Subgroups — C₁₃₇×#20820 #46870206

Order 35072.

810 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2314×C₁₃₇	C₂	1	2	yes	maximal
3	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1836×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1809×C₁₃₇	C₂	1	2	yes	maximal
7	#1807×C₁₃₇	C₂	1	2	yes	maximal
8	#1776×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×#106×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	#405×C₁₃₇	C₄	1	8	yes	complemented
18	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	#388×C₁₃₇	—	2	4	no	—
23	#387×C₁₃₇	—	2	4	no	—
24	#384×C₁₃₇	C₂²	1	2	yes	—
25	#382×C₁₃₇	C₂²	1	2	yes	—
26	#380×C₁₃₇	C₂²	1	4	yes	—
27	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₃₇	—	2	2	no	—
32	C₂×M₁₆×C₁₃₇	—	2	2	no	—
33	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃₇	—	2	2	no	—
35	C₂²×C₈×C₁₃₇	—	2	2	no	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
41	#117×C₁₃₇	C₂×C₄	1	4	yes	—
42	#117×C₁₃₇	Q₈	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃₇	—	2	4	no	—
50	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#107×C₁₃₇	—	2	4	no	—
53	#106×C₁₃₇	—	2	4	no	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	4	no	—
56	#97×C₁₃₇	C₂×C₄	1	4	yes	—
57	#97×C₁₃₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
60	#95×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	4	no	—
62	#95×C₁₃₇	—	2	4	no	—
63	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	—	2	1	no	—
65	C₂²×C₄×C₁₃₇	—	2	1	no	—
66	C₂²×C₄×C₁₃₇	—	2	1	no	—
67	C₂²×C₄×C₁₃₇	—	2	1	no	—
68	C₂²×C₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₁₃₇	—	2	8	no	—
77	C₂×C₈×C₁₃₇	—	2	1	no	—
78	C₂×C₈×C₁₃₇	—	2	1	no	—
79	C₂×C₈×C₁₃₇	—	2	1	no	—
80	C₂×C₈×C₁₃₇	—	2	1	no	—
81	C₂×C₈×C₁₃₇	—	2	8	no	—
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃₇	#41	1	2	yes	—
87	C₂×C₈×C₁₃₇	#41	1	2	yes	—
88	C₂×C₈×C₁₃₇	#41	1	2	yes	—
89	C₂×C₈×C₁₃₇	#41	1	2	yes	—
90	C₄⋊C₄×C₁₃₇	—	2	4	no	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	—	2	4	no	—
94	C₄⋊C₄×C₁₃₇	—	8	4	no	—
95	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
96	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
97	C₂²⋊C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
98	C₄²×C₁₃₇	—	8	4	no	—
99	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
103	C₂³×C₁₃₇	—	2	1	no	—
104	C₂³×C₁₃₇	—	2	1	no	—
105	C₂³×C₁₃₇	—	2	4	no	—
106	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	—	2	1	no	—
109	C₂×C₄×C₁₃₇	—	2	1	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	2	4	no	—
112	C₂×C₄×C₁₃₇	—	2	2	no	—
113	C₂×C₄×C₁₃₇	—	2	2	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	—	4	4	no	—
119	C₂×C₄×C₁₃₇	—	4	4	no	—
120	C₂×C₄×C₁₃₇	—	4	4	no	—
121	C₂×C₄×C₁₃₇	—	4	4	no	—
122	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
131	C₈×C₁₃₇	—	2	2	no	—
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	#117	1	4	yes	—
134	C₈×C₁₃₇	#117	1	4	yes	—
135	C₂²×C₁₃₇	—	2	4	no	—
136	C₂²×C₁₃₇	—	2	2	no	—
137	C₂²×C₁₃₇	—	4	2	no	—
138	C₂²×C₁₃₇	—	4	2	no	—
139	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₁₃₇	#532	1	1	yes	characteristic
141	C₂²×C₁₃₇	#532	1	1	yes	characteristic
142	C₂²×C₁₃₇	#577	1	4	yes	—
143	C₄×C₁₃₇	—	2	4	no	—
144	C₄×C₁₃₇	—	2	4	no	—
145	C₄×C₁₃₇	—	2	2	no	—
146	C₄×C₁₃₇	—	8	4	no	—
147	C₄×C₁₃₇	—	8	4	no	—
148	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
149	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
150	C₄×C₁₃₇	#533	1	1	yes	characteristic
151	C₄×C₁₃₇	#533	1	1	yes	characteristic
152	C₂×C₁₃₇	—	4	2	no	—
153	C₂×C₁₃₇	#2285	1	1	yes	characteristic
154	C₂×C₁₃₇	#2912	1	2	yes	—
155	C₂×C₁₃₇	#3173	1	2	yes	—
156	C₂×C₁₃₇	#3174	1	2	yes	—
157	#20820	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₇	#20820	1	1	yes	characteristic, direct product factor, Sylow
159	C₂×#405	C₂×C₁₃₇	1	1	yes	characteristic
160	#2314	C₂×C₁₃₇	1	2	yes	—
161	#1846	C₂×C₁₃₇	1	1	yes	characteristic, complemented
