Subgroups — C₁₃₇×#28709 #46878095

Order 35072.

774 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#2076×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2075×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1989×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1809×C₁₃₇	C₂	1	4	yes	maximal, complemented
8	C₄×#107×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	2	yes	—
15	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
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	#107×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	2	no	—
102	C₂³×C₁₃₇	—	2	2	no	—
103	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
104	D₄×C₁₃₇	—	4	4	no	—
105	C₂×C₄×C₁₃₇	—	2	2	no	—
106	C₂×C₄×C₁₃₇	—	2	1	no	—
107	C₂×C₄×C₁₃₇	—	2	1	no	—
108	C₂×C₄×C₁₃₇	—	2	2	no	—
109	C₂×C₄×C₁₃₇	—	2	4	no	—
110	C₂×C₄×C₁₃₇	—	4	2	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	4	no	—
113	C₂×C₄×C₁₃₇	—	4	2	no	—
114	C₂×C₄×C₁₃₇	—	4	2	no	—
115	C₂×C₄×C₁₃₇	—	4	4	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	2	no	—
120	C₂×C₄×C₁₃₇	—	4	2	no	—
121	C₂×C₄×C₁₃₇	#121	1	2	yes	—
122	C₂×C₄×C₁₃₇	#121	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
125	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₁₃₇	—	2	2	no	—
128	C₈×C₁₃₇	—	4	2	no	—
129	C₈×C₁₃₇	#124	1	4	yes	—
130	C₂²×C₁₃₇	—	4	4	no	—
131	C₂²×C₁₃₇	—	4	4	no	—
132	C₂²×C₁₃₇	—	4	2	no	—
133	C₂²×C₁₃₇	—	4	2	no	—
134	C₂²×C₁₃₇	—	4	2	no	—
135	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
136	C₂²×C₁₃₇	#547	1	2	yes	—
137	C₂²×C₁₃₇	#575	1	2	yes	—
138	C₂²×C₁₃₇	#576	1	1	yes	characteristic
139	C₂²×C₁₃₇	#578	1	1	yes	characteristic
140	C₄×C₁₃₇	—	2	2	no	—
141	C₄×C₁₃₇	—	2	2	no	—
142	C₄×C₁₃₇	—	4	2	no	—
143	C₄×C₁₃₇	—	8	4	no	—
144	C₄×C₁₃₇	—	8	2	no	—
145	C₄×C₁₃₇	—	8	2	no	—
146	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
147	C₄×C₁₃₇	#536	1	1	yes	characteristic
148	C₄×C₁₃₇	#556	1	2	yes	—
149	C₂×C₁₃₇	—	8	2	no	—
150	C₂×C₁₃₇	#2547	1	1	yes	characteristic
151	C₂×C₁₃₇	#3054	1	1	yes	characteristic
152	C₂×C₁₃₇	#3061	1	1	yes	characteristic
153	C₂×C₁₃₇	#3352	1	2	yes	—
154	C₂×C₁₃₇	#3355	1	2	yes	—
155	#28709	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₁₃₇	#28709	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
158	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
159	#2562	C₂×C₁₃₇	1	2	yes	—
160	#2076	C₂×C₁₃₇	1	2	yes	complemented
161	#2075	C₂×C₁₃₇	1	2	yes	complemented
162	#1989	C₂×C₁₃₇	1	2	yes	complemented
163	#1809	C₂×C₁₃₇	1	4	yes	complemented
164	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
165	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
166	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
167	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
168	#499	—	2	4	no	—
169	#436	—	2	4	no	—
170	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
171	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
172	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
173	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
177	#400	—	2	2	no	—
178	#394	—	2	2	no	—
179	#388	—	2	4	no	—
180	#380	C₂²×C₁₃₇	1	4	yes	—
181	#380	C₂²×C₁₃₇	1	4	yes	—
182	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
183	#340	C₂²×C₁₃₇	1	4	yes	—
184	#340	C₂²×C₁₃₇	1	4	yes	—
185	#340	C₂²×C₁₃₇	1	2	yes	—
186	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	2	8	no	—
190	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
193	#125	—	2	4	no	—
194	#125	—	2	4	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	4	no	—
205	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
208	#107	—	2	2	no	—
209	#107	—	2	2	no	—
210	#107	—	2	4	no	—
211	#106	—	2	4	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
215	#95	—	2	2	no	—
216	#95	—	2	4	no	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
232	M₁₆	—	4	8	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
238	C₂×C₈	#41×C₁₃₇	1	4	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	4	8	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₄²	—	8	2	no	—
252	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
253	C₄²	#41×C₁₃₇	1	2	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	4	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
260	D₄	—	4	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	#121×C₁₃₇	1	2	yes	—
278	C₂×C₄	#121×C₁₃₇	1	2	yes	—
279	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
281	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
283	C₈	—	2	2	no	—
284	C₈	—	4	2	no	—
285	C₈	#124×C₁₃₇	1	4	yes	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
292	C₂²	#547×C₁₃₇	1	2	yes	—
293	C₂²	#575×C₁₃₇	1	2	yes	—
294	C₂²	#576×C₁₃₇	1	1	yes	characteristic
295	C₂²	#578×C₁₃₇	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
303	C₄	#536×C₁₃₇	1	1	yes	characteristic
304	C₄	#556×C₁₃₇	1	2	yes	—
305	C₂	—	8	2	no	—
306	C₂	#2547×C₁₃₇	1	1	yes	characteristic
307	C₂	#3054×C₁₃₇	1	1	yes	characteristic
308	C₂	#3061×C₁₃₇	1	1	yes	characteristic
309	C₂	#3352×C₁₃₇	1	2	yes	—
310	C₂	#3355×C₁₃₇	1	2	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. #2076×C₁₃₇	C₂	semidirect
5. #2075×C₁₃₇	C₂	semidirect
6. #1989×C₁₃₇	C₂	semidirect
7. #1809×C₁₃₇	C₂	semidirect
8. C₄×#107×C₁₃₇	C₂	semidirect
155. #28709	C₁₃₇	direct (N × H)
156. C₁₃₇	#28709	direct (N × H)
157. C₂×#499	C₂×C₁₃₇	semidirect
160. #2076	C₂×C₁₃₇	semidirect
161. #2075	C₂×C₁₃₇	semidirect
162. #1989	C₂×C₁₃₇	semidirect
163. #1809	C₂×C₁₃₇	semidirect
164. C₄×#107	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
#2076×C₁₃₇	2
#2075×C₁₃₇	2
#1989×C₁₃₇	2
#1809×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28709	137

Subgroups — C137×#28709 #46878095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28709 #46878095