Subgroups — C₁₃₇×#20787 #46870173

Order 35072.

616 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1841×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1833×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1827×C₁₃₇	C₂	1	2	yes	maximal
5	#1813×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1812×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1772×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1757×C₁₃₇	C₂	1	2	yes	maximal
9	#1747×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#131×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
14	#427×C₁₃₇	C₂²	1	2	yes	—
15	#413×C₁₃₇	C₂²	1	2	yes	—
16	#410×C₁₃₇	C₂²	1	2	yes	—
17	#405×C₁₃₇	C₄	1	2	yes	complemented
18	#405×C₁₃₇	C₄	1	2	yes	complemented
19	#405×C₁₃₇	C₂²	1	4	yes	—
20	#405×C₁₃₇	C₂²	1	1	yes	characteristic
21	#405×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₁₃₇	C₄	1	2	yes	complemented
23	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₇	C₄	1	2	yes	complemented
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#344×C₁₃₇	C₂²	1	4	yes	—
27	#339×C₁₃₇	C₂²	1	4	yes	—
28	#338×C₁₃₇	C₂²	1	2	yes	—
29	H(3,ℤ/4)×C₁₃₇	C₂²	1	2	yes	—
30	H(3,ℤ/4)×C₁₃₇	C₂²	1	4	yes	—
31	C₂×#41×C₁₃₇	C₂³	1	1	yes	characteristic
32	#131×C₁₃₇	—	2	2	no	—
33	#131×C₁₃₇	—	2	2	no	—
34	#131×C₁₃₇	—	2	2	no	—
35	#131×C₁₃₇	—	2	2	no	—
36	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
39	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₁₃₇	—	2	1	no	—
46	C₄×D₄×C₁₃₇	—	2	2	no	—
47	C₄×D₄×C₁₃₇	—	2	1	no	—
48	#117×C₁₃₇	C₂×C₄	1	2	yes	—
49	#117×C₁₃₇	C₂³	1	4	yes	—
50	#117×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	#108×C₁₃₇	—	2	2	no	—
54	#108×C₁₃₇	—	2	2	no	—
55	#107×C₁₃₇	—	2	2	no	—
56	#106×C₁₃₇	—	2	2	no	—
57	#101×C₁₃₇	—	2	2	no	—
58	#100×C₁₃₇	—	2	2	no	—
59	#99×C₁₃₇	—	2	4	no	—
60	#97×C₁₃₇	—	2	4	no	—
61	#97×C₁₃₇	C₂×C₄	1	2	yes	—
62	#97×C₁₃₇	C₂×C₄	1	2	yes	—
63	#97×C₁₃₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₁₃₇	—	2	4	no	—
65	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
72	#41×C₁₃₇	—	2	2	no	—
73	#41×C₁₃₇	—	2	2	no	—
74	C₂×Q₈×C₁₃₇	#41	1	1	yes	characteristic
75	C₂×D₄×C₁₃₇	#41	1	2	yes	—
76	C₂×D₄×C₁₃₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	—	2	2	no	—
78	C₂²×C₄×C₁₃₇	—	2	1	no	—
79	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₁₃₇	—	2	2	no	—
82	M₁₆×C₁₃₇	—	2	2	no	—
83	M₁₆×C₁₃₇	—	2	4	no	—
84	M₁₆×C₁₃₇	—	2	4	no	—
85	M₁₆×C₁₃₇	—	2	4	no	—
86	M₁₆×C₁₃₇	—	2	4	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	2	2	no	—
89	C₂×C₈×C₁₃₇	—	2	2	no	—
90	C₂×C₈×C₁₃₇	—	2	2	no	—
91	C₂×C₈×C₁₃₇	—	2	4	no	—
92	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₇	#41	1	2	yes	—
99	C₂×C₈×C₁₃₇	#41	1	2	yes	—
100	C₂×C₈×C₁₃₇	#41	1	2	yes	—
101	C₂×C₈×C₁₃₇	#41	1	2	yes	—
102	C₄⋊C₄×C₁₃₇	—	2	2	no	—
103	C₄⋊C₄×C₁₃₇	—	2	1	no	—
104	C₄⋊C₄×C₁₃₇	—	2	4	no	—
105	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
106	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
107	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
108	C₄²×C₁₃₇	—	2	2	no	—
109	C₄²×C₁₃₇	—	2	1	no	—
110	C₄²×C₁₃₇	—	2	4	no	—
111	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
114	C₂³×C₁₃₇	#117	1	2	yes	—
115	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
116	Q₈×C₁₃₇	—	2	2	no	—
117	D₄×C₁₃₇	—	2	4	no	—
118	D₄×C₁₃₇	—	2	2	no	—
119	C₂×C₄×C₁₃₇	—	2	1	no	—
120	C₂×C₄×C₁₃₇	—	2	2	no	—
121	C₂×C₄×C₁₃₇	—	2	2	no	—
122	C₂×C₄×C₁₃₇	—	2	2	no	—
