Subgroups — C₁₃₇×#33647 #46883033

Order 35072.

584 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3050×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2237×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#2234×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#2159×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2139×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2112×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	#525×C₁₃₇	—	2	1	no	—
11	C₂×#117×C₁₃₇	—	2	1	no	—
12	C₂×M₃₂×C₁₃₇	C₂²	1	2	yes	—
13	#467×C₁₃₇	C₂²	1	2	yes	—
14	#466×C₁₃₇	C₂²	1	2	yes	—
15	#465×C₁₃₇	—	2	2	no	—
16	#427×C₁₃₇	—	2	2	no	—
17	#427×C₁₃₇	C₂²	1	1	yes	characteristic
18	#427×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₇	—	2	1	no	—
21	#416×C₁₃₇	—	2	1	no	—
22	#367×C₁₃₇	C₂²	1	4	yes	complemented
23	#366×C₁₃₇	C₂²	1	4	yes	complemented
24	#358×C₁₃₇	C₂²	1	4	yes	complemented
25	#357×C₁₃₇	C₂²	1	4	yes	complemented
26	C₂²⋊C₁₆×C₁₃₇	C₂²	1	4	yes	—
27	C₂×#41×C₁₃₇	—	2	1	no	—
28	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
29	#135×C₁₃₇	—	2	4	no	—
30	C₂×Q₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
32	C₂×D₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₇	—	2	4	no	—
34	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	#123×C₁₃₇	—	4	1	no	—
39	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
40	#121×C₁₃₇	—	2	2	no	—
41	#120×C₁₃₇	—	4	1	no	—
42	C₄×D₄×C₁₃₇	—	4	2	no	—
43	#117×C₁₃₇	—	2	1	no	—
44	#117×C₁₃₇	—	2	4	no	—
45	#117×C₁₃₇	—	4	2	no	—
46	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	4	1	no	—
48	C₂×C₄²×C₁₃₇	—	4	1	no	—
49	M₃₂×C₁₃₇	—	2	4	no	—
50	C₂×C₁₆×C₁₃₇	C₂³	1	4	yes	—
51	#107×C₁₃₇	—	2	2	no	—
52	#107×C₁₃₇	C₂³	1	2	yes	—
53	#107×C₁₃₇	C₂³	1	4	yes	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	1	no	—
56	#103×C₁₃₇	—	2	2	no	—
57	#102×C₁₃₇	—	2	2	no	—
58	C₂⁴×C₁₃₇	D₈	1	1	yes	characteristic
59	#41×C₁₃₇	—	4	2	no	—
60	#41×C₁₃₇	—	4	2	no	—
61	C₂×Q₈×C₁₃₇	—	2	1	no	—
62	C₂×D₄×C₁₃₇	—	2	1	no	—
63	C₂×D₄×C₁₃₇	—	4	2	no	—
64	C₂×D₄×C₁₃₇	—	4	1	no	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	4	2	no	—
68	C₂²×C₄×C₁₃₇	—	4	1	no	—
69	C₂²×C₄×C₁₃₇	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₇	D₈	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₁₃₇	—	2	2	no	—
74	SD₁₆×C₁₃₇	—	4	2	no	—
75	D₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	4	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₇	#41	1	4	yes	—
82	C₂×C₈×C₁₃₇	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	2	no	—
86	C₄⋊C₄×C₁₃₇	—	4	2	no	—
87	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
88	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
89	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
91	C₄²×C₁₃₇	—	4	4	no	—
92	C₄²×C₁₃₇	—	4	1	no	—
93	C₁₆×C₁₃₇	—	2	4	no	—
94	C₂³×C₁₃₇	—	2	2	no	—
95	C₂³×C₁₃₇	—	4	2	no	—
96	C₂³×C₁₃₇	—	4	1	no	—
97	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
98	C₂³×C₁₃₇	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₁₃₇	—	4	1	no	—
100	Q₈×C₁₃₇	—	4	1	no	—
101	D₄×C₁₃₇	—	4	1	no	—
102	D₄×C₁₃₇	—	4	1	no	—
103	D₄×C₁₃₇	—	8	4	no	—
104	D₄×C₁₃₇	—	8	2	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	2	no	—
108	C₂×C₄×C₁₃₇	—	2	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	1	no	—
112	C₂×C₄×C₁₃₇	—	8	4	no	—
113	C₂×C₄×C₁₃₇	—	8	2	no	—
114	C₂×C₄×C₁₃₇	#123	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	#137	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃₇	—	2	2	no	—
120	C₈×C₁₃₇	#123	1	2	yes	—
121	C₈×C₁₃₇	C₂×#41	1	2	yes	—
122	C₂²×C₁₃₇	—	2	2	no	—
123	C₂²×C₁₃₇	—	2	2	no	—
124	C₂²×C₁₃₇	—	2	2	no	—
125	C₂²×C₁₃₇	—	2	2	no	—
126	C₂²×C₁₃₇	—	4	4	no	—
127	C₂²×C₁₃₇	—	8	1	no	—
128	C₂²×C₁₃₇	—	8	2	no	—
129	C₂²×C₁₃₇	#480	1	2	yes	—
130	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	8	1	no	—
133	C₄×C₁₃₇	—	8	4	no	—
134	C₄×C₁₃₇	—	8	2	no	—
135	C₄×C₁₃₇	#480	1	2	yes	—
136	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
137	C₄×C₁₃₇	C₂×#137	1	1	yes	characteristic
138	C₂×C₁₃₇	—	2	2	no	—
139	C₂×C₁₃₇	—	4	2	no	—
140	C₂×C₁₃₇	—	16	1	no	—
141	C₂×C₁₃₇	C₂×#480	1	1	yes	characteristic
142	C₂×C₁₃₇	#3414	1	2	yes	—
143	#33647	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₇	#33647	1	1	yes	characteristic, direct product factor, Sylow
145	#3050	C₂×C₁₃₇	1	1	yes	characteristic
