Subgroups — C₁₃₇×#19661 #46869047

Order 35072.

990 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3205×C₁₃₇	C₂	1	4	yes	maximal, complemented
2	#3147×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#493×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1721×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1712×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1709×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₃₇	—	2	2	no	—
9	C₂²×D₈×C₁₃₇	—	2	2	no	—
10	#532×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
12	#495×C₁₃₇	—	2	2	no	—
13	#494×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
14	#493×C₁₃₇	—	2	2	no	—
15	#493×C₁₃₇	C₂²	1	2	yes	—
16	#493×C₁₃₇	C₂²	1	2	yes	—
17	#493×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
18	#493×C₁₃₇	C₂²	1	8	yes	complemented
19	#492×C₁₃₇	C₂²	1	2	yes	complemented
20	#486×C₁₃₇	C₂²	1	8	yes	complemented
21	#459×C₁₃₇	—	2	8	no	—
22	#447×C₁₃₇	—	2	8	no	—
23	#431×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	#322×C₁₃₇	C₂²	1	2	yes	complemented
26	C₈²×C₁₃₇	C₂²	1	2	yes	complemented
27	C₂×#41×C₁₃₇	—	2	2	no	—
28	C₂²×D₄×C₁₃₇	—	2	2	no	—
29	C₂²×D₄×C₁₃₇	—	2	4	no	—
30	#135×C₁₃₇	—	4	8	no	—
31	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
32	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
33	C₂×D₈×C₁₃₇	—	2	4	no	—
34	C₂×D₈×C₁₃₇	—	2	4	no	—
35	C₂×D₈×C₁₃₇	—	2	2	no	—
36	C₂×D₈×C₁₃₇	—	2	16	no	—
37	C₂×D₈×C₁₃₇	—	4	8	no	—
38	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
39	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
40	#127×C₁₃₇	—	2	4	no	—
41	#127×C₁₃₇	—	2	2	no	—
42	#127×C₁₃₇	C₂³	1	4	yes	—
43	#127×C₁₃₇	C₂³	1	2	yes	—
44	#124×C₁₃₇	—	2	2	no	—
45	#121×C₁₃₇	—	2	4	no	—
46	C₄×D₄×C₁₃₇	—	2	4	no	—
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	—
49	#102×C₁₃₇	—	2	16	no	—
50	C₂²⋊C₈×C₁₃₇	D₄	1	4	yes	—
51	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
52	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₃₇	C₂³	1	4	yes	—
55	C₂⁴×C₁₃₇	—	4	2	no	—
56	#41×C₁₃₇	—	4	8	no	—
57	C₂×Q₈×C₁₃₇	—	2	2	no	—
58	C₂×D₄×C₁₃₇	—	2	8	no	—
59	C₂×D₄×C₁₃₇	—	2	4	no	—
60	C₂×D₄×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	4	2	no	—
63	C₂×D₄×C₁₃₇	—	4	4	no	—
64	C₂×D₄×C₁₃₇	—	4	2	no	—
65	C₂×D₄×C₁₃₇	—	4	8	no	—
66	C₂×D₄×C₁₃₇	—	8	8	no	—
67	C₂²×C₄×C₁₃₇	—	4	2	no	—
68	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₁₃₇	—	4	4	no	—
71	SD₁₆×C₁₃₇	—	4	8	no	—
72	D₈×C₁₃₇	—	4	8	no	—
73	D₈×C₁₃₇	—	4	8	no	—
74	D₈×C₁₃₇	—	4	4	no	—
75	D₈×C₁₃₇	—	8	16	no	—
76	C₂×C₈×C₁₃₇	D₈	1	8	yes	complemented
77	C₂×C₈×C₁₃₇	C₂×D₄	1	8	yes	—
78	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₁₃₇	—	2	4	no	—
81	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
82	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
84	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
85	C₂³×C₁₃₇	—	4	2	no	—
86	C₂³×C₁₃₇	—	4	4	no	—
87	C₂³×C₁₃₇	—	8	8	no	—
