Subgroups — C₁₃₇×#20463 #46869849

Order 35072.

1222 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄³×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1767×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	#1749×C₁₃₇	C₂	1	4	yes	maximal
5	#1744×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₄×#97×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄²×C₁₃₇	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
11	#431×C₁₃₇	C₄	1	16	yes	complemented
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	4	yes	—
14	C₂×#97×C₁₃₇	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
16	C₄³×C₁₃₇	C₄	1	2	yes	—
17	C₄³×C₁₃₇	C₄	1	2	yes	—
18	C₄³×C₁₃₇	C₂²	1	2	yes	—
19	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
21	#336×C₁₃₇	C₂²	1	8	yes	—
22	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₇	C₂³	1	8	yes	—
27	C₂×C₄²×C₁₃₇	—	2	8	no	—
28	C₂×C₄²×C₁₃₇	—	2	8	no	—
29	C₂×C₄²×C₁₃₇	—	2	8	no	—
30	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
45	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	16	yes	—
46	#97×C₁₃₇	—	2	16	no	—
47	C₄×C₈×C₁₃₇	C₂×C₄	1	16	yes	complemented
48	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₇	—	2	8	no	—
50	C₂²×C₄×C₁₃₇	—	2	8	no	—
51	C₂²×C₄×C₁₃₇	—	2	8	no	—
52	C₂²×C₄×C₁₃₇	—	2	4	no	—
53	C₂²×C₄×C₁₃₇	—	2	4	no	—
54	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
55	C₂²×C₄×C₁₃₇	C₄²	1	4	yes	—
56	C₂²×C₄×C₁₃₇	M₁₆	1	8	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₃₇	—	2	16	no	—
67	C₂×C₈×C₁₃₇	C₂²×C₄	1	16	yes	—
68	C₄²×C₁₃₇	—	2	8	no	—
69	C₄²×C₁₃₇	—	2	8	no	—
70	C₄²×C₁₃₇	—	2	4	no	—
71	C₄²×C₁₃₇	—	2	8	no	—
72	C₄²×C₁₃₇	—	2	4	no	—
73	C₄²×C₁₃₇	—	2	4	no	—
74	C₄²×C₁₃₇	—	2	4	no	—
75	C₄²×C₁₃₇	C₄²	1	4	yes	—
76	C₄²×C₁₃₇	C₄²	1	4	yes	—
77	C₄²×C₁₃₇	M₁₆	1	8	yes	—
78	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
79	C₄²×C₁₃₇	C₂²×C₄	1	4	yes	—
80	C₄²×C₁₃₇	#41	1	8	yes	—
81	C₂³×C₁₃₇	—	2	2	no	—
82	C₂³×C₁₃₇	—	2	2	no	—
83	C₂³×C₁₃₇	C₂×C₄²	1	2	yes	—
84	C₂³×C₁₃₇	C₂×M₁₆	1	4	yes	—
85	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	—	2	32	no	—
87	C₂×C₄×C₁₃₇	—	2	4	no	—
88	C₂×C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄×C₁₃₇	—	2	4	no	—
91	C₂×C₄×C₁₃₇	—	2	4	no	—
92	C₂×C₄×C₁₃₇	—	2	4	no	—
93	C₂×C₄×C₁₃₇	—	2	8	no	—
94	C₂×C₄×C₁₃₇	—	2	8	no	—
95	C₂×C₄×C₁₃₇	#97	1	16	yes	complemented
96	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
97	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
99	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₁₃₇	#117	1	8	yes	—
102	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
103	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₁₃₇	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₁₃₇	C₂³×C₄	1	2	yes	—
107	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
108	C₈×C₁₃₇	—	2	16	no	—
109	C₂²×C₁₃₇	—	2	2	no	—
110	C₂²×C₁₃₇	—	2	8	no	—
111	C₂²×C₁₃₇	—	2	2	no	—
112	C₂²×C₁₃₇	C₂×#97	1	4	yes	—
113	C₂²×C₁₃₇	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
115	C₂²×C₁₃₇	C₂²×M₁₆	1	2	yes	—
116	C₂²×C₁₃₇	C₂²×M₁₆	1	2	yes	—
117	C₄×C₁₃₇	—	2	2	no	—
118	C₄×C₁₃₇	—	2	8	no	—
119	C₄×C₁₃₇	—	2	8	no	—
120	C₄×C₁₃₇	—	2	2	no	—
121	C₄×C₁₃₇	C₂×#97	1	4	yes	—
122	C₄×C₁₃₇	C₂×#97	1	4	yes	—
123	C₄×C₁₃₇	C₄×M₁₆	1	4	yes	—
124	C₄×C₁₃₇	C₄×#41	1	4	yes	—
125	C₂×C₁₃₇	—	2	4	no	—
126	C₂×C₁₃₇	C₄×#117	1	1	yes	characteristic
127	C₂×C₁₃₇	C₂²×#97	1	1	yes	characteristic
128	C₂×C₁₃₇	C₂×C₄×M₁₆	1	1	yes	characteristic
129	C₂×C₁₃₇	#2920	1	4	yes	—
130	#20463	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₁₃₇	#20463	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#431	C₂×C₁₃₇	1	2	yes	—
133	C₂×C₄³	C₂×C₁₃₇	1	1	yes	characteristic
134	#1767	C₂×C₁₃₇	1	4	yes	complemented
135	#1749	C₂×C₁₃₇	1	4	yes	—
136	#1744	C₂×C₁₃₇	1	2	yes	complemented
137	C₄×#97	C₂×C₁₃₇	1	2	yes	complemented
138	C₂²×C₄²	C₄×C₁₃₇	1	2	yes	—
139	C₂²×C₄²	C₄×C₁₃₇	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃₇	1	2	yes	—
141	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
142	#431	C₄×C₁₃₇	1	16	yes	complemented
143	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
144	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	4	yes	—
145	C₂×#97	C₂²×C₁₃₇	1	8	yes	—
146	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
