Finite Group Databasebeta

← C137×#18404 #46867790

Subgroups — C137×#18404 #46867790

Order 35072.

910 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3502×C137C211yesmaximal, characteristic
2#3209×C137C212yesmaximal
3#2567×C137C211yesmaximal, characteristic
4#1605×C137C218yesmaximal
5#1503×C137C212yesmaximal
6#1461×C137C211yesmaximal, characteristic
7Q82×C137—22no—
8#557×C137—22no—
9#557×C137—24no—
10#544×C137—21no—
11#544×C137C2214yes—
12#533×C137—22no—
13#533×C137C2212yes—
14C2×C4⋊Q8×C137C2211yescharacteristic
15#519×C137—21no—
16C2×C4×Q8×C137C2211yescharacteristic
17#494×C137—22no—
18#487×C137—22no—
19#462×C137—24no—
20#451×C137C2218yes—
21#431×C137C2212yes—
22#395×C137C2214yes—
23#391×C137—21no—
24#389×C137—22no—
25#384×C137—21no—
26C4×C4⋊C4×C137C2211yescharacteristic
27#328×C137C2218yes—
28#324×C137C2214yes—
29C22×Q8×C137D411yescharacteristic
30C22×Q8×C137C2312yes—
31C2×Q16×C137—28no—
32C4⋊Q8×C137—22no—
33C4⋊Q8×C137—22no—
34C4⋊Q8×C137—24no—
35C4⋊Q8×C137—28no—
36C4⋊Q8×C137—22no—
37C4⋊Q8×C137—22no—
38C4⋊Q8×C137D412yes—
39C4⋊Q8×C137D412yes—
40#125×C137—24no—
41#125×C137—24no—
42#125×C137—28no—
43#125×C137—22no—
44C22⋊Q8×C137—22no—
45C22⋊Q8×C137—24no—
46C22⋊Q8×C137C2314yes—
47C4×Q8×C137—24no—
48C4×Q8×C137—24no—
49C4×Q8×C137—24no—
50C4×Q8×C137—24no—
51C4×Q8×C137—22no—
52C4×Q8×C137—22no—
53C4×Q8×C137—22no—
54C4×Q8×C137D414yes—
55#117×C137—24no—
56#117×C137—22no—
57C2×C4⋊C4×C137—21no—
58C2×C4⋊C4×C137—21no—
59C2×C4⋊C4×C137—21no—
60C2×C4⋊C4×C137—24no—
61C2×C4⋊C4×C137D411yescharacteristic
62C2×C4⋊C4×C137C2312yes—
63C2×C42×C137—21no—
64C2×C42×C137D412yes—
65C2×C42×C137C2311yescharacteristic
66C4⋊C8×C137—22no—
67#103×C137—28no—
68C22⋊C8×C137C2314yes—
69C4×C8×C137—22no—
70#95×C137—24no—
71#95×C137C2312yes—
72C2×Q8×C137—24no—
73C2×Q8×C137—22no—
74C2×Q8×C137—22no—
75C2×Q8×C137—22no—
76C2×Q8×C137—24no—
77C2×Q8×C137—82no—
78C2×Q8×C137Q1614yescomplemented
79C2×Q8×C137C2×D414yes—
80C22×C4×C137—22no—
81C22×C4×C137—22no—
82C22×C4×C137C2×D411yescharacteristic
83C22×C4×C137C2×D412yes—
84C22×C4×C137C2×D412yes—
85C22×C4×C137C2×D411yescharacteristic
86C22×C4×C137C2411yescharacteristic
87Q16×C137—88no—
88C2×C8×C137—24no—
89C4⋊C4×C137—24no—
90C4⋊C4×C137—24no—
91C4⋊C4×C137—24no—
92C4⋊C4×C137—24no—
93C4⋊C4×C137—28no—
94C4⋊C4×C137—22no—
95C4⋊C4×C137—22no—
96C4⋊C4×C137—22no—
97C4⋊C4×C137—22no—
98C4⋊C4×C137—24no—
99C4⋊C4×C137—24no—
100C4⋊C4×C137—24no—
101C4⋊C4×C137—41no—
102C4⋊C4×C137—42no—
103C4⋊C4×C137—41no—
104C4⋊C4×C137—84no—
105C4⋊C4×C137C2×D414yes—
106C22⋊C4×C137—24no—
107C42×C137—28no—
108C42×C137—22no—
109C42×C137—22no—
110C42×C137—24no—
111C42×C137—42no—
112C42×C137C2×D412yes—
113C42×C137C2×D412yes—
114C23×C137C22×D411yescharacteristic
115Q8×C137—28no—
116Q8×C137—44no—
117Q8×C137—48no—
118Q8×C137—44no—
119Q8×C137—84no—
120C2×C4×C137—22no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—24no—
124C2×C4×C137—24no—
125C2×C4×C137—24no—
126C2×C4×C137—22no—
127C2×C4×C137—41no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137—44no—
132C2×C4×C137—42no—
133C2×C4×C137#12011yescharacteristic
134C2×C4×C137#12012yes—
135C2×C4×C137#12011yescharacteristic
136C2×C4×C137C2×Q1614yes—
137C2×C4×C137C2×Q1612yes—
138C2×C4×C137#13712yes—
139C2×C4×C137C22×D412yes—
140C8×C137—44no—
141C22×C137—21no—
142C22×C137—21no—
143C22×C137C2×#12011yescharacteristic
144C22×C137C22×Q1611yescharacteristic
145C22×C137C2×#13711yescharacteristic
146C4×C137—24no—
147C4×C137—21no—
148C4×C137—22no—
149C4×C137—44no—
150C4×C137—44no—
151C4×C137—42no—
