Finite Group Databasebeta

← C137×#13560 #46862946

Subgroups — C137×#13560 #46862946

Order 35072.

486 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 90 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#3018×C137C211yesmaximal, characteristic
2#2126×C137C211yesmaximal, characteristic, complemented
3#1543×C137C211yesmaximal, characteristic, complemented
4#1402×C137C214yesmaximal, complemented
5#585×C137C411yescharacteristic, complemented
6#575×C137—21no—
7#532×C137C2211yescharacteristic
8#456×C137—21no—
9#455×C137C412yescomplemented
10#454×C137—21no—
11#453×C137C2212yes—
12#431×C137—21no—
13C2×#104×C137—21no—
14#413×C137C411yescharacteristic, complemented
15#410×C137—21no—
16#354×C137—21no—
17#353×C137C2212yescomplemented
18#333×C137—21no—
19#331×C137C2212yescomplemented
202−1+4×C137—21no—
212−1+4×C137—41no—
22H(5,𝔽2)×C137—41no—
23H(5,𝔽2)×C137—41no—
24H(5,𝔽2)×C137C2×C412yes—
25C2×#41×C137—21no—
26C2×#41×C137—41no—
27C2×#41×C137—42no—
28C2×#41×C137C2×C411yescharacteristic
29C2×#41×C137D411yescharacteristic
30#137×C137—22no—
31#137×C137—41no—
32AGL(1,ℤ/8)×C137—22no—
33AGL(1,ℤ/8)×C137—41no—
34#135×C137—42no—
35C2×M16×C137—21no—
36C4⋊Q8×C137D411yescharacteristic
37#127×C137C2311yescharacteristic
38#124×C137C2×C411yescharacteristic
39#124×C137D411yescharacteristic
40#121×C137—41no—
41C4×D4×C137—41no—
42#117×C137—41no—
43C2×C42×C137D411yescharacteristic
44C4⋊C8×C137—22no—
45#104×C137—22no—
46#104×C137—22no—
47#101×C137—21no—
48#100×C137C2×C412yes—
49#99×C137—22no—
50#99×C137—41no—
51C22⋊C8×C137—41no—
52C4×C8×C137—41no—
53#41×C137—22no—
54#41×C137—22no—
55#41×C137—41no—
56#41×C137—42no—
57#41×C137—41no—
58#41×C137—41no—
59#41×C137—41no—
60#41×C137—41no—
61#41×C137—42no—
62#41×C137—42no—
63#41×C137—44no—
64#41×C137—42no—
65#41×C137—41no—
66#41×C137—42no—
67C2×Q8×C137—21no—
68C2×Q8×C137—42no—
69C2×Q8×C137—41no—
70C2×Q8×C137C22⋊C411yescharacteristic
71C2×Q8×C137C22⋊C411yescharacteristic
72C2×D4×C137—21no—
73C2×D4×C137—21no—
74C2×D4×C137—21no—
75C2×D4×C137—41no—
76C2×D4×C137—42no—
77C2×D4×C137—44no—
78C2×D4×C137—41no—
79C2×D4×C137—42no—
80C2×D4×C137—41no—
81C2×D4×C137C22×C411yescharacteristic
82C2×D4×C137C2×D411yescharacteristic
83C22×C4×C137—21no—
84C22×C4×C137—21no—
85C22×C4×C137—21no—
86C22×C4×C137—41no—
87C22×C4×C137—42no—
88C22×C4×C137C22⋊C411yescharacteristic
89Q16×C137—42no—
90SD16×C137—42no—
91SD16×C137—42no—
92D8×C137—42no—
93M16×C137—22no—
94M16×C137—41no—
95C2×C8×C137—41no—
96C2×C8×C137—42no—
97C4⋊C4×C137—41no—
98C4⋊C4×C137—81no—
99C22⋊C4×C137—41no—
100C22⋊C4×C137—41no—
101C22⋊C4×C137—42no—
102C42×C137—21no—
103C42×C137—81no—
104C42×C137C22⋊C411yescharacteristic
105C42×C137C2×D411yescharacteristic
106C23×C137—21no—
107C23×C137—21no—
108C23×C137—41no—
109C23×C137—42no—
110C23×C137#9911yescharacteristic
111Q8×C137—22no—
112Q8×C137—41no—
113Q8×C137—41no—
114Q8×C137—42no—
115Q8×C137—41no—
116D4×C137—22no—
117D4×C137—41no—
118D4×C137—41no—
119D4×C137—42no—
120D4×C137—42no—
121D4×C137—42no—
122D4×C137—44no—
123D4×C137—42no—
124D4×C137—41no—
125D4×C137—82no—
126D4×C137—81no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—21no—
130C2×C4×C137—21no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—42no—
134C2×C4×C137—42no—
135C2×C4×C137—41no—
136C2×C4×C137—44no—
137C2×C4×C137—41no—
138C2×C4×C137—41no—
139C2×C4×C137—41no—
140C2×C4×C137—41no—
141C2×C4×C137—42no—
142C2×C4×C137—42no—
143C2×C4×C137—82no—
144C2×C4×C137#9911yescharacteristic
145C2×C4×C137C2×C22⋊C411yescharacteristic
146C8×C137—42no—
147C8×C137—82no—
148C22×C137—21no—
149C22×C137—41no—
150C22×C137—42no—
151C22×C137—41no—
152C22×C137—41no—
153C22×C137—81no—
154C22×C137—81no—
155C22×C137—161no—
156C22×C137—162no—
157C22×C137C2×#9911yescharacteristic
158C4×C137—22no—
159C4×C137—21no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137—42no—
164C4×C137—162no—
165C4×C137AΣL(1,𝔽16)11yescharacteristic
166C4×C137AΣL(1,𝔽16)11yescharacteristic
167C2×C137—21no—
168C2×C137—41no—
169C2×C137—81no—
