Subgroups — C₁₃₇×#20815 #46870201

Order 35072.

862 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1842×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1831×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1798×C₁₃₇	C₂	1	4	yes	maximal
7	#1768×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	#432×C₁₃₇	—	2	4	no	—
11	#431×C₁₃₇	—	2	4	no	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	8	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	2	yes	—
23	#376×C₁₃₇	C₂²	1	2	yes	—
24	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
26	#336×C₁₃₇	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
32	#117×C₁₃₇	—	2	8	no	—
33	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
42	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
43	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
47	C₄⋊C₈×C₁₃₇	—	2	4	no	—
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊C₈×C₁₃₇	—	2	16	no	—
50	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
51	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
52	#97×C₁₃₇	—	2	4	no	—
53	C₄×C₈×C₁₃₇	—	2	4	no	—
54	#95×C₁₃₇	C₂×C₄	1	4	yes	—
55	#95×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	4	2	no	—
60	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	—	2	4	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	—	2	8	no	—
75	C₂×C₈×C₁₃₇	—	2	8	no	—
76	C₄⋊C₄×C₁₃₇	—	2	8	no	—
77	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
78	C₄²×C₁₃₇	—	2	4	no	—
79	C₄²×C₁₃₇	—	2	4	no	—
80	C₄²×C₁₃₇	—	2	4	no	—
81	C₄²×C₁₃₇	—	4	2	no	—
82	C₄²×C₁₃₇	—	4	2	no	—
83	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
84	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
88	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
89	C₄²×C₁₃₇	#41	1	8	yes	—
90	C₂³×C₁₃₇	—	2	2	no	—
91	C₂³×C₁₃₇	—	2	2	no	—
92	C₂³×C₁₃₇	—	2	2	no	—
93	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₄×C₁₃₇	—	2	4	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	4	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	4	no	—
103	C₂×C₄×C₁₃₇	—	2	8	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃₇	#117	1	8	yes	—
114	C₂×C₄×C₁₃₇	#131	1	4	yes	—
115	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
118	C₈×C₁₃₇	—	4	4	no	—
119	C₈×C₁₃₇	—	4	4	no	—
120	C₂²×C₁₃₇	—	2	2	no	—
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	2	2	no	—
123	C₂²×C₁₃₇	—	4	4	no	—
124	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₃₇	C₂×#117	1	2	yes	—
126	C₂²×C₁₃₇	C₂×#131	1	2	yes	—
127	C₂²×C₁₃₇	#568	1	2	yes	—
128	C₄×C₁₃₇	—	2	2	no	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	—	2	2	no	—
131	C₄×C₁₃₇	—	2	8	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	#424	1	4	yes	—
135	C₄×C₁₃₇	#532	1	2	yes	—
136	C₄×C₁₃₇	#533	1	2	yes	—
137	C₂×C₁₃₇	—	4	2	no	—
138	C₂×C₁₃₇	#2285	1	1	yes	characteristic
139	C₂×C₁₃₇	C₂×#424	1	1	yes	characteristic
140	C₂×C₁₃₇	#2905	1	1	yes	characteristic
141	C₂×C₁₃₇	#2925	1	4	yes	—
142	#20815	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₇	#20815	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#432	C₂×C₁₃₇	1	1	yes	characteristic
145	C₂×#431	C₂×C₁₃₇	1	1	yes	characteristic
146	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
147	#1842	C₂×C₁₃₇	1	2	yes	complemented
148	#1831	C₂×C₁₃₇	1	2	yes	complemented
149	#1798	C₂×C₁₃₇	1	4	yes	—
150	#1768	C₂×C₁₃₇	1	4	yes	complemented
151	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
152	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
153	#432	—	2	4	no	—
154	#431	—	2	4	no	—
155	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	8	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
159	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
162	C₄×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
163	C₄×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
164	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
165	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	2	yes	—
166	#376	C₂²×C₁₃₇	1	2	yes	—
167	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
169	#336	C₂²×C₁₃₇	1	8	yes	—
170	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
174	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
175	#117	—	2	8	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
182	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
184	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
185	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
186	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
187	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
190	C₄⋊C₈	—	2	4	no	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	16	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	C₂²⋊C₈	—	2	4	no	—
195	#97	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	#95	C₂×C₄×C₁₃₇	1	4	yes	—
198	#95	C₂×C₄×C₁₃₇	1	4	yes	—
199	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
204	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	8	no	—
219	C₄⋊C₄	—	2	8	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
227	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
228	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
231	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
232	C₄²	#41×C₁₃₇	1	8	yes	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
237	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
256	C₂×C₄	#117×C₁₃₇	1	8	yes	—
257	C₂×C₄	#131×C₁₃₇	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	4	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
268	C₂²	C₂×#117×C₁₃₇	1	2	yes	—
269	C₂²	C₂×#131×C₁₃₇	1	2	yes	—
270	C₂²	#568×C₁₃₇	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	8	no	—
276	C₄	—	4	4	no	—
277	C₄	#424×C₁₃₇	1	4	yes	—
278	C₄	#532×C₁₃₇	1	2	yes	—
279	C₄	#533×C₁₃₇	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	#2285×C₁₃₇	1	1	yes	characteristic
282	C₂	C₂×#424×C₁₃₇	1	1	yes	characteristic
283	C₂	#2905×C₁₃₇	1	1	yes	characteristic
284	C₂	#2925×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #1842×C₁₃₇	C₂	semidirect
5. #1831×C₁₃₇	C₂	semidirect
7. #1768×C₁₃₇	C₂	semidirect
142. #20815	C₁₃₇	direct (N × H)
143. C₁₃₇	#20815	direct (N × H)
147. #1842	C₂×C₁₃₇	semidirect
148. #1831	C₂×C₁₃₇	semidirect
150. #1768	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₁₃₇	2
C₂×#431×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1842×C₁₃₇	2
#1831×C₁₃₇	2
#1798×C₁₃₇	2
#1768×C₁₃₇	2
#20815	137

Subgroups — C137×#20815 #46870201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20815 #46870201