Subgroups — C₁₃₇×#28710 #46878096

Order 35072.

774 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2562×C₁₃₇	C₂	1	2	yes	maximal
4	#2077×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1989×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1809×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1809×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	#499×C₁₃₇	—	2	4	no	—
13	#436×C₁₃₇	—	2	4	no	—
14	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#400×C₁₃₇	—	2	2	no	—
22	#394×C₁₃₇	—	2	2	no	—
23	#388×C₁₃₇	—	2	4	no	—
24	#380×C₁₃₇	C₂²	1	4	yes	—
25	#380×C₁₃₇	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
27	#340×C₁₃₇	C₂²	1	2	yes	—
28	#340×C₁₃₇	C₂²	1	4	yes	—
29	#340×C₁₃₇	C₂²	1	4	yes	—
30	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
32	C₂×M₁₆×C₁₃₇	—	2	4	no	—
33	C₂×M₁₆×C₁₃₇	—	2	8	no	—
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	#125×C₁₃₇	—	2	4	no	—
38	#125×C₁₃₇	—	2	4	no	—
39	C₄×D₄×C₁₃₇	—	2	2	no	—
40	C₄×D₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃₇	—	2	2	no	—
51	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
52	#107×C₁₃₇	—	2	2	no	—
53	#107×C₁₃₇	—	2	2	no	—
54	#106×C₁₃₇	—	2	4	no	—
55	#106×C₁₃₇	—	2	4	no	—
56	C₄⋊C₈×C₁₃₇	—	2	2	no	—
57	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
58	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
59	#95×C₁₃₇	—	2	2	no	—
60	#95×C₁₃₇	—	2	4	no	—
61	#95×C₁₃₇	—	2	4	no	—
62	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
63	C₂×D₄×C₁₃₇	—	2	2	no	—
64	C₂×D₄×C₁₃₇	—	4	2	no	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	—	2	4	no	—
69	C₂²×C₄×C₁₃₇	—	2	4	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₁₃₇	—	4	8	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃₇	#41	1	4	yes	—
83	C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₄⋊C₄×C₁₃₇	—	2	2	no	—
86	C₄⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄⋊C₄×C₁₃₇	—	2	2	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	4	no	—
90	C₄⋊C₄×C₁₃₇	—	4	4	no	—
91	C₄⋊C₄×C₁₃₇	—	4	4	no	—
92	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
93	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
94	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
95	C₄²×C₁₃₇	—	8	2	no	—
96	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
97	C₄²×C₁₃₇	#41	1	2	yes	—
98	C₂³×C₁₃₇	—	2	2	no	—
99	C₂³×C₁₃₇	—	2	4	no	—
100	C₂³×C₁₃₇	—	2	4	no	—
101	C₂³×C₁₃₇	—	2	4	no	—
102	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	D₄×C₁₃₇	—	4	4	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	4	no	—
108	C₂×C₄×C₁₃₇	—	4	2	no	—
109	C₂×C₄×C₁₃₇	—	4	2	no	—
110	C₂×C₄×C₁₃₇	—	4	4	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	4	no	—
114	C₂×C₄×C₁₃₇	—	4	4	no	—
115	C₂×C₄×C₁₃₇	—	4	4	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	#121	1	2	yes	—
120	C₂×C₄×C₁₃₇	#121	1	2	yes	—
121	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₁₃₇	—	2	2	no	—
126	C₈×C₁₃₇	—	4	2	no	—
127	C₈×C₁₃₇	#124	1	4	yes	—
128	C₂²×C₁₃₇	—	4	8	no	—
129	C₂²×C₁₃₇	—	4	2	no	—
130	C₂²×C₁₃₇	—	4	2	no	—
131	C₂²×C₁₃₇	—	4	2	no	—
132	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
133	C₂²×C₁₃₇	#547	1	2	yes	—
134	C₂²×C₁₃₇	#575	1	2	yes	—
135	C₂²×C₁₃₇	#577	1	2	yes	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	—	2	2	no	—
138	C₄×C₁₃₇	—	4	2	no	—
139	C₄×C₁₃₇	—	8	4	no	—
140	C₄×C₁₃₇	—	8	2	no	—
141	C₄×C₁₃₇	—	8	2	no	—
142	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
143	C₄×C₁₃₇	#536	1	1	yes	characteristic
144	C₄×C₁₃₇	#556	1	2	yes	—
145	C₂×C₁₃₇	—	8	2	no	—
146	C₂×C₁₃₇	#2547	1	1	yes	characteristic
147	C₂×C₁₃₇	#3054	1	1	yes	characteristic
148	C₂×C₁₃₇	#3060	1	1	yes	characteristic
149	C₂×C₁₃₇	#3351	1	4	yes	—
