Finite Group Databasebeta

← C137×#8544 #46857930

Subgroups — C137×#8544 #46857930

Order 35072.

1222 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#124×C137C211yesmaximal, characteristic
2#1819×C137C214yesmaximal, complemented
3#1781×C137C212yesmaximal
4C23×Q8×C137C411yescharacteristic, complemented
5C23×D4×C137C2211yescharacteristic
6C2×#124×C137—24no—
7C2×#124×C137—28no—
8C22×C22⋊C4×C137C412yescomplemented
9C22×C22⋊C4×C137C2212yes—
10C22×C42×C137C411yescharacteristic, complemented
11C2×#99×C137—24no—
12#379×C137C2214yes—
13#342×C137—28no—
14C25×C137C2×C411yescharacteristic
15C25×C137C2311yescharacteristic
16C22×Q8×C137—24no—
17C22×Q8×C137—22no—
18C22×Q8×C137—44no—
19C22×D4×C137—24no—
20C22×D4×C137—44no—
21C22×D4×C137D414yes—
22C23×C4×C137C2×C411yescharacteristic
23C23×C4×C137C2×C412yes—
24C23×C4×C137C2×C412yes—
25#124×C137—232no—
26C2×C22⋊C4×C137—24no—
27C2×C22⋊C4×C137—24no—
28C2×C22⋊C4×C137—24no—
29C2×C22⋊C4×C137—24no—
30C2×C22⋊C4×C137—24no—
31C2×C22⋊C4×C137—24no—
32C2×C22⋊C4×C137—28no—
33C2×C22⋊C4×C137D418yes—
34C2×C42×C137—24no—
35C2×C42×C137—22no—
36#99×C137—48no—
37C24×C137—22no—
38C24×C137—42no—
39C24×C137—42no—
40C24×C137—44no—
41C24×C137—44no—
42C24×C137C22⋊C412yes—
43C24×C137C22×C411yescharacteristic
44C24×C137C2×D412yes—
45C24×C137C2×D414yes—
46C2×Q8×C137—28no—
47C2×Q8×C137—416no—
48C2×Q8×C137—44no—
49C2×Q8×C137—44no—
50C2×D4×C137—28no—
51C2×D4×C137—28no—
52C2×D4×C137—44no—
53C2×D4×C137—416no—
54C22×C4×C137—22no—
55C22×C4×C137—24no—
56C22×C4×C137—22no—
57C22×C4×C137—22no—
58C22×C4×C137—22no—
59C22×C4×C137—22no—
60C22×C4×C137—44no—
61C22×C4×C137—44no—
62C22×C4×C137—44no—
63C22×C4×C137C22⋊C412yes—
64C22×C4×C137C22⋊C414yes—
65C22×C4×C137C22⋊C414yes—
66C22⋊C4×C137—216no—
67C22⋊C4×C137—216no—
68C22⋊C4×C137—48no—
69C22⋊C4×C137—416no—
70C42×C137—28no—
71C23×C137—22no—
72C23×C137—24no—
73C23×C137—24no—
74C23×C137—28no—
75C23×C137—22no—
76C23×C137—24no—
77C23×C137—44no—
78C23×C137—44no—
79C23×C137—44no—
80C23×C137—44no—
81C23×C137—44no—
82C23×C137—44no—
83C23×C137—44no—
84C23×C137—88no—
85C23×C137—88no—
86C23×C137C2×C22⋊C411yescharacteristic
87C23×C137C2×C22⋊C412yes—
88C23×C137#12014yes—
89Q8×C137—416no—
90D4×C137—416no—
91C2×C4×C137—28no—
92C2×C4×C137—22no—
93C2×C4×C137—22no—
94C2×C4×C137—24no—
95C2×C4×C137—24no—
96C2×C4×C137—44no—
97C2×C4×C137—44no—
98C2×C4×C137—44no—
99C2×C4×C137—44no—
100C2×C4×C137—44no—
101C2×C4×C137—44no—
102C2×C4×C137—84no—
103C2×C4×C137—88no—
104C2×C4×C137#9914yes—
105C22×C137—28no—
106C22×C137—22no—
107C22×C137—22no—
108C22×C137—22no—
109C22×C137—24no—
110C22×C137—28no—
111C22×C137—42no—
112C22×C137—44no—
113C22×C137—84no—
114C22×C137—88no—
115C22×C137—84no—
116C22×C137—88no—
117C22×C137#37911yescharacteristic
118C22×C137C2×#9912yes—
119C22×C137#41014yes—
120C4×C137—24no—
121C4×C137—44no—
122C4×C137—44no—
123C4×C137—164no—
124C2×C137—22no—
125C2×C137—24no—
126C2×C137—44no—
127C2×C137—82no—
