Finite Group Databasebeta

← C137×#8765 #46858151

Subgroups — C137×#8765 #46858151

Order 35072.

498 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 142 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#2349×C137C211yesmaximal, characteristic
2#1800×C137C211yesmaximal, characteristic
3#1460×C137C211yesmaximal, characteristic
4#1303×C137C212yesmaximal, complemented
5#1286×C137C212yesmaximal, complemented
6C2×C4×D4×C137—21no—
7C2×C4×D4×C137C2211yescharacteristic
8#431×C137C2212yes—
9#423×C137C2212yes—
10C2×#100×C137—21no—
11#389×C137—21no—
12#385×C137—22no—
13#382×C137C2211yescharacteristic
14#380×C137C412yes—
15C4×C4⋊C4×C137C2211yescharacteristic
16C2×#95×C137C412yes—
17#343×C137—21no—
18#323×C137—22no—
19C22×D4×C137D411yescharacteristic
20C23×C4×C137—22no—
21C23×C4×C137C2×C412yes—
22C2×M16×C137—22no—
23C4×D4×C137—22no—
24C4×D4×C137—22no—
25C4×D4×C137—44no—
26C2×C4⋊C4×C137—21no—
27C2×C4⋊C4×C137—21no—
28C2×C4⋊C4×C137—22no—
29C2×C4⋊C4×C137D411yescharacteristic
30C2×C22⋊C4×C137—22no—
31C2×C22⋊C4×C137D412yes—
32C2×C42×C137—21no—
33C2×C42×C137C2311yescharacteristic
34C4⋊C8×C137—22no—
35C4⋊C8×C137—22no—
36C4⋊C8×C137—22no—
37#100×C137—42no—
38C22⋊C8×C137—22no—
39C4×C8×C137—22no—
40#95×C137—22no—
41#95×C137—21no—
42#95×C137C2×C411yescharacteristic
43#95×C137C2×C412yes—
44#95×C137C2×C411yescharacteristic
45C24×C137C22⋊C412yes—
46C2×D4×C137—22no—
47C2×D4×C137—22no—
48C2×D4×C137SD1612yes—
49C2×D4×C137Q1612yes—
50C22×C4×C137—21no—
51C22×C4×C137—22no—
52C22×C4×C137—22no—
53C22×C4×C137—22no—
54C22×C4×C137—21no—
55C22×C4×C137—44no—
56C22×C4×C137—44no—
57C22×C4×C137—82no—
58C22×C4×C137—82no—
59C22×C4×C137C22⋊C412yes—
60C22×C4×C137C22×C411yescharacteristic
61C22×C4×C137C2×D411yescharacteristic
62C22×C4×C137C2×D411yescharacteristic
63M16×C137—42no—
64M16×C137—42no—
65C2×C8×C137—42no—
66C2×C8×C137—42no—
67C4⋊C4×C137—22no—
68C4⋊C4×C137—42no—
69C4⋊C4×C137—42no—
70C4⋊C4×C137—82no—
71C22⋊C4×C137—24no—
72C22⋊C4×C137—44no—
73C42×C137—42no—
74C42×C137#4112yes—
75C42×C137#4112yes—
76C23×C137—22no—
77C23×C137—22no—
78C23×C137—42no—
79C23×C137—42no—
80C23×C137#10314yes—
81C23×C137C2×C22⋊C411yescharacteristic
82D4×C137—24no—
83D4×C137—24no—
84C2×C4×C137—22no—
85C2×C4×C137—22no—
86C2×C4×C137—41no—
87C2×C4×C137—41no—
88C2×C4×C137—41no—
89C2×C4×C137—41no—
90C2×C4×C137—42no—
91C2×C4×C137—41no—
92C2×C4×C137—41no—
93C2×C4×C137—42no—
94C2×C4×C137—42no—
95C2×C4×C137—41no—
96C2×C4×C137—42no—
97C2×C4×C137—82no—
98C2×C4×C137—82no—
99C2×C4×C137—88no—
100C2×C4×C137#11711yescharacteristic
101C2×C4×C137#11711yescharacteristic
102C2×C4×C137#12312yes—
103C2×C4×C137#12312yes—
104C2×C4×C137C2×SD1611yescharacteristic
105C2×C4×C137C2×Q1611yescharacteristic
106C2×C4×C137AGL(1,ℤ/8)12yes—
107C8×C137—82no—
108C8×C137—82no—
109C22×C137—21no—
110C22×C137—21no—
111C22×C137—24no—
112C22×C137—44no—
113C22×C137—44no—
114C22×C137—42no—
115C22×C137—42no—
116C22×C137#38111yescharacteristic
117C22×C137C2×#10311yescharacteristic
118C22×C137#41811yescharacteristic
119C4×C137—22no—
120C4×C137—21no—
121C4×C137—21no—
122C4×C137—42no—