147	C₄³	C₄×C₁₃₇	1	2	yes	—
148	C₄³	C₄×C₁₃₇	1	2	yes	—
149	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
151	C₄³	C₂²×C₁₃₇	1	2	yes	—
152	#336	C₂²×C₁₃₇	1	8	yes	—
153	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
154	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
155	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
156	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₁₃₇	1	8	yes	—
158	C₂×C₄²	—	2	8	no	—
159	C₂×C₄²	—	2	8	no	—
160	C₂×C₄²	—	2	8	no	—
161	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
162	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
164	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
165	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
166	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₁₃₇	1	8	yes	—
170	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
171	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
175	C₄⋊C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
176	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	16	yes	—
177	#97	—	2	16	no	—
178	C₄×C₈	C₂×C₄×C₁₃₇	1	16	yes	complemented
179	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
186	C₂²×C₄	C₄²×C₁₃₇	1	4	yes	—
187	C₂²×C₄	M₁₆×C₁₃₇	1	8	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
191	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
197	C₂×C₈	—	2	16	no	—
198	C₂×C₈	C₂²×C₄×C₁₃₇	1	16	yes	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	8	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	8	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	2	4	no	—
206	C₄²	C₄²×C₁₃₇	1	4	yes	—
207	C₄²	C₄²×C₁₃₇	1	4	yes	—
208	C₄²	M₁₆×C₁₃₇	1	8	yes	—
209	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
210	C₄²	C₂²×C₄×C₁₃₇	1	4	yes	—
211	C₄²	#41×C₁₃₇	1	8	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	C₂×C₄²×C₁₃₇	1	2	yes	—
215	C₂³	C₂×M₁₆×C₁₃₇	1	4	yes	—
216	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
217	C₂×C₄	—	2	32	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	#97×C₁₃₇	1	16	yes	complemented
227	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
228	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
229	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
230	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
231	C₂×C₄	C₂×C₄²×C₁₃₇	1	2	yes	—
232	C₂×C₄	#117×C₁₃₇	1	8	yes	—
233	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
234	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
235	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
236	C₂×C₄	C₂×M₁₆×C₁₃₇	1	4	yes	—
237	C₂×C₄	C₂³×C₄×C₁₃₇	1	2	yes	—
238	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
239	C₈	—	2	16	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	C₂×#97×C₁₃₇	1	4	yes	—
244	C₂²	C₂²×C₄²×C₁₃₇	1	1	yes	characteristic
245	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
246	C₂²	C₂²×M₁₆×C₁₃₇	1	2	yes	—
247	C₂²	C₂²×M₁₆×C₁₃₇	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	8	no	—
252	C₄	C₂×#97×C₁₃₇	1	4	yes	—
253	C₄	C₂×#97×C₁₃₇	1	4	yes	—
254	C₄	C₄×M₁₆×C₁₃₇	1	4	yes	—
255	C₄	C₄×#41×C₁₃₇	1	4	yes	—
256	C₂	—	2	4	no	—
257	C₂	C₄×#117×C₁₃₇	1	1	yes	characteristic
258	C₂	C₂²×#97×C₁₃₇	1	1	yes	characteristic
259	C₂	C₂×C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
260	C₂	#2920×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂⁴×C₁₃₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1767×C₁₃₇	C₂	semidirect
5. #1744×C₁₃₇	C₂	semidirect
6. C₄×#97×C₁₃₇	C₂	semidirect
11. #431×C₁₃₇	C₄	semidirect
44. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
47. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
95. C₂×C₄×C₁₃₇	#97	semidirect
130. #20463	C₁₃₇	direct (N × H)
131. C₁₃₇	#20463	direct (N × H)
134. #1767	C₂×C₁₃₇	semidirect
136. #1744	C₂×C₁₃₇	semidirect
137. C₄×#97	C₂×C₁₃₇	semidirect
142. #431	C₄×C₁₃₇	semidirect
175. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
178. C₄×C₈	C₂×C₄×C₁₃₇	semidirect
226. C₂×C₄	#97×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₃₇	2
C₂×C₄³×C₁₃₇	2
#1767×C₁₃₇	2
#1749×C₁₃₇	2
#1744×C₁₃₇	2
C₄×#97×C₁₃₇	2
#20463	137

Subgroups — C137×#20463 #46869849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20463 #46869849