88	C₂³×C₁₃₇	—	8	4	no	—
89	C₂³×C₁₃₇	#127	1	1	yes	characteristic
90	Q₈×C₁₃₇	—	4	4	no	—
91	D₄×C₁₃₇	—	4	8	no	—
92	D₄×C₁₃₇	—	4	4	no	—
93	D₄×C₁₃₇	—	4	8	no	—
94	D₄×C₁₃₇	—	4	16	no	—
95	D₄×C₁₃₇	—	8	8	no	—
96	D₄×C₁₃₇	—	8	4	no	—
97	D₄×C₁₃₇	—	8	4	no	—
98	C₂×C₄×C₁₃₇	—	2	4	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	8	4	no	—
101	C₂×C₄×C₁₃₇	#127	1	2	yes	—
102	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
106	C₈×C₁₃₇	—	2	8	no	—
107	C₈×C₁₃₇	C₂×D₈	1	4	yes	—
108	C₈×C₁₃₇	C₂×D₈	1	4	yes	—
109	C₂²×C₁₃₇	—	2	2	no	—
110	C₂²×C₁₃₇	—	8	2	no	—
111	C₂²×C₁₃₇	—	8	4	no	—
112	C₂²×C₁₃₇	—	8	8	no	—
113	C₂²×C₁₃₇	—	8	4	no	—
114	C₂²×C₁₃₇	—	16	8	no	—
115	C₂²×C₁₃₇	#493	1	2	yes	—
116	C₂²×C₁₃₇	C₂×#127	1	1	yes	characteristic
117	C₄×C₁₃₇	—	2	2	no	—
118	C₄×C₁₃₇	—	8	2	no	—
119	C₄×C₁₃₇	#493	1	2	yes	—
120	C₄×C₁₃₇	C₂²×D₈	1	2	yes	—
121	C₄×C₁₃₇	#576	1	4	yes	—
122	C₂×C₁₃₇	—	2	2	no	—
123	C₂×C₁₃₇	—	16	2	no	—
124	C₂×C₁₃₇	—	16	4	no	—
125	C₂×C₁₃₇	C₂×#493	1	1	yes	characteristic
126	C₂×C₁₃₇	#3155	1	2	yes	—
127	#19661	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₁₃₇	#19661	1	1	yes	characteristic, direct product factor, Sylow
129	#3205	C₂×C₁₃₇	1	4	yes	complemented
130	#3147	C₂×C₁₃₇	1	1	yes	characteristic, complemented
131	C₂×#493	C₂×C₁₃₇	1	1	yes	characteristic, complemented
132	#1721	C₂×C₁₃₇	1	4	yes	complemented
133	#1712	C₂×C₁₃₇	1	2	yes	complemented
134	#1709	C₂×C₁₃₇	1	2	yes	complemented
135	#1450	C₂×C₁₃₇	1	1	yes	characteristic, complemented
136	C₂×#135	—	2	2	no	—
137	C₂²×D₈	—	2	2	no	—
138	#532	C₂²×C₁₃₇	1	2	yes	—
139	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
140	#495	—	2	2	no	—
141	#494	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
142	#493	—	2	2	no	—
143	#493	C₂²×C₁₃₇	1	2	yes	—
144	#493	C₂²×C₁₃₇	1	2	yes	—
145	#493	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
146	#493	C₂²×C₁₃₇	1	8	yes	complemented
147	#492	C₂²×C₁₃₇	1	2	yes	complemented
148	#486	C₂²×C₁₃₇	1	8	yes	complemented
149	#459	—	2	8	no	—
150	#447	—	2	8	no	—
151	#431	C₂²×C₁₃₇	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
153	#322	C₂²×C₁₃₇	1	2	yes	complemented
154	C₈²	C₂²×C₁₃₇	1	2	yes	complemented
155	C₂×#41	—	2	2	no	—
156	C₂²×D₄	—	2	2	no	—
157	C₂²×D₄	—	2	4	no	—
158	#135	—	4	8	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	16	no	—
165	C₂×D₈	—	4	8	no	—
166	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
167	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
168	#127	—	2	4	no	—
169	#127	—	2	2	no	—
170	#127	C₂³×C₁₃₇	1	4	yes	—
171	#127	C₂³×C₁₃₇	1	2	yes	—
172	#124	—	2	2	no	—
173	#121	—	2	4	no	—
174	C₄×D₄	—	2	4	no	—
175	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
176	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	—
177	#102	—	2	16	no	—
178	C₂²⋊C₈	D₄×C₁₃₇	1	4	yes	—
179	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
180	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
