Subgroups — C₁₃₇×#28708 #46878094

Order 35072.

806 conjugacy classes of subgroups.

The subgroup lattice has 332 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×C₈×D₄×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	#1808×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1808×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#107×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×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	C₈×D₄×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	C₂²×C₈×C₁₃₇	—	2	4	no	—
33	C₂²×C₈×C₁₃₇	—	2	4	no	—
34	C₂²×C₈×C₁₃₇	—	2	4	no	—
35	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
38	#125×C₁₃₇	—	2	4	no	—
39	#125×C₁₃₇	—	2	4	no	—
40	C₄×D₄×C₁₃₇	—	2	2	no	—
41	C₄×D₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃₇	—	2	2	no	—
52	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
53	#107×C₁₃₇	—	2	2	no	—
54	#107×C₁₃₇	—	2	2	no	—
55	#107×C₁₃₇	—	2	4	no	—
56	#106×C₁₃₇	—	2	4	no	—
57	C₄⋊C₈×C₁₃₇	—	2	2	no	—
58	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
59	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
60	#95×C₁₃₇	—	2	2	no	—
61	#95×C₁₃₇	—	2	4	no	—
62	#95×C₁₃₇	—	2	4	no	—
63	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
64	C₂×D₄×C₁₃₇	—	2	4	no	—
65	C₂×D₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	2	2	no	—
68	C₂²×C₄×C₁₃₇	—	2	2	no	—
69	C₂²×C₄×C₁₃₇	—	2	2	no	—
70	C₂²×C₄×C₁₃₇	—	2	2	no	—
71	C₂²×C₄×C₁₃₇	—	2	2	no	—
72	C₂²×C₄×C₁₃₇	—	2	4	no	—
73	C₂²×C₄×C₁₃₇	—	2	2	no	—
74	C₂²×C₄×C₁₃₇	—	2	2	no	—
75	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	—	2	2	no	—
80	C₂×C₈×C₁₃₇	—	2	2	no	—
81	C₂×C₈×C₁₃₇	—	2	2	no	—
82	C₂×C₈×C₁₃₇	—	2	2	no	—
83	C₂×C₈×C₁₃₇	—	4	8	no	—
84	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₃₇	#41	1	4	yes	—
87	C₄⋊C₄×C₁₃₇	—	2	2	no	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	4	no	—
91	C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₄⋊C₄×C₁₃₇	—	2	2	no	—
93	C₄⋊C₄×C₁₃₇	—	2	4	no	—
94	C₄⋊C₄×C₁₃₇	—	4	4	no	—
95	C₄⋊C₄×C₁₃₇	—	4	4	no	—
96	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
97	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
98	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
99	C₄²×C₁₃₇	—	8	2	no	—
100	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₄²×C₁₃₇	#41	1	2	yes	—
102	C₂³×C₁₃₇	—	2	2	no	—
103	C₂³×C₁₃₇	—	2	2	no	—
104	C₂³×C₁₃₇	—	2	2	no	—
105	C₂³×C₁₃₇	—	2	2	no	—
106	C₂³×C₁₃₇	—	2	2	no	—
107	C₂³×C₁₃₇	—	2	2	no	—
108	C₂³×C₁₃₇	#135	1	4	yes	—
109	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₁₃₇	—	2	8	no	—
111	C₂×C₄×C₁₃₇	—	2	2	no	—
112	C₂×C₄×C₁₃₇	—	2	1	no	—
113	C₂×C₄×C₁₃₇	—	2	1	no	—
114	C₂×C₄×C₁₃₇	—	2	2	no	—
115	C₂×C₄×C₁₃₇	—	2	2	no	—
116	C₂×C₄×C₁₃₇	—	2	2	no	—
117	C₂×C₄×C₁₃₇	—	4	2	no	—
118	C₂×C₄×C₁₃₇	—	4	2	no	—
119	C₂×C₄×C₁₃₇	—	4	2	no	—
120	C₂×C₄×C₁₃₇	—	4	2	no	—
121	C₂×C₄×C₁₃₇	—	4	2	no	—
122	C₂×C₄×C₁₃₇	—	4	2	no	—
123	C₂×C₄×C₁₃₇	—	4	4	no	—
124	C₂×C₄×C₁₃₇	—	4	4	no	—
125	C₂×C₄×C₁₃₇	—	4	4	no	—
126	C₂×C₄×C₁₃₇	—	4	4	no	—
127	C₂×C₄×C₁₃₇	—	4	2	no	—
128	C₂×C₄×C₁₃₇	—	4	2	no	—
129	C₂×C₄×C₁₃₇	#121	1	2	yes	—
130	C₂×C₄×C₁₃₇	#121	1	2	yes	—
131	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
135	C₈×C₁₃₇	—	2	2	no	—
136	C₈×C₁₃₇	—	4	2	no	—
137	C₈×C₁₃₇	#124	1	4	yes	—
138	C₂²×C₁₃₇	—	2	4	no	—
139	C₂²×C₁₃₇	—	2	4	no	—
140	C₂²×C₁₃₇	—	2	4	no	—
141	C₂²×C₁₃₇	—	4	2	no	—
142	C₂²×C₁₃₇	—	4	2	no	—
143	C₂²×C₁₃₇	—	4	2	no	—
144	C₂²×C₁₃₇	—	4	2	no	—
145	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
146	C₂²×C₁₃₇	#547	1	2	yes	—
147	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
148	C₂²×C₁₃₇	#576	1	1	yes	characteristic
149	C₂²×C₁₃₇	#578	1	1	yes	characteristic
150	C₄×C₁₃₇	—	2	2	no	—
151	C₄×C₁₃₇	—	2	2	no	—
152	C₄×C₁₃₇	—	4	2	no	—
153	C₄×C₁₃₇	—	8	4	no	—
154	C₄×C₁₃₇	—	8	2	no	—
155	C₄×C₁₃₇	—	8	2	no	—
156	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
157	C₄×C₁₃₇	#536	1	1	yes	characteristic
158	C₄×C₁₃₇	#556	1	2	yes	—