123	C₂×C₄×C₁₃₇	—	2	1	no	—
124	C₂×C₄×C₁₃₇	—	4	2	no	—
125	C₂×C₄×C₁₃₇	—	4	4	no	—
126	C₂×C₄×C₁₃₇	—	4	4	no	—
127	C₂×C₄×C₁₃₇	#117	1	2	yes	—
128	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
132	C₈×C₁₃₇	—	2	2	no	—
133	C₈×C₁₃₇	—	2	2	no	—
134	C₈×C₁₃₇	—	4	4	no	—
135	C₈×C₁₃₇	#117	1	1	yes	characteristic
136	C₈×C₁₃₇	#117	1	1	yes	characteristic
137	C₈×C₁₃₇	#117	1	1	yes	characteristic
138	C₈×C₁₃₇	#117	1	2	yes	—
139	C₈×C₁₃₇	#117	1	2	yes	—
140	C₈×C₁₃₇	#117	1	1	yes	characteristic
141	C₂²×C₁₃₇	—	2	2	no	—
142	C₂²×C₁₃₇	—	4	1	no	—
143	C₂²×C₁₃₇	—	4	2	no	—
144	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
145	C₂²×C₁₃₇	#529	1	2	yes	—
146	C₄×C₁₃₇	—	2	2	no	—
147	C₄×C₁₃₇	—	2	2	no	—
148	C₄×C₁₃₇	—	4	2	no	—
149	C₄×C₁₃₇	—	4	1	no	—
150	C₄×C₁₃₇	—	8	4	no	—
151	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
152	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
153	C₄×C₁₃₇	#529	1	2	yes	—
154	C₂×C₁₃₇	—	2	1	no	—
155	C₂×C₁₃₇	—	2	2	no	—
156	C₂×C₁₃₇	—	4	2	no	—
157	C₂×C₁₃₇	#2283	1	1	yes	characteristic
158	#20787	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₁₃₇	#20787	1	1	yes	characteristic, direct product factor, Sylow
160	C₄×#131	C₂×C₁₃₇	1	1	yes	characteristic
161	#1841	C₂×C₁₃₇	1	2	yes	complemented
162	#1833	C₂×C₁₃₇	1	2	yes	complemented
163	#1827	C₂×C₁₃₇	1	2	yes	—
164	#1813	C₂×C₁₃₇	1	1	yes	characteristic
165	#1812	C₂×C₁₃₇	1	1	yes	characteristic
166	#1772	C₂×C₁₃₇	1	2	yes	complemented
167	#1757	C₂×C₁₃₇	1	2	yes	—
168	#1747	C₂×C₁₃₇	1	2	yes	complemented
169	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×#131	C₂²×C₁₃₇	1	1	yes	characteristic
171	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
173	#427	C₂²×C₁₃₇	1	2	yes	—
174	#413	C₂²×C₁₃₇	1	2	yes	—
175	#410	C₂²×C₁₃₇	1	2	yes	—
176	#405	C₄×C₁₃₇	1	2	yes	complemented
177	#405	C₄×C₁₃₇	1	2	yes	complemented
178	#405	C₂²×C₁₃₇	1	4	yes	—
179	#405	C₂²×C₁₃₇	1	1	yes	characteristic
180	#405	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₄×M₁₆	C₄×C₁₃₇	1	2	yes	complemented
182	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₄×C₁₃₇	1	2	yes	complemented
184	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
185	#344	C₂²×C₁₃₇	1	4	yes	—
186	#339	C₂²×C₁₃₇	1	4	yes	—
187	#338	C₂²×C₁₃₇	1	2	yes	—
188	H(3,ℤ/4)	C₂²×C₁₃₇	1	2	yes	—
189	H(3,ℤ/4)	C₂²×C₁₃₇	1	4	yes	—
190	C₂×#41	C₂³×C₁₃₇	1	1	yes	characteristic
191	#131	—	2	2	no	—
192	#131	—	2	2	no	—
193	#131	—	2	2	no	—
194	#131	—	2	2	no	—
195	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
196	C₂×M₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
197	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
198	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
199	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
200	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
201	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
204	C₄×Q₈	—	2	1	no	—
205	C₄×D₄	—	2	2	no	—
206	C₄×D₄	—	2	1	no	—
207	#117	C₂×C₄×C₁₃₇	1	2	yes	—
208	#117	C₂³×C₁₃₇	1	4	yes	—
209	#117	C₂³×C₁₃₇	1	1	yes	characteristic
210	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
211	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
212	#108	—	2	2	no	—
213	#108	—	2	2	no	—
214	#107	—	2	2	no	—
215	#106	—	2	2	no	—
216	#101	—	2	2	no	—
217	#100	—	2	2	no	—
218	#99	—	2	4	no	—
219	#97	—	2	4	no	—
220	#97	C₂×C₄×C₁₃₇	1	2	yes	—
221	#97	C₂×C₄×C₁₃₇	1	2	yes	—
222	#97	C₂×C₄×C₁₃₇	1	2	yes	—
223	C₄×C₈	—	2	4	no	—