181	C₄×C₈	C₂³×C₁₃₇	1	4	yes	—
182	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
183	C₂⁴	—	4	2	no	—
184	#41	—	4	8	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	—	8	8	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
198	Q₁₆	—	4	4	no	—
199	SD₁₆	—	4	8	no	—
200	D₈	—	4	8	no	—
201	D₈	—	4	8	no	—
202	D₈	—	4	4	no	—
203	D₈	—	8	16	no	—
204	C₂×C₈	D₈×C₁₃₇	1	8	yes	complemented
205	C₂×C₈	C₂×D₄×C₁₃₇	1	8	yes	—
206	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
212	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
213	C₂³	—	4	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	8	no	—
216	C₂³	—	8	4	no	—
217	C₂³	#127×C₁₃₇	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	D₄	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	16	no	—
223	D₄	—	8	8	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#127×C₁₃₇	1	2	yes	—
230	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
231	C₂×C₄	C₂×D₈×C₁₃₇	1	4	yes	—
232	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
234	C₈	—	2	8	no	—
235	C₈	C₂×D₈×C₁₃₇	1	4	yes	—
236	C₈	C₂×D₈×C₁₃₇	1	4	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	8	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	16	8	no	—
243	C₂²	#493×C₁₃₇	1	2	yes	—
244	C₂²	C₂×#127×C₁₃₇	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	2	no	—
247	C₄	#493×C₁₃₇	1	2	yes	—
248	C₄	C₂²×D₈×C₁₃₇	1	2	yes	—
249	C₄	#576×C₁₃₇	1	4	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	16	4	no	—
252	C₂	—	16	2	no	—
253	C₂	C₂×#493×C₁₃₇	1	1	yes	characteristic
254	C₂	#3155×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ 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. #3205×C₁₃₇	C₂	semidirect
2. #3147×C₁₃₇	C₂	semidirect
3. C₂×#493×C₁₃₇	C₂	semidirect
4. #1721×C₁₃₇	C₂	semidirect
5. #1712×C₁₃₇	C₂	semidirect
6. #1709×C₁₃₇	C₂	semidirect
7. #1450×C₁₃₇	C₂	semidirect
13. #494×C₁₃₇	C₂²	semidirect
17. #493×C₁₃₇	C₂²	semidirect
18. #493×C₁₃₇	C₂²	semidirect
19. #492×C₁₃₇	C₂²	semidirect
20. #486×C₁₃₇	C₂²	semidirect
25. #322×C₁₃₇	C₂²	semidirect
26. C₈²×C₁₃₇	C₂²	semidirect
76. C₂×C₈×C₁₃₇	D₈	semidirect
127. #19661	C₁₃₇	direct (N × H)
128. C₁₃₇	#19661	direct (N × H)
129. #3205	C₂×C₁₃₇	semidirect
130. #3147	C₂×C₁₃₇	semidirect
131. C₂×#493	C₂×C₁₃₇	semidirect
132. #1721	C₂×C₁₃₇	semidirect
133. #1712	C₂×C₁₃₇	semidirect
134. #1709	C₂×C₁₃₇	semidirect
135. #1450	C₂×C₁₃₇	semidirect
141. #494	C₂²×C₁₃₇	semidirect
145. #493	C₂²×C₁₃₇	semidirect
146. #493	C₂²×C₁₃₇	semidirect
147. #492	C₂²×C₁₃₇	semidirect
148. #486	C₂²×C₁₃₇	semidirect
153. #322	C₂²×C₁₃₇	semidirect
154. C₈²	C₂²×C₁₃₇	semidirect
204. C₂×C₈	D₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3205×C₁₃₇	2
#3147×C₁₃₇	2
C₂×#493×C₁₃₇	2
#1721×C₁₃₇	2
#1712×C₁₃₇	2
#1709×C₁₃₇	2
#1450×C₁₃₇	2
#19661	137

Subgroups — C137×#19661 #46869047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19661 #46869047