152C4×C137—44no—
153C4×C137—82no—
154C4×C137#45114yes—
155C4×C137#45612yes—
156C2×C137—21no—
157C2×C137—21no—
158C2×C137C2×#45111yescharacteristic
159C2×C137C2×#45611yescharacteristic
160C2×C137#302711yescharacteristic
161#18404C13711yesmaximal, characteristic, direct product factor, Sylow
162C137#1840411yescharacteristic, direct product factor, Sylow
163#3502C2×C13711yescharacteristic
164#3209C2×C13712yes—
165#2567C2×C13711yescharacteristic
166#1605C2×C13718yes—
167#1503C2×C13712yes—
168#1461C2×C13711yescharacteristic
169Q82—22no—
170#557—22no—
171#557—24no—
172#544—21no—
173#544C22×C13714yes—
174#533—22no—
175#533C22×C13712yes—
176C2×C4⋊Q8C22×C13711yescharacteristic
177#519—21no—
178C2×C4×Q8C22×C13711yescharacteristic
179#494—22no—
180#487—22no—
181#462—24no—
182#451C22×C13718yes—
183#431C22×C13712yes—
184#395C22×C13714yes—
185#391—21no—
186#389—22no—
187#384—21no—
188C4×C4⋊C4C22×C13711yescharacteristic
189#328C22×C13718yes—
190#324C22×C13714yes—
191C22×Q8D4×C13711yescharacteristic
192C22×Q8C23×C13712yes—
193C2×Q16—28no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—24no—
197C4⋊Q8—28no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8D4×C13712yes—
201C4⋊Q8D4×C13712yes—
202#125—24no—
203#125—24no—
204#125—28no—
205#125—22no—
206C22⋊Q8—22no—
207C22⋊Q8—24no—
208C22⋊Q8C23×C13714yes—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8D4×C13714yes—
217#117—24no—
218#117—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4D4×C13711yescharacteristic
224C2×C4⋊C4C23×C13712yes—
225C2×C42—21no—
226C2×C42D4×C13712yes—
227C2×C42C23×C13711yescharacteristic
228C4⋊C8—22no—
229#103—28no—
230C22⋊C8C23×C13714yes—
231C4×C8—22no—
232#95—24no—
233#95C23×C13712yes—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—82no—
240C2×Q8Q16×C13714yescomplemented
241C2×Q8C2×D4×C13714yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×D4×C13711yescharacteristic
245C22×C4C2×D4×C13712yes—
246C22×C4C2×D4×C13712yes—
247C22×C4C2×D4×C13711yescharacteristic
248C22×C4C24×C13711yescharacteristic
249Q16—88no—
250C2×C8—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—28no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—84no—
267C4⋊C4C2×D4×C13714yes—
268C22⋊C4—24no—
269C42—28no—
270C42—22no—
271C42—22no—
272C42—24no—
273C42—42no—
274C42C2×D4×C13712yes—
275C42C2×D4×C13712yes—
276C23C22×D4×C13711yescharacteristic
277Q8—28no—
278Q8—44no—
279Q8—48no—
280Q8—44no—
281Q8—84no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4#120×C13711yescharacteristic
296C2×C4#120×C13712yes—
297C2×C4#120×C13711yescharacteristic
298C2×C4C2×Q16×C13714yes—
299C2×C4C2×Q16×C13712yes—
300C2×C4#137×C13712yes—
301C2×C4C22×D4×C13712yes—
302C8—44no—
303C22—21no—
304C22—21no—
305C22C2×#120×C13711yescharacteristic
306C22C22×Q16×C13711yescharacteristic
307C22C2×#137×C13711yescharacteristic
308C4—24no—
309C4—21no—
310C4—22no—
311C4—44no—
312C4—44no—
313C4—42no—
314C4—44no—
315C4—82no—
316C4#451×C13714yes—
317C4#456×C13712yes—
318C2—21no—
319C2—21no—
320C2C2×#451×C13711yescharacteristic
321C2C2×#456×C13711yescharacteristic
322C2#3027×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC23×C137C22×D4
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
78. C2×Q8×C137Q16semidirect
161. #18404C137direct (N × H)
162. C137#18404direct (N × H)
240. C2×Q8Q16×C137semidirect

Maximal subgroups

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

subgroupindex
#3502×C1372
#3209×C1372
#2567×C1372
#1605×C1372
#1503×C1372
#1461×C1372
#18404137