Subgroups — C₁₃₇×#28674 #46878060

Order 35072.

674 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3166×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3138×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#490×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2961×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2063×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1998×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1931×C₁₃₇	C₂	1	4	yes	maximal, complemented
9	#1778×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
13	#501×C₁₃₇	—	2	2	no	—
14	#499×C₁₃₇	—	2	2	no	—
15	#490×C₁₃₇	—	2	2	no	—
16	#490×C₁₃₇	—	2	4	no	—
17	#486×C₁₃₇	—	2	2	no	—
18	#446×C₁₃₇	—	2	4	no	—
19	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
20	#428×C₁₃₇	C₂²	1	4	yes	—
21	#424×C₁₃₇	C₂²	1	2	yes	—
22	#423×C₁₃₇	C₂²	1	1	yes	characteristic
23	#418×C₁₃₇	C₂²	1	4	yes	—
24	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
27	#405×C₁₃₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
31	#343×C₁₃₇	C₂²	1	4	yes	—
32	#341×C₁₃₇	C₂²	1	2	yes	—
33	C₂²×D₄×C₁₃₇	—	2	2	no	—
34	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
35	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
36	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
41	#127×C₁₃₇	—	2	1	no	—
42	#127×C₁₃₇	—	2	1	no	—
43	#127×C₁₃₇	—	2	2	no	—
44	#125×C₁₃₇	—	2	2	no	—
45	#125×C₁₃₇	—	2	1	no	—
46	#125×C₁₃₇	—	2	1	no	—
47	#125×C₁₃₇	D₄	1	2	yes	—
48	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
49	C₄×Q₈×C₁₃₇	—	2	2	no	—
50	#117×C₁₃₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
54	#108×C₁₃₇	—	2	4	no	—
55	#107×C₁₃₇	—	2	4	no	—
56	#106×C₁₃₇	—	2	4	no	—
57	C₄⋊C₈×C₁₃₇	—	2	4	no	—
58	C₄⋊C₈×C₁₃₇	—	2	2	no	—
59	#102×C₁₃₇	—	2	4	no	—
60	#102×C₁₃₇	—	2	4	no	—
61	#102×C₁₃₇	—	2	8	no	—
62	#100×C₁₃₇	—	2	8	no	—
63	#97×C₁₃₇	—	2	2	no	—
64	#97×C₁₃₇	—	2	1	no	—
65	#97×C₁₃₇	—	2	1	no	—
66	#97×C₁₃₇	D₄	1	4	yes	—
67	C₄×C₈×C₁₃₇	—	2	2	no	—
68	C₄×C₈×C₁₃₇	—	2	1	no	—
69	C₄×C₈×C₁₃₇	—	2	1	no	—
70	C₂⁴×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×D₄×C₁₃₇	—	2	2	no	—
73	C₂×D₄×C₁₃₇	—	2	2	no	—
74	C₂×D₄×C₁₃₇	—	2	4	no	—
75	C₂×D₄×C₁₃₇	—	2	4	no	—
76	C₂×D₄×C₁₃₇	—	4	2	no	—
77	C₂×D₄×C₁₃₇	—	4	2	no	—
78	C₂×D₄×C₁₃₇	—	8	4	no	—
79	C₂²×C₄×C₁₃₇	—	2	2	no	—
80	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
81	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
82	M₁₆×C₁₃₇	—	2	2	no	—
83	M₁₆×C₁₃₇	—	2	4	no	—
84	M₁₆×C₁₃₇	—	2	2	no	—
85	M₁₆×C₁₃₇	—	2	2	no	—
86	M₁₆×C₁₃₇	#41	1	4	yes	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	—	2	2	no	—
92	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄⋊C₄×C₁₃₇	—	2	2	no	—
96	C₄⋊C₄×C₁₃₇	—	2	2	no	—
97	C₄⋊C₄×C₁₃₇	—	2	2	no	—
98	C₄⋊C₄×C₁₃₇	—	4	2	no	—
99	C₄⋊C₄×C₁₃₇	—	4	2	no	—
100	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
101	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	—	2	2	no	—
103	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
106	C₂³×C₁₃₇	—	4	2	no	—
107	C₂³×C₁₃₇	—	4	4	no	—
108	C₂³×C₁₃₇	—	8	4	no	—
109	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
110	Q₈×C₁₃₇	—	8	2	no	—
111	D₄×C₁₃₇	—	4	2	no	—
112	D₄×C₁₃₇	—	4	2	no	—
113	D₄×C₁₃₇	—	4	4	no	—
114	D₄×C₁₃₇	—	8	4	no	—
115	D₄×C₁₃₇	—	8	4	no	—
116	C₂×C₄×C₁₃₇	—	2	4	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	8	2	no	—
120	C₂×C₄×C₁₃₇	#121	1	2	yes	—
121	C₂×C₄×C₁₃₇	#121	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
126	C₈×C₁₃₇	—	2	2	no	—
127	C₈×C₁₃₇	—	2	2	no	—
128	C₈×C₁₃₇	—	2	4	no	—
129	C₈×C₁₃₇	—	4	2	no	—
130	C₈×C₁₃₇	—	4	2	no	—
131	C₂²×C₁₃₇	—	2	2	no	—
132	C₂²×C₁₃₇	—	8	2	no	—
133	C₂²×C₁₃₇	—	8	4	no	—
134	C₂²×C₁₃₇	—	16	4	no	—
135	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
136	C₂²×C₁₃₇	#556	1	2	yes	—
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	2	2	no	—
139	C₄×C₁₃₇	—	4	4	no	—
140	C₄×C₁₃₇	—	8	2	no	—
141	C₄×C₁₃₇	#532	1	1	yes	characteristic
142	C₄×C₁₃₇	#536	1	1	yes	characteristic
143	C₄×C₁₃₇	#548	1	2	yes	—
144	C₂×C₁₃₇	—	2	2	no	—
145	C₂×C₁₃₇	—	16	2	no	—
146	C₂×C₁₃₇	#2545	1	1	yes	characteristic
147	C₂×C₁₃₇	#3068	1	2	yes	—