146	C₂×#427	C₂×C₁₃₇	1	1	yes	characteristic, complemented
147	#2237	C₂×C₁₃₇	1	4	yes	complemented
148	#2234	C₂×C₁₃₇	1	4	yes	complemented
149	#2159	C₂×C₁₃₇	1	2	yes	complemented
150	#2139	C₂×C₁₃₇	1	2	yes	complemented
151	#2112	C₂×C₁₃₇	1	1	yes	characteristic, complemented
152	C₂×#135	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
154	#525	—	2	1	no	—
155	C₂×#117	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₁₃₇	1	2	yes	—
157	#467	C₂²×C₁₃₇	1	2	yes	—
158	#466	C₂²×C₁₃₇	1	2	yes	—
159	#465	—	2	2	no	—
160	#427	—	2	2	no	—
161	#427	C₂²×C₁₃₇	1	4	yes	—
162	#427	C₂²×C₁₃₇	1	1	yes	characteristic
163	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
164	C₂×#106	—	2	1	no	—
165	#416	—	2	1	no	—
166	#367	C₂²×C₁₃₇	1	4	yes	complemented
167	#366	C₂²×C₁₃₇	1	4	yes	complemented
168	#358	C₂²×C₁₃₇	1	4	yes	complemented
169	#357	C₂²×C₁₃₇	1	4	yes	complemented
170	C₂²⋊C₁₆	C₂²×C₁₃₇	1	4	yes	—
171	C₂×#41	—	2	1	no	—
172	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×D₈	C₂³×C₁₃₇	1	1	yes	characteristic
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
180	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
181	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
182	#123	—	4	1	no	—
183	C₂²⋊Q₈	—	2	2	no	—
184	#121	—	2	2	no	—
185	#120	—	4	1	no	—
186	C₄×D₄	—	4	2	no	—
187	#117	—	2	4	no	—
188	#117	—	2	1	no	—
189	#117	—	4	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	4	1	no	—
192	C₂×C₄²	—	4	1	no	—
193	M₃₂	—	2	4	no	—
194	C₂×C₁₆	C₂³×C₁₃₇	1	4	yes	—
195	#107	—	2	2	no	—
196	#107	C₂³×C₁₃₇	1	4	yes	—
197	#107	C₂³×C₁₃₇	1	2	yes	—
198	#106	—	2	4	no	—
199	#106	—	2	1	no	—
200	#103	—	2	2	no	—
201	#102	—	2	2	no	—
202	C₂⁴	D₈×C₁₃₇	1	1	yes	characteristic
203	#41	—	4	2	no	—
204	#41	—	4	2	no	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×D₄	—	2	1	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	D₈×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₄	D₈×C₁₃₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	4	2	no	—
219	D₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₃₇	1	4	yes	—
226	C₂×C₈	C₂⁴×C₁₃₇	1	1	yes	characteristic
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	4	no	—
236	C₄²	—	4	1	no	—
237	C₁₆	—	2	4	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	1	no	—
241	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
242	C₂³	C₂×D₈×C₁₃₇	1	1	yes	characteristic
243	Q₈	—	4	1	no	—
244	Q₈	—	4	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	D₄	—	8	4	no	—
248	D₄	—	8	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	#123×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	#137×C₁₃₇	1	2	yes	—
262	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	#123×C₁₃₇	1	2	yes	—
265	C₈	C₂×#41×C₁₃₇	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	1	no	—
272	C₂²	—	8	2	no	—
273	C₂²	#480×C₁₃₇	1	2	yes	—
274	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	8	1	no	—
277	C₄	—	8	4	no	—
278	C₄	—	8	2	no	—
279	C₄	#480×C₁₃₇	1	2	yes	—
280	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
281	C₄	C₂×#137×C₁₃₇	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	—	16	1	no	—
285	C₂	C₂×#480×C₁₃₇	1	1	yes	characteristic
286	C₂	#3414×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₇
prime radical	#3050×C₁₃₇	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ C₂³×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₁₃₇	C₂	semidirect
3. #2237×C₁₃₇	C₂	semidirect
4. #2234×C₁₃₇	C₂	semidirect
5. #2159×C₁₃₇	C₂	semidirect
6. #2139×C₁₃₇	C₂	semidirect
7. #2112×C₁₃₇	C₂	semidirect
22. #367×C₁₃₇	C₂²	semidirect
23. #366×C₁₃₇	C₂²	semidirect
24. #358×C₁₃₇	C₂²	semidirect
25. #357×C₁₃₇	C₂²	semidirect
143. #33647	C₁₃₇	direct (N × H)
144. C₁₃₇	#33647	direct (N × H)
146. C₂×#427	C₂×C₁₃₇	semidirect
147. #2237	C₂×C₁₃₇	semidirect
148. #2234	C₂×C₁₃₇	semidirect
149. #2159	C₂×C₁₃₇	semidirect
150. #2139	C₂×C₁₃₇	semidirect
151. #2112	C₂×C₁₃₇	semidirect
166. #367	C₂²×C₁₃₇	semidirect
167. #366	C₂²×C₁₃₇	semidirect
168. #358	C₂²×C₁₃₇	semidirect
169. #357	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3050×C₁₃₇	2
C₂×#427×C₁₃₇	2
#2237×C₁₃₇	2
#2234×C₁₃₇	2
#2159×C₁₃₇	2
#2139×C₁₃₇	2
#2112×C₁₃₇	2
#33647	137

Subgroups — C137×#33647 #46883033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#33647 #46883033