224	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
225	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
226	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
227	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
228	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
229	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
230	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	C₂×Q₈	#41×C₁₃₇	1	1	yes	characteristic
234	C₂×D₄	#41×C₁₃₇	1	2	yes	—
235	C₂×D₄	#41×C₁₃₇	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
240	M₁₆	—	2	2	no	—
241	M₁₆	—	2	2	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	2	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
252	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
253	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
255	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
257	C₂×C₈	#41×C₁₃₇	1	2	yes	—
258	C₂×C₈	#41×C₁₃₇	1	2	yes	—
259	C₂×C₈	#41×C₁₃₇	1	2	yes	—
260	C₂×C₈	#41×C₁₃₇	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	4	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	4	no	—
270	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
271	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
272	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
273	C₂³	#117×C₁₃₇	1	2	yes	—
274	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
275	Q₈	—	2	2	no	—
276	D₄	—	2	4	no	—
277	D₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	#117×C₁₃₇	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
288	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
289	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
290	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	4	no	—
294	C₈	#117×C₁₃₇	1	1	yes	characteristic
295	C₈	#117×C₁₃₇	1	1	yes	characteristic
296	C₈	#117×C₁₃₇	1	1	yes	characteristic
297	C₈	#117×C₁₃₇	1	1	yes	characteristic
298	C₈	#117×C₁₃₇	1	2	yes	—
299	C₈	#117×C₁₃₇	1	2	yes	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
304	C₂²	#529×C₁₃₇	1	2	yes	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	8	4	no	—
310	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
311	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
312	C₄	#529×C₁₃₇	1	2	yes	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	—	4	2	no	—
316	C₂	#2283×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₇	#117
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	#2283
prime radical	C₂×#41×C₁₃₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₇ ◁ #405×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1841×C₁₃₇	C₂	semidirect
3. #1833×C₁₃₇	C₂	semidirect
7. #1772×C₁₃₇	C₂	semidirect
9. #1747×C₁₃₇	C₂	semidirect
17. #405×C₁₃₇	C₄	semidirect
18. #405×C₁₃₇	C₄	semidirect
22. C₄×M₁₆×C₁₃₇	C₄	semidirect
24. C₂×C₄×C₈×C₁₃₇	C₄	semidirect
158. #20787	C₁₃₇	direct (N × H)
159. C₁₃₇	#20787	direct (N × H)
161. #1841	C₂×C₁₃₇	semidirect
162. #1833	C₂×C₁₃₇	semidirect
166. #1772	C₂×C₁₃₇	semidirect
168. #1747	C₂×C₁₃₇	semidirect
176. #405	C₄×C₁₃₇	semidirect
177. #405	C₄×C₁₃₇	semidirect
181. C₄×M₁₆	C₄×C₁₃₇	semidirect
183. C₂×C₄×C₈	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#131×C₁₃₇	2
#1841×C₁₃₇	2
#1833×C₁₃₇	2
#1827×C₁₃₇	2
#1813×C₁₃₇	2
#1812×C₁₃₇	2
#1772×C₁₃₇	2
#1757×C₁₃₇	2
#1747×C₁₃₇	2
#20787	137

Subgroups — C137×#20787 #46870173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20787 #46870173