Finite Group Databasebeta

← C137×#18221 #46867607

Subgroups — C137×#18221 #46867607

Order 35072.

890 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3502×C137C211yesmaximal, characteristic
2#3104×C137C212yesmaximal
3#2581×C137C211yesmaximal, characteristic
4#1617×C137C214yesmaximal
5#1605×C137C214yesmaximal
6#1518×C137C212yesmaximal
7#1461×C137C211yesmaximal, characteristic
8Q82×C137—22no—
9#557×C137—22no—
10#557×C137—24no—
11#544×C137—21no—
12#544×C137C2214yes—
13#533×C137—22no—
14#533×C137C2212yes—
15#519×C137—21no—
16C2×C4×Q8×C137C2211yescharacteristic
17C2×C4×Q8×C137C2211yescharacteristic
18#484×C137C2214yes—
19#477×C137—22no—
20#462×C137—22no—
21#451×C137C2214yes—
22C4×Q16×C137—24no—
23#431×C137C2212yes—
24#400×C137C2212yes—
25#395×C137C2212yes—
26#391×C137—21no—
27#389×C137—21no—
28#387×C137—22no—
29#382×C137C2211yescharacteristic
30#328×C137C2218yes—
31#324×C137C2214yes—
32C22×Q8×C137D411yescharacteristic
33C22×Q8×C137C2311yescharacteristic
34C2×Q16×C137—24no—
35C4⋊Q8×C137—22no—
36C4⋊Q8×C137—22no—
37C4⋊Q8×C137—24no—
38C4⋊Q8×C137—28no—
39C4⋊Q8×C137D412yes—
40C4⋊Q8×C137D412yes—
41#125×C137—24no—
42#125×C137—24no—
43#125×C137—28no—
44#125×C137—22no—
45C22⋊Q8×C137—22no—
46C22⋊Q8×C137—24no—
47C22⋊Q8×C137C2314yes—
48C4×Q8×C137—24no—
49C4×Q8×C137—24no—
50C4×Q8×C137—24no—
51C4×Q8×C137—24no—
52C4×Q8×C137—22no—
53C4×Q8×C137—22no—
54C4×Q8×C137—22no—
55C4×Q8×C137—22no—
56C4×Q8×C137—22no—
57C4×Q8×C137D414yes—
58#117×C137—24no—
59#117×C137—22no—
60C2×C4⋊C4×C137—21no—
61C2×C4⋊C4×C137—21no—
62C2×C4⋊C4×C137—22no—
63C2×C4⋊C4×C137D411yescharacteristic
64C2×C4⋊C4×C137C2311yescharacteristic
65C2×C4⋊C4×C137C2312yes—
66C2×C42×C137—21no—
67C2×C42×C137D412yes—
68C2×C42×C137C2311yescharacteristic
69#107×C137—24no—
70C4⋊C8×C137—22no—
71#103×C137—24no—
72#103×C137—24no—
73C22⋊C8×C137C2314yes—
74C4×C8×C137—22no—
75#95×C137—21no—
76#95×C137—22no—
77#95×C137—22no—
78#95×C137—22no—
79#95×C137C2312yes—
80C2×Q8×C137—24no—
81C2×Q8×C137—22no—
82C2×Q8×C137—22no—
83C2×Q8×C137—22no—
84C2×Q8×C137—22no—
85C2×Q8×C137—42no—
86C2×Q8×C137Q1614yescomplemented
87C2×Q8×C137C2×D414yes—
88C22×C4×C137—21no—
89C22×C4×C137—22no—
90C22×C4×C137—21no—
91C22×C4×C137C2×D412yes—
92C22×C4×C137C2×D412yes—
93C22×C4×C137C2×D411yescharacteristic
94C22×C4×C137C2×D411yescharacteristic
95C22×C4×C137C2411yescharacteristic
96Q16×C137—48no—
97C2×C8×C137—24no—
98C4⋊C4×C137—24no—
99C4⋊C4×C137—24no—
100C4⋊C4×C137—24no—
101C4⋊C4×C137—28no—
102C4⋊C4×C137—22no—
103C4⋊C4×C137—22no—
104C4⋊C4×C137—22no—
105C4⋊C4×C137—22no—
106C4⋊C4×C137—24no—
107C4⋊C4×C137—24no—
108C4⋊C4×C137—24no—
109C4⋊C4×C137—22no—
110C4⋊C4×C137—44no—
111C4⋊C4×C137—41no—
112C4⋊C4×C137—41no—
113C4⋊C4×C137—42no—
114C4⋊C4×C137C2×D414yes—
115C22⋊C4×C137—24no—
116C42×C137—28no—
117C42×C137—22no—
118C42×C137—22no—
119C42×C137—24no—
120C42×C137—42no—
121C42×C137C2×D412yes—
122C42×C137C2×D412yes—
123C23×C137C22×D411yescharacteristic
124Q8×C137—28no—
125Q8×C137—24no—
126Q8×C137—44no—
127Q8×C137—44no—
128Q8×C137—42no—
129Q8×C137—84no—
130C2×C4×C137—22no—
131C2×C4×C137—22no—
132C2×C4×C137—22no—
133C2×C4×C137—24no—
134C2×C4×C137—24no—
135C2×C4×C137—22no—
136C2×C4×C137—41no—
137C2×C4×C137—41no—
138C2×C4×C137—41no—
139C2×C4×C137—41no—
140C2×C4×C137—42no—
141C2×C4×C137—42no—
142C2×C4×C137—42no—
143C2×C4×C137—42no—
144C2×C4×C137—41no—
145C2×C4×C137—42no—
146C2×C4×C137#12011yescharacteristic
147C2×C4×C137#12011yescharacteristic
148C2×C4×C137#12012yes—
149C2×C4×C137C2×Q1614yes—
150C2×C4×C137C2×Q1612yes—
151C2×C4×C137#13712yes—
152C2×C4×C137C22×D412yes—
153C8×C137—44no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137C2×#12011yescharacteristic
157C22×C137C22×Q1611yescharacteristic