148	#28674	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₁₃₇	#28674	1	1	yes	characteristic, direct product factor, Sylow
150	#3166	C₂×C₁₃₇	1	1	yes	characteristic, complemented
151	#3138	C₂×C₁₃₇	1	1	yes	characteristic
152	C₂×#490	C₂×C₁₃₇	1	1	yes	characteristic
153	#2961	C₂×C₁₃₇	1	1	yes	characteristic, complemented
154	#2063	C₂×C₁₃₇	1	4	yes	complemented
155	#1998	C₂×C₁₃₇	1	1	yes	characteristic, complemented
156	#1965	C₂×C₁₃₇	1	1	yes	characteristic
157	#1931	C₂×C₁₃₇	1	4	yes	complemented
158	#1778	C₂×C₁₃₇	1	1	yes	characteristic, complemented
159	#533	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
162	#501	—	2	2	no	—
163	#499	—	2	2	no	—
164	#490	—	2	2	no	—
165	#490	—	2	4	no	—
166	#486	—	2	2	no	—
167	#446	—	2	4	no	—
168	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
169	#428	C₂²×C₁₃₇	1	4	yes	—
170	#424	C₂²×C₁₃₇	1	2	yes	—
171	#423	C₂²×C₁₃₇	1	1	yes	characteristic
172	#418	C₂²×C₁₃₇	1	4	yes	—
173	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
174	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
175	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
176	#405	C₂²×C₁₃₇	1	2	yes	—
177	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
180	#343	C₂²×C₁₃₇	1	4	yes	—
181	#341	C₂²×C₁₃₇	1	2	yes	—
182	C₂²×D₄	—	2	2	no	—
183	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
184	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
185	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
186	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
190	#127	—	2	1	no	—
191	#127	—	2	1	no	—
192	#127	—	2	2	no	—
193	#125	—	2	2	no	—
194	#125	—	2	1	no	—
195	#125	—	2	1	no	—
196	#125	D₄×C₁₃₇	1	2	yes	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	#117	C₂³×C₁₃₇	1	2	yes	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
202	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
203	#108	—	2	4	no	—
204	#107	—	2	4	no	—
205	#106	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	#102	—	2	4	no	—
209	#102	—	2	4	no	—
210	#102	—	2	8	no	—
211	#100	—	2	8	no	—
212	#97	—	2	2	no	—
213	#97	—	2	1	no	—
214	#97	—	2	1	no	—
215	#97	D₄×C₁₃₇	1	4	yes	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₂⁴	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	8	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	#41×C₁₃₇	1	4	yes	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
242	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	8	4	no	—
258	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
259	Q₈	—	8	2	no	—
260	D₄	—	4	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	D₄	—	8	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	#121×C₁₃₇	1	2	yes	—
270	C₂×C₄	#121×C₁₃₇	1	2	yes	—
271	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
274	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
275	C₈	—	2	2	no	—
276	C₈	—	2	2	no	—
277	C₈	—	2	4	no	—
278	C₈	—	4	2	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	8	4	no	—
283	C₂²	—	16	4	no	—
284	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
285	C₂²	#556×C₁₃₇	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	2	no	—
290	C₄	#532×C₁₃₇	1	1	yes	characteristic
291	C₄	#536×C₁₃₇	1	1	yes	characteristic
292	C₄	#548×C₁₃₇	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	16	2	no	—
295	C₂	#2545×C₁₃₇	1	1	yes	characteristic
296	C₂	#3068×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3166×C₁₃₇	C₂	semidirect
4. #2961×C₁₃₇	C₂	semidirect
5. #2063×C₁₃₇	C₂	semidirect
6. #1998×C₁₃₇	C₂	semidirect
8. #1931×C₁₃₇	C₂	semidirect
9. #1778×C₁₃₇	C₂	semidirect
148. #28674	C₁₃₇	direct (N × H)
149. C₁₃₇	#28674	direct (N × H)
150. #3166	C₂×C₁₃₇	semidirect
153. #2961	C₂×C₁₃₇	semidirect
154. #2063	C₂×C₁₃₇	semidirect
155. #1998	C₂×C₁₃₇	semidirect
157. #1931	C₂×C₁₃₇	semidirect
158. #1778	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3166×C₁₃₇	2
#3138×C₁₃₇	2
C₂×#490×C₁₃₇	2
#2961×C₁₃₇	2
#2063×C₁₃₇	2
#1998×C₁₃₇	2
#1965×C₁₃₇	2
#1931×C₁₃₇	2
#1778×C₁₃₇	2
#28674	137

Subgroups — C137×#28674 #46878060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28674 #46878060