150	#28710	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₇	#28710	1	1	yes	characteristic, direct product factor, Sylow
152	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
153	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
154	#2562	C₂×C₁₃₇	1	2	yes	—
155	#2077	C₂×C₁₃₇	1	4	yes	complemented
156	#1989	C₂×C₁₃₇	1	2	yes	complemented
157	#1809	C₂×C₁₃₇	1	2	yes	complemented
158	#1809	C₂×C₁₃₇	1	2	yes	complemented
159	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
160	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
162	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
163	#499	—	2	4	no	—
164	#436	—	2	4	no	—
165	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
166	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
167	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
172	#400	—	2	2	no	—
173	#394	—	2	2	no	—
174	#388	—	2	4	no	—
175	#380	C₂²×C₁₃₇	1	4	yes	—
176	#380	C₂²×C₁₃₇	1	4	yes	—
177	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
178	#340	C₂²×C₁₃₇	1	4	yes	—
179	#340	C₂²×C₁₃₇	1	4	yes	—
180	#340	C₂²×C₁₃₇	1	2	yes	—
181	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
182	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	8	no	—
185	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
186	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
188	#125	—	2	4	no	—
189	#125	—	2	4	no	—
190	C₄×D₄	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
203	#107	—	2	2	no	—
204	#107	—	2	2	no	—
205	#106	—	2	4	no	—
206	#106	—	2	4	no	—
207	C₄⋊C₈	—	2	2	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
210	#95	—	2	2	no	—
211	#95	—	2	4	no	—
212	#95	—	2	4	no	—
213	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
226	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
227	M₁₆	—	4	8	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₂×C₈	#41×C₁₃₇	1	4	yes	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	4	8	no	—
245	C₂²⋊C₄	—	8	4	no	—
246	C₄²	—	8	2	no	—
247	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
248	C₄²	#41×C₁₃₇	1	2	yes	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
254	D₄	—	4	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	#121×C₁₃₇	1	2	yes	—
271	C₂×C₄	#121×C₁₃₇	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
274	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	4	2	no	—
278	C₈	#124×C₁₃₇	1	4	yes	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	8	no	—
283	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
284	C₂²	#547×C₁₃₇	1	2	yes	—
285	C₂²	#575×C₁₃₇	1	2	yes	—
286	C₂²	#577×C₁₃₇	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	—	8	4	no	—
293	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
294	C₄	#536×C₁₃₇	1	1	yes	characteristic
295	C₄	#556×C₁₃₇	1	2	yes	—
296	C₂	—	8	2	no	—
297	C₂	#2547×C₁₃₇	1	1	yes	characteristic
298	C₂	#3054×C₁₃₇	1	1	yes	characteristic
299	C₂	#3060×C₁₃₇	1	1	yes	characteristic
300	C₂	#3351×C₁₃₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₁₃₇	C₂	semidirect
4. #2077×C₁₃₇	C₂	semidirect
5. #1989×C₁₃₇	C₂	semidirect
6. #1809×C₁₃₇	C₂	semidirect
7. #1809×C₁₃₇	C₂	semidirect
8. C₄×#106×C₁₃₇	C₂	semidirect
150. #28710	C₁₃₇	direct (N × H)
151. C₁₃₇	#28710	direct (N × H)
152. C₂×#499	C₂×C₁₃₇	semidirect
155. #2077	C₂×C₁₃₇	semidirect
156. #1989	C₂×C₁₃₇	semidirect
157. #1809	C₂×C₁₃₇	semidirect
158. #1809	C₂×C₁₃₇	semidirect
159. C₄×#106	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×#499×C₁₃₇	2
C₂×#436×C₁₃₇	2
#2562×C₁₃₇	2
#2077×C₁₃₇	2
#1989×C₁₃₇	2
#1809×C₁₃₇	2
C₄×#106×C₁₃₇	2
#28710	137

Subgroups — C137×#28710 #46878096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28710 #46878096