158C22×C137C2×#13711yescharacteristic
159C4×C137—24no—
160C4×C137—21no—
161C4×C137—21no—
162C4×C137—22no—
163C4×C137—44no—
164C4×C137—42no—
165C4×C137—42no—
166C4×C137—44no—
167C4×C137—82no—
168C4×C137—81no—
169C4×C137#45114yes—
170C2×C137—21no—
171C2×C137—21no—
172C2×C137C2×#45111yescharacteristic
173C2×C137#302011yescharacteristic
174C2×C137#302511yescharacteristic
175#18221C13711yesmaximal, characteristic, direct product factor, Sylow
176C137#1822111yescharacteristic, direct product factor, Sylow
177#3502C2×C13711yescharacteristic
178#3104C2×C13712yes—
179#2581C2×C13711yescharacteristic
180#1617C2×C13714yes—
181#1605C2×C13714yes—
182#1518C2×C13712yes—
183#1461C2×C13711yescharacteristic
184Q82—22no—
185#557—22no—
186#557—24no—
187#544—21no—
188#544C22×C13714yes—
189#533—22no—
190#533C22×C13712yes—
191#519—21no—
192C2×C4×Q8C22×C13711yescharacteristic
193C2×C4×Q8C22×C13711yescharacteristic
194#484C22×C13714yes—
195#477—22no—
196#462—22no—
197#451C22×C13714yes—
198C4×Q16—24no—
199#431C22×C13712yes—
200#400C22×C13712yes—
201#395C22×C13712yes—
202#391—21no—
203#389—21no—
204#387—22no—
205#382C22×C13711yescharacteristic
206#328C22×C13718yes—
207#324C22×C13714yes—
208C22×Q8D4×C13711yescharacteristic
209C22×Q8C23×C13711yescharacteristic
210C2×Q16—24no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—24no—
214C4⋊Q8—28no—
215C4⋊Q8D4×C13712yes—
216C4⋊Q8D4×C13712yes—
217#125—24no—
218#125—24no—
219#125—28no—
220#125—22no—
221C22⋊Q8—22no—
222C22⋊Q8—24no—
223C22⋊Q8C23×C13714yes—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8D4×C13714yes—
234#117—24no—
235#117—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4D4×C13711yescharacteristic
240C2×C4⋊C4C23×C13711yescharacteristic
241C2×C4⋊C4C23×C13712yes—
242C2×C42—21no—
243C2×C42D4×C13712yes—
244C2×C42C23×C13711yescharacteristic
245#107—24no—
246C4⋊C8—22no—
247#103—24no—
248#103—24no—
249C22⋊C8C23×C13714yes—
250C4×C8—22no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95C23×C13712yes—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—42no—
262C2×Q8Q16×C13714yescomplemented
263C2×Q8C2×D4×C13714yes—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4C2×D4×C13712yes—
268C22×C4C2×D4×C13712yes—
269C22×C4C2×D4×C13711yescharacteristic
270C22×C4C2×D4×C13711yescharacteristic
271C22×C4C24×C13711yescharacteristic
272Q16—48no—
273C2×C8—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—44no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4C2×D4×C13714yes—
291C22⋊C4—24no—
292C42—28no—
293C42—22no—
294C42—22no—
295C42—24no—
296C42—42no—
297C42C2×D4×C13712yes—
298C42C2×D4×C13712yes—
299C23C22×D4×C13711yescharacteristic
300Q8—28no—
301Q8—24no—
302Q8—44no—
303Q8—42no—
304Q8—44no—
305Q8—84no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4#120×C13711yescharacteristic
323C2×C4#120×C13711yescharacteristic
324C2×C4#120×C13712yes—
325C2×C4C2×Q16×C13714yes—
326C2×C4C2×Q16×C13712yes—
327C2×C4#137×C13712yes—
328C2×C4C22×D4×C13712yes—
329C8—44no—
330C22—21no—
331C22—21no—
332C22C2×#120×C13711yescharacteristic
333C22C22×Q16×C13711yescharacteristic
334C22C2×#137×C13711yescharacteristic
335C4—24no—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—44no—
340C4—42no—
341C4—42no—
342C4—44no—
343C4—82no—
344C4—81no—
345C4#451×C13714yes—
346C2—21no—
347C2—21no—
348C2C2×#451×C13711yescharacteristic
349C2#3020×C13711yescharacteristic
350C2#3025×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
86. C2×Q8×C137Q16semidirect
175. #18221C137direct (N × H)
176. C137#18221direct (N × H)
262. 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
#3104×C1372
#2581×C1372
#1617×C1372
#1605×C1372
#1518×C1372
#1461×C1372
#18221137