159	C₂×C₁₃₇	—	4	4	no	—
160	C₂×C₁₃₇	#2547	1	1	yes	characteristic
161	C₂×C₁₃₇	#3050	1	1	yes	characteristic
162	C₂×C₁₃₇	#3061	1	1	yes	characteristic
163	C₂×C₁₃₇	#3336	1	2	yes	—
164	C₂×C₁₃₇	#3339	1	2	yes	—
165	#28708	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₇	#28708	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
168	C₂×C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
169	#2562	C₂×C₁₃₇	1	2	yes	—
170	#2077	C₂×C₁₃₇	1	4	yes	complemented
171	#1989	C₂×C₁₃₇	1	2	yes	complemented
172	#1808	C₂×C₁₃₇	1	2	yes	complemented
173	#1808	C₂×C₁₃₇	1	2	yes	complemented
174	C₄×#107	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
176	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
177	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
178	#499	—	2	4	no	—
179	C₈×D₄	—	2	4	no	—
180	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
182	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
183	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
186	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
187	#400	—	2	2	no	—
188	#394	—	2	2	no	—
189	#388	—	2	4	no	—
190	#380	C₂²×C₁₃₇	1	4	yes	—
191	#380	C₂²×C₁₃₇	1	4	yes	—
192	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
193	#340	C₂²×C₁₃₇	1	4	yes	—
194	#340	C₂²×C₁₃₇	1	4	yes	—
195	#340	C₂²×C₁₃₇	1	2	yes	—
196	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	—	2	4	no	—
201	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
204	#125	—	2	4	no	—
205	#125	—	2	4	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	4	no	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	2	4	no	—
222	#106	—	2	4	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₂²⋊C₈	—	2	4	no	—
225	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
226	#95	—	2	2	no	—
227	#95	—	2	4	no	—
228	#95	—	2	4	no	—
229	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	4	8	no	—
250	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
252	C₂×C₈	#41×C₁₃₇	1	4	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	4	8	no	—
264	C₂²⋊C₄	—	8	4	no	—
265	C₄²	—	8	2	no	—
266	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
267	C₄²	#41×C₁₃₇	1	2	yes	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	#135×C₁₃₇	1	4	yes	—
275	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
276	D₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	#121×C₁₃₇	1	2	yes	—
296	C₂×C₄	#121×C₁₃₇	1	2	yes	—
297	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
299	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
300	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₈	#124×C₁₃₇	1	4	yes	—
304	C₂²	—	2	4	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
312	C₂²	#547×C₁₃₇	1	2	yes	—
313	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
314	C₂²	#576×C₁₃₇	1	1	yes	characteristic
315	C₂²	#578×C₁₃₇	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	4	no	—
322	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
323	C₄	#536×C₁₃₇	1	1	yes	characteristic
324	C₄	#556×C₁₃₇	1	2	yes	—
325	C₂	—	4	4	no	—
326	C₂	#2547×C₁₃₇	1	1	yes	characteristic
327	C₂	#3050×C₁₃₇	1	1	yes	characteristic
328	C₂	#3061×C₁₃₇	1	1	yes	characteristic
329	C₂	#3336×C₁₃₇	1	2	yes	—
330	C₂	#3339×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. #2077×C₁₃₇	C₂	semidirect
5. #1989×C₁₃₇	C₂	semidirect
6. #1808×C₁₃₇	C₂	semidirect
7. #1808×C₁₃₇	C₂	semidirect
8. C₄×#107×C₁₃₇	C₂	semidirect
165. #28708	C₁₃₇	direct (N × H)
166. C₁₃₇	#28708	direct (N × H)
167. C₂×#499	C₂×C₁₃₇	semidirect
170. #2077	C₂×C₁₃₇	semidirect
171. #1989	C₂×C₁₃₇	semidirect
172. #1808	C₂×C₁₃₇	semidirect
173. #1808	C₂×C₁₃₇	semidirect
174. 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×C₈×D₄×C₁₃₇	2
#2562×C₁₃₇	2
#2077×C₁₃₇	2
#1989×C₁₃₇	2
#1808×C₁₃₇	2
C₄×#107×C₁₃₇	2
#28708	137

Subgroups — C137×#28708 #46878094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28708 #46878094