162	#1844	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	#1836	C₂×C₁₃₇	1	2	yes	complemented
164	#1809	C₂×C₁₃₇	1	2	yes	—
165	#1807	C₂×C₁₃₇	1	2	yes	—
166	#1776	C₂×C₁₃₇	1	2	yes	complemented
167	C₄×#106	C₂×C₁₃₇	1	2	yes	complemented
168	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
169	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
172	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
173	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
174	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
175	#405	C₄×C₁₃₇	1	8	yes	complemented
176	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
177	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
180	#388	—	2	4	no	—
181	#387	—	2	4	no	—
182	#384	C₂²×C₁₃₇	1	2	yes	—
183	#382	C₂²×C₁₃₇	1	2	yes	—
184	#380	C₂²×C₁₃₇	1	4	yes	—
185	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
186	#340	C₂²×C₁₃₇	1	4	yes	—
187	#340	C₂²×C₁₃₇	1	4	yes	—
188	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
195	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
199	#117	C₂×C₄×C₁₃₇	1	4	yes	—
200	#117	Q₈×C₁₃₇	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
205	C₂×C₂²⋊C₄	—	2	4	no	—
206	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	#106	—	2	4	no	—
213	#106	—	2	4	no	—
214	#97	C₂×C₄×C₁₃₇	1	4	yes	—
215	#97	C₂×C₄×C₁₃₇	1	4	yes	—
216	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
217	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
218	#95	—	2	4	no	—
219	#95	—	2	4	no	—
220	#95	—	2	4	no	—
221	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
234	M₁₆	—	2	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
244	C₂×C₈	#41×C₁₃₇	1	2	yes	—
245	C₂×C₈	#41×C₁₃₇	1	2	yes	—
246	C₂×C₈	#41×C₁₃₇	1	2	yes	—
247	C₂×C₈	#41×C₁₃₇	1	2	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	8	4	no	—
253	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
254	C₂²⋊C₄	—	8	4	no	—
255	C₂²⋊C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
256	C₄²	—	8	4	no	—
257	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
258	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
259	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
260	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
265	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
281	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
282	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
283	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	2	no	—
291	C₈	#117×C₁₃₇	1	4	yes	—
292	C₈	#117×C₁₃₇	1	4	yes	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	4	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
298	C₂²	#532×C₁₃₇	1	1	yes	characteristic
299	C₂²	#532×C₁₃₇	1	1	yes	characteristic
300	C₂²	#577×C₁₃₇	1	4	yes	—
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	8	4	no	—
305	C₄	—	8	4	no	—
306	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
307	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
308	C₄	#533×C₁₃₇	1	1	yes	characteristic
309	C₄	#533×C₁₃₇	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	#2285×C₁₃₇	1	1	yes	characteristic
312	C₂	#2912×C₁₃₇	1	2	yes	—
313	C₂	#3173×C₁₃₇	1	2	yes	—
314	C₂	#3174×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1846×C₁₃₇	C₂	semidirect
4. #1844×C₁₃₇	C₂	semidirect
5. #1836×C₁₃₇	C₂	semidirect
8. #1776×C₁₃₇	C₂	semidirect
9. C₄×#106×C₁₃₇	C₂	semidirect
17. #405×C₁₃₇	C₄	semidirect
157. #20820	C₁₃₇	direct (N × H)
158. C₁₃₇	#20820	direct (N × H)
161. #1846	C₂×C₁₃₇	semidirect
162. #1844	C₂×C₁₃₇	semidirect
163. #1836	C₂×C₁₃₇	semidirect
166. #1776	C₂×C₁₃₇	semidirect
167. C₄×#106	C₂×C₁₃₇	semidirect
175. #405	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#405×C₁₃₇	2
#2314×C₁₃₇	2
#1846×C₁₃₇	2
#1844×C₁₃₇	2
#1836×C₁₃₇	2
#1809×C₁₃₇	2
#1807×C₁₃₇	2
#1776×C₁₃₇	2
C₄×#106×C₁₃₇	2
#20820	137

Subgroups — C137×#20820 #46870206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20820 #46870206