128C2×C137#178111yescharacteristic
129C2×C137#178212yes—
130#8544C13711yesmaximal, characteristic, direct product factor, Sylow
131C137#854411yescharacteristic, direct product factor, Sylow
132C22×#124C2×C13711yescharacteristic
133#1819C2×C13714yescomplemented
134#1781C2×C13712yes—
135C23×Q8C4×C13711yescharacteristic, complemented
136C23×D4C22×C13711yescharacteristic
137C2×#124—24no—
138C2×#124—28no—
139C22×C22⋊C4C4×C13712yescomplemented
140C22×C22⋊C4C22×C13712yes—
141C22×C42C4×C13711yescharacteristic, complemented
142C2×#99—24no—
143#379C22×C13714yes—
144#342—28no—
145C25C2×C4×C13711yescharacteristic
146C25C23×C13711yescharacteristic
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—44no—
150C22×D4—24no—
151C22×D4—44no—
152C22×D4D4×C13714yes—
153C23×C4C2×C4×C13711yescharacteristic
154C23×C4C2×C4×C13712yes—
155C23×C4C2×C4×C13712yes—
156#124—232no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4D4×C13718yes—
165C2×C42—24no—
166C2×C42—22no—
167#99—48no—
168C24—22no—
169C24—42no—
170C24—42no—
171C24—44no—
172C24—44no—
173C24C22⋊C4×C13712yes—
174C24C22×C4×C13711yescharacteristic
175C24C2×D4×C13712yes—
176C24C2×D4×C13714yes—
177C2×Q8—28no—
178C2×Q8—416no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×D4—28no—
182C2×D4—28no—
183C2×D4—44no—
184C2×D4—416no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4C22⋊C4×C13712yes—
195C22×C4C22⋊C4×C13714yes—
196C22×C4C22⋊C4×C13714yes—
197C22⋊C4—216no—
198C22⋊C4—216no—
199C22⋊C4—48no—
200C22⋊C4—416no—
201C42—28no—
202C23—22no—
203C23—24no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—24no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—88no—
216C23—88no—
217C23C2×C22⋊C4×C13711yescharacteristic
218C23C2×C22⋊C4×C13712yes—
219C23#120×C13714yes—
220Q8—416no—
221D4—416no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—84no—
234C2×C4—88no—
235C2×C4#99×C13714yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22—28no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—88no—
246C22—84no—
247C22—88no—
248C22#379×C13711yescharacteristic
249C22C2×#99×C13712yes—
250C22#410×C13714yes—
251C4—24no—
252C4—44no—
253C4—44no—
254C4—164no—
255C2—22no—
256C2—24no—
257C2—44no—
258C2—82no—
259C2#1781×C13711yescharacteristic
260C2#1782×C13712yes—
subgroupG/H
center, socleC22×C137#379
commutatorC24C22×C4×C137
Frattini, cosocleC25C23×C137
prime radicalC23×D4×C137C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C25×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1819×C137C2semidirect
4. C23×Q8×C137C4semidirect
8. C22×C22⋊C4×C137C4semidirect
10. C22×C42×C137C4semidirect
130. #8544C137direct (N × H)
131. C137#8544direct (N × H)
133. #1819C2×C137semidirect
135. C23×Q8C4×C137semidirect
139. C22×C22⋊C4C4×C137semidirect
141. C22×C42C4×C137semidirect

Maximal subgroups

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

subgroupindex
C22×#124×C1372
#1819×C1372
#1781×C1372
#8544137