Subgroups — C₁₃₇×#28673 #46878059

Order 35072.

622 conjugacy classes of subgroups.

The subgroup lattice has 284 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
2	#3139×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#491×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2961×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2064×C₁₃₇	C₂	1	4	yes	maximal
6	#1998×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1966×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1930×C₁₃₇	C₂	1	4	yes	maximal
9	#1778×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×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	#491×C₁₃₇	—	2	2	no	—
16	#491×C₁₃₇	—	2	4	no	—
17	#487×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	#419×C₁₃₇	C₂²	1	4	yes	—
24	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#101×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×Q₈×C₁₃₇	C₂³	1	2	yes	—
34	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
35	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
36	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
40	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
41	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₇	D₄	1	2	yes	—
43	#125×C₁₃₇	—	2	2	no	—
44	#125×C₁₃₇	—	2	1	no	—
45	#125×C₁₃₇	—	2	1	no	—
46	#125×C₁₃₇	D₄	1	2	yes	—
47	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
48	C₄×Q₈×C₁₃₇	—	2	2	no	—
49	#117×C₁₃₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
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	#103×C₁₃₇	—	2	4	no	—
60	#103×C₁₃₇	—	2	4	no	—
61	#103×C₁₃₇	—	2	8	no	—
62	#101×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	C₂×Q₈×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×Q₈×C₁₃₇	—	2	2	no	—
73	C₂×Q₈×C₁₃₇	—	2	4	no	—
74	C₂×Q₈×C₁₃₇	—	2	4	no	—
75	C₂²×C₄×C₁₃₇	—	2	2	no	—
76	C₂²×C₄×C₁₃₇	—	2	2	no	—
77	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
78	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₃₇	—	2	2	no	—
80	M₁₆×C₁₃₇	—	2	4	no	—
81	M₁₆×C₁₃₇	—	2	2	no	—
82	M₁₆×C₁₃₇	—	2	2	no	—
83	M₁₆×C₁₃₇	#41	1	4	yes	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	2	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₄⋊C₄×C₁₃₇	—	4	2	no	—
96	C₄⋊C₄×C₁₃₇	—	4	2	no	—
97	C₄⋊C₄×C₁₃₇	—	4	2	no	—
98	C₄⋊C₄×C₁₃₇	—	4	2	no	—
99	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
100	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
101	C₄²×C₁₃₇	—	2	2	no	—
102	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
105	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
106	Q₈×C₁₃₇	—	4	2	no	—
107	Q₈×C₁₃₇	—	4	2	no	—
108	Q₈×C₁₃₇	—	4	4	no	—
109	Q₈×C₁₃₇	—	8	2	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	8	2	no	—
116	C₂×C₄×C₁₃₇	#121	1	2	yes	—
117	C₂×C₄×C₁₃₇	#121	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
121	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
122	C₈×C₁₃₇	—	2	2	no	—
123	C₈×C₁₃₇	—	2	2	no	—
124	C₈×C₁₃₇	—	2	4	no	—
125	C₈×C₁₃₇	—	4	2	no	—
126	C₈×C₁₃₇	—	4	2	no	—
127	C₂²×C₁₃₇	—	2	2	no	—
128	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
129	C₂²×C₁₃₇	#556	1	2	yes	—
130	C₄×C₁₃₇	—	2	2	no	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	4	4	no	—
133	C₄×C₁₃₇	—	8	2	no	—
134	C₄×C₁₃₇	—	8	2	no	—
135	C₄×C₁₃₇	#532	1	1	yes	characteristic
136	C₄×C₁₃₇	#536	1	1	yes	characteristic
137	C₄×C₁₃₇	#548	1	2	yes	—
138	C₂×C₁₃₇	—	2	2	no	—
139	C₂×C₁₃₇	#2545	1	1	yes	characteristic
140	C₂×C₁₃₇	#3069	1	2	yes	—
141	#28673	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₇	#28673	1	1	yes	characteristic, direct product factor, Sylow
143	#3166	C₂×C₁₃₇	1	1	yes	characteristic
144	#3139	C₂×C₁₃₇	1	1	yes	characteristic
145	C₂×#491	C₂×C₁₃₇	1	1	yes	characteristic
146	#2961	C₂×C₁₃₇	1	1	yes	characteristic
147	#2064	C₂×C₁₃₇	1	4	yes	—
148	#1998	C₂×C₁₃₇	1	1	yes	characteristic
149	#1966	C₂×C₁₃₇	1	1	yes	characteristic
150	#1930	C₂×C₁₃₇	1	4	yes	—
151	#1778	C₂×C₁₃₇	1	1	yes	characteristic
152	#533	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
155	#501	—	2	2	no	—
156	#499	—	2	2	no	—
157	#491	—	2	2	no	—
158	#491	—	2	4	no	—
159	#487	—	2	2	no	—
160	#446	—	2	4	no	—
161	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
162	#428	C₂²×C₁₃₇	1	4	yes	—
163	#424	C₂²×C₁₃₇	1	2	yes	—
164	#423	C₂²×C₁₃₇	1	1	yes	characteristic
165	#419	C₂²×C₁₃₇	1	4	yes	—
166	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
167	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#101	C₂²×C₁₃₇	1	4	yes	—
169	#405	C₂²×C₁₃₇	1	2	yes	—
170	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
173	#343	C₂²×C₁₃₇	1	4	yes	—
174	#341	C₂²×C₁₃₇	1	2	yes	—
175	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
176	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
177	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
178	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	D₄×C₁₃₇	1	2	yes	—
185	#125	—	2	2	no	—
186	#125	—	2	1	no	—
187	#125	—	2	1	no	—
188	#125	D₄×C₁₃₇	1	2	yes	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	#117	C₂³×C₁₃₇	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
196	#108	—	2	4	no	—
197	#107	—	2	4	no	—
198	#106	—	2	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	#103	—	2	4	no	—
202	#103	—	2	4	no	—
203	#103	—	2	8	no	—
204	#101	—	2	8	no	—
205	#97	—	2	2	no	—
206	#97	—	2	1	no	—
207	#97	—	2	1	no	—
208	#97	D₄×C₁₃₇	1	4	yes	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	#41×C₁₃₇	1	4	yes	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
247	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	Q₈	—	8	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	#121×C₁₃₇	1	2	yes	—
259	C₂×C₄	#121×C₁₃₇	1	2	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
263	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	4	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
271	C₂²	#556×C₁₃₇	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	2	no	—
277	C₄	#532×C₁₃₇	1	1	yes	characteristic
278	C₄	#536×C₁₃₇	1	1	yes	characteristic
279	C₄	#548×C₁₃₇	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	#2545×C₁₃₇	1	1	yes	characteristic
282	C₂	#3069×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×#41
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
141. #28673	C₁₃₇	direct (N × H)
142. C₁₃₇	#28673	direct (N × H)

Maximal subgroups

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

subgroup	index
#3166×C₁₃₇	2
#3139×C₁₃₇	2
C₂×#491×C₁₃₇	2
#2961×C₁₃₇	2
#2064×C₁₃₇	2
#1998×C₁₃₇	2
#1966×C₁₃₇	2
#1930×C₁₃₇	2
#1778×C₁₃₇	2
#28673	137

Subgroups — C137×#28673 #46878059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28673 #46878059