170C2×C137—81no—
171C2×C137—82no—
172C2×C137C2×AΣL(1,𝔽16)11yescharacteristic
173#13560C13711yesmaximal, characteristic, direct product factor, Sylow
174C137#1356011yescharacteristic, direct product factor, Sylow
175#3018C2×C13711yescharacteristic
176#2126C2×C13711yescharacteristic, complemented
177#1543C2×C13711yescharacteristic, complemented
178#1402C2×C13714yescomplemented
179#585C4×C13711yescharacteristic, complemented
180#575—21no—
181#532C22×C13711yescharacteristic
182#456—21no—
183#455C4×C13712yescomplemented
184#454—21no—
185#453C22×C13712yes—
186#431—21no—
187C2×#104—21no—
188#413C4×C13711yescharacteristic, complemented
189#410—21no—
190#354—21no—
191#353C22×C13712yescomplemented
192#333—21no—
193#331C22×C13712yescomplemented
1942−1+4—21no—
1952−1+4—41no—
196H(5,𝔽2)—41no—
197H(5,𝔽2)—41no—
198H(5,𝔽2)C2×C4×C13712yes—
199C2×#41—21no—
200C2×#41—41no—
201C2×#41—42no—
202C2×#41C2×C4×C13711yescharacteristic
203C2×#41D4×C13711yescharacteristic
204#137—22no—
205#137—41no—
206AGL(1,ℤ/8)—22no—
207AGL(1,ℤ/8)—41no—
208#135—42no—
209C2×M16—21no—
210C4⋊Q8D4×C13711yescharacteristic
211#127C23×C13711yescharacteristic
212#124C2×C4×C13711yescharacteristic
213#124D4×C13711yescharacteristic
214#121—41no—
215C4×D4—41no—
216#117—41no—
217C2×C42D4×C13711yescharacteristic
218C4⋊C8—22no—
219#104—22no—
220#104—22no—
221#101—21no—
222#100C2×C4×C13712yes—
223#99—22no—
224#99—41no—
225C22⋊C8—41no—
226C4×C8—41no—
227#41—22no—
228#41—22no—
229#41—41no—
230#41—42no—
231#41—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235#41—42no—
236#41—42no—
237#41—44no—
238#41—42no—
239#41—41no—
240#41—42no—
241C2×Q8—21no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8C22⋊C4×C13711yescharacteristic
245C2×Q8C22⋊C4×C13711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4C22×C4×C13711yescharacteristic
256C2×D4C2×D4×C13711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4C22⋊C4×C13711yescharacteristic
263Q16—42no—
264SD16—42no—
265SD16—42no—
266D8—42no—
267M16—22no—
268M16—41no—
269C2×C8—41no—
270C2×C8—42no—
271C4⋊C4—41no—
272C4⋊C4—81no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C42—21no—
277C42—81no—
278C42C22⋊C4×C13711yescharacteristic
279C42C2×D4×C13711yescharacteristic
280C23—21no—
281C23—21no—
282C23—41no—
283C23—42no—
284C23#99×C13711yescharacteristic
285Q8—22no—
286Q8—41no—
287Q8—41no—
288Q8—42no—
289Q8—41no—
290D4—22no—
291D4—41no—
292D4—41no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—44no—
297D4—42no—
298D4—41no—
299D4—82no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4#99×C13711yescharacteristic
319C2×C4C2×C22⋊C4×C13711yescharacteristic
320C8—42no—
321C8—82no—
322C22—21no—
323C22—41no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—161no—
330C22—162no—
331C22C2×#99×C13711yescharacteristic
332C4—22no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—42no—
338C4—162no—
339C4AΣL(1,𝔽16)×C13711yescharacteristic
340C4AΣL(1,𝔽16)×C13711yescharacteristic
341C2—21no—
342C2—41no—
343C2—81no—
344C2—81no—
345C2—82no—
346C2C2×AΣL(1,𝔽16)×C13711yescharacteristic
subgroupG/H
centerC4×C137AΣL(1,𝔽16)
commutatorC2×D4C22×C4×C137
Frattini, cosocle#127C23×C137
socleC2×C137C2×AΣL(1,𝔽16)
prime radical#3018×C137C2
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C137 ◁ C2×C4×C137 ◁ C22×C4×C137 ◁ #532×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2126×C137C2semidirect
3. #1543×C137C2semidirect
4. #1402×C137C2semidirect
5. #585×C137C4semidirect
9. #455×C137C4semidirect
14. #413×C137C4semidirect
17. #353×C137C22semidirect
19. #331×C137C22semidirect
173. #13560C137direct (N × H)
174. C137#13560direct (N × H)
176. #2126C2×C137semidirect
177. #1543C2×C137semidirect
178. #1402C2×C137semidirect
179. #585C4×C137semidirect
183. #455C4×C137semidirect
188. #413C4×C137semidirect
191. #353C22×C137semidirect
193. #331C22×C137semidirect

Maximal subgroups

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

subgroupindex
#3018×C1372
#2126×C1372
#1543×C1372
#1402×C1372
#13560137