sylow
130C2×#123—1371no—
131#518—1371no—
132C2×#99—1371no—
133#381—1372no—
134#342—1372no—
135C22×D4—1371no—
136C23×C4—1371no—
137#123—2741no—
138#123—2741no—
139#123—1371no—
140#123—1371no—
141#123—1371no—
142#123#1058773911yescharacteristic, complemented
143C4×D4—2744no—
144#117—2741no—
145C2×C4⋊C4—2741no—
146C2×C22⋊C4—2741no—
147C2×C22⋊C4—2741no—
148C2×C22⋊C4—1372no—
149C2×C22⋊C4—1372no—
150C2×C22⋊C4—1371no—
151#99—2741no—
152#99—2741no—
153#95—2744no—
154C24—1371no—
155C24—1371no—
156C2×D4—2742no—
157C2×D4—2742no—
158C2×D4—1371no—
159C2×D4—1371no—
160C2×D4—1371no—
161C2×D4C2×#1058773911yescharacteristic
162C22×C4—2742no—
163C22×C4—2741no—
164C22×C4—2741no—
165C22×C4—2741no—
166C22×C4—2742no—
167C22×C4—2742no—
168C22×C4—2742no—
169C22×C4—1371no—
170C22×C4—1371no—
171C22×C4—1374no—
172C22×C4C2×#1058773911yescharacteristic, complemented
173C4⋊C4—21no—
174C4⋊C4—2741no—
175C4⋊C4—2741no—
176C4⋊C4—2742no—
177C4⋊C4—2741no—
178C22⋊C4—21no—
179C22⋊C4—2741no—
180C22⋊C4—2741no—
181C22⋊C4—2741no—
182C22⋊C4—2742no—
183C22⋊C4—2742no—
184C22⋊C4—2742no—
185C22⋊C4—2741no—
186C22⋊C4—2742no—
187C22⋊C4—2742no—
188C22⋊C4—1371no—
189C22⋊C4C2×#1058773911yescharacteristic, complemented
190C42—2742no—
191C23—2741no—
192C23—2741no—
193C23—5482no—
194C23—5482no—
195C23—1371no—
196C23—1372no—
197C23—1371no—
198C23—1371no—
199C23—1372no—
200C23C22⋊#1058773911yescharacteristic
201C23C22×#1058773911yescharacteristic
202D4—22no—
203D4—2742no—
204D4—2742no—
205D4—2742no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—2741no—
210C2×C4—2741no—
211C2×C4—2744no—
212C2×C4—2741no—
213C2×C4—2741no—
214C2×C4—2742no—
215C2×C4—5482no—
216C2×C4—5481no—
217C2×C4—5482no—
218C2×C4—5481no—
219C2×C4—5484no—
220C2×C4—5484no—
221C2×C4—5482no—
222C2×C4—10962no—
223C2×C4—1371no—
224C2×C4C22⋊#1058773911yescharacteristic
225C22—21no—
226C22—41no—
227C22—41no—
228C22—2744no—
229C22—2742no—
230C22—2741no—
231C22—5481no—
232C22—5482no—
233C22—5481no—
234C22—5482no—
235C22—10961no—
236C22—1371no—
237C22—1372no—
238C22—1371no—
239C22C2×C22⋊#1058773911yescharacteristic
240C22D4×#1058773912yes—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—2741no—
245C4—5481no—
246C4—5484no—
247C4—5481no—
248C4—10962no—
249C2—21no—
250C2—22no—
251C2—41no—
252C2—2742no—
253C2—2741no—
254C2—5481no—
255C2—1372no—
256C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#123×C137C4
socleC2×C137#386
prime radicalC2×D4×D137C22
cosocleC23×D137C23
hypercenter#123#10587739
hypocenterC137#1890
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #22578346C2semidirect
4. #22578280C2semidirect
17. #17984072C4semidirect
18. #17984071C4semidirect
20. #17984069C4semidirect
24. #123×C137C4semidirect
42. C22×Dic137C2×C4semidirect
53. C22⋊Dic137C2×C4semidirect
58. C22×C4×C137C2×C4semidirect
61. C22⋊C4×C137C2×C4semidirect
128. C137#1890semidirect
142. #123#10587739semidirect
172. C22×C4C2×#10587739semidirect
189. C22⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784702
D137⋊#992
#225783462
#225782802
#123×D1372
#1890137