123C4×C137—42no—
124C4×C137—81no—
125C4×C137—82no—
126C4×C137#48112yes—
127C4×C137#48412yes—
128C2×C137—21no—
129C2×C137—21no—
130C2×C137—44no—
131C2×C137#182411yescharacteristic
132C2×C137#182711yescharacteristic
133C2×C137#182911yescharacteristic
134#8765C13711yesmaximal, characteristic, direct product factor, Sylow
135C137#876511yescharacteristic, direct product factor, Sylow
136#2349C2×C13711yescharacteristic
137#1800C2×C13711yescharacteristic
138#1460C2×C13711yescharacteristic
139#1303C2×C13712yescomplemented
140#1286C2×C13712yescomplemented
141C2×C4×D4—21no—
142C2×C4×D4C22×C13711yescharacteristic
143#431C22×C13712yes—
144#423C22×C13712yes—
145C2×#100—21no—
146#389—21no—
147#385—22no—
148#382C22×C13711yescharacteristic
149#380C4×C13712yes—
150C4×C4⋊C4C22×C13711yescharacteristic
151C2×#95C4×C13712yes—
152#343—21no—
153#323—22no—
154C22×D4D4×C13711yescharacteristic
155C23×C4—22no—
156C23×C4C2×C4×C13712yes—
157C2×M16—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4D4×C13711yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4D4×C13712yes—
167C2×C42—21no—
168C2×C42C23×C13711yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172#100—42no—
173C22⋊C8—22no—
174C4×C8—22no—
175#95—22no—
176#95—21no—
177#95C2×C4×C13711yescharacteristic
178#95C2×C4×C13712yes—
179#95C2×C4×C13711yescharacteristic
180C24C22⋊C4×C13712yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4SD16×C13712yes—
184C2×D4Q16×C13712yes—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4C22⋊C4×C13712yes—
195C22×C4C22×C4×C13711yescharacteristic
196C22×C4C2×D4×C13711yescharacteristic
197C22×C4C2×D4×C13711yescharacteristic
198M16—42no—
199M16—42no—
200C2×C8—42no—
201C2×C8—42no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C22⋊C4—24no—
207C22⋊C4—44no—
208C42—42no—
209C42#41×C13712yes—
210C42#41×C13712yes—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23#103×C13714yes—
216C23C2×C22⋊C4×C13711yescharacteristic
217D4—24no—
218D4—24no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—88no—
235C2×C4#117×C13711yescharacteristic
236C2×C4#117×C13711yescharacteristic
237C2×C4#123×C13712yes—
238C2×C4#123×C13712yes—
239C2×C4C2×SD16×C13711yescharacteristic
240C2×C4C2×Q16×C13711yescharacteristic
241C2×C4AGL(1,ℤ/8)×C13712yes—
242C8—82no—
243C8—82no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—44no—
249C22—42no—
250C22—42no—
251C22#381×C13711yescharacteristic
252C22C2×#103×C13711yescharacteristic
253C22#418×C13711yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—81no—
261C4#481×C13712yes—
262C4#484×C13712yes—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#1824×C13711yescharacteristic
267C2#1827×C13711yescharacteristic
268C2#1829×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#381
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1303×C137C2semidirect
5. #1286×C137C2semidirect
134. #8765C137direct (N × H)
135. C137#8765direct (N × H)
139. #1303C2×C137semidirect
140. #1286C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2349×C1372
#1800×C1372
#1460×C1372
#1303×C1372
#1286×C1372
#8765137