Subgroups — C₁₃₇×#21277 #46870663

Order 35072.

950 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2293×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1972×C₁₃₇	C₂	1	2	yes	maximal
4	#1943×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1936×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#102×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
11	C₂×#117×C₁₃₇	—	2	2	no	—
12	C₄×D₈×C₁₃₇	C₄	1	8	yes	complemented
13	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
15	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
16	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	#390×C₁₃₇	C₂²	1	4	yes	—
22	#389×C₁₃₇	C₂²	1	4	yes	—
23	#388×C₁₃₇	—	2	4	no	—
24	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
25	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
26	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
27	#340×C₁₃₇	C₂²	1	4	yes	—
28	#340×C₁₃₇	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃₇	—	2	2	no	—
31	C₂×D₈×C₁₃₇	—	2	4	no	—
32	C₂×D₈×C₁₃₇	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄×D₄×C₁₃₇	—	2	4	no	—
37	C₄×D₄×C₁₃₇	C₂×C₄	1	8	yes	—
38	#117×C₁₃₇	—	4	8	no	—
39	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
44	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
45	C₂×C₄²×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
47	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
50	#107×C₁₃₇	C₂×C₄	1	4	yes	—
51	#106×C₁₃₇	—	2	4	no	—
52	#102×C₁₃₇	—	2	8	no	—
53	#102×C₁₃₇	C₂×C₄	1	8	yes	—
54	#97×C₁₃₇	—	2	4	no	—
55	#97×C₁₃₇	—	2	4	no	—
56	C₄×C₈×C₁₃₇	—	2	2	no	—
57	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
58	#95×C₁₃₇	—	2	4	no	—
59	#95×C₁₃₇	—	2	2	no	—
60	C₂⁴×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	2	4	no	—
63	C₂×D₄×C₁₃₇	—	2	2	no	—
64	C₂×D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	8	no	—
67	C₂²×C₄×C₁₃₇	—	2	4	no	—
68	C₂²×C₄×C₁₃₇	—	4	4	no	—
69	C₂²×C₄×C₁₃₇	—	4	4	no	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	D₈×C₁₃₇	—	2	8	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	2	2	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃₇	#41	1	4	yes	—
81	C₂×C₈×C₁₃₇	#41	1	4	yes	—
82	C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	2	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	8	no	—
86	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
87	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
88	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
89	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
90	C₄²×C₁₃₇	—	2	4	no	—
91	C₄²×C₁₃₇	—	2	2	no	—
92	C₄²×C₁₃₇	—	2	2	no	—
93	C₄²×C₁₃₇	—	4	2	no	—
94	C₄²×C₁₃₇	—	4	4	no	—
95	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
96	C₄²×C₁₃₇	#41	1	2	yes	—
97	C₂³×C₁₃₇	—	2	4	no	—
98	C₂³×C₁₃₇	—	4	4	no	—
99	C₂³×C₁₃₇	—	4	2	no	—
100	C₂³×C₁₃₇	—	4	2	no	—
101	C₂³×C₁₃₇	—	4	4	no	—
102	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	D₄×C₁₃₇	—	2	4	no	—
104	D₄×C₁₃₇	—	2	4	no	—
105	D₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	4	no	—
108	C₂×C₄×C₁₃₇	—	2	4	no	—
109	C₂×C₄×C₁₃₇	—	2	4	no	—
110	C₂×C₄×C₁₃₇	—	2	4	no	—
111	C₂×C₄×C₁₃₇	—	4	2	no	—
112	C₂×C₄×C₁₃₇	—	4	2	no	—
113	C₂×C₄×C₁₃₇	—	4	16	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	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
120	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
122	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
124	C₈×C₁₃₇	—	2	2	no	—
125	C₈×C₁₃₇	—	2	4	no	—
126	C₈×C₁₃₇	#117	1	4	yes	complemented
127	C₂²×C₁₃₇	—	4	4	no	—
128	C₂²×C₁₃₇	—	4	4	no	—
129	C₂²×C₁₃₇	—	4	4	no	—
130	C₂²×C₁₃₇	—	8	4	no	—
131	C₂²×C₁₃₇	—	8	4	no	—
132	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂²×C₁₃₇	#532	1	2	yes	—
134	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₂²×C₁₃₇	#575	1	2	yes	—
136	C₄×C₁₃₇	—	2	2	no	—
137	C₄×C₁₃₇	—	4	4	no	—
138	C₄×C₁₃₇	—	4	4	no	—
139	C₄×C₁₃₇	—	8	4	no	—
140	C₄×C₁₃₇	#442	1	4	yes	—
141	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
142	C₄×C₁₃₇	C₄×#41	1	1	yes	characteristic
143	C₄×C₁₃₇	#529	1	2	yes	—
144	C₂×C₁₃₇	—	8	4	no	—
145	C₂×C₁₃₇	#2293	1	1	yes	characteristic
146	C₂×C₁₃₇	C₂×#442	1	1	yes	characteristic
147	C₂×C₁₃₇	#2943	1	1	yes	characteristic
148	C₂×C₁₃₇	#3171	1	4	yes	—
149	#21277	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₇	#21277	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×C₄×D₈	C₂×C₁₃₇	1	1	yes	characteristic
152	#2293	C₂×C₁₃₇	1	2	yes	complemented
153	#1972	C₂×C₁₃₇	1	2	yes	—
154	#1943	C₂×C₁₃₇	1	2	yes	complemented
155	#1936	C₂×C₁₃₇	1	4	yes	complemented
156	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
157	C₄×#102	C₂×C₁₃₇	1	2	yes	complemented
158	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic, complemented
159	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#117	—	2	2	no	—
162	C₄×D₈	C₄×C₁₃₇	1	8	yes	complemented
163	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
164	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
165	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
166	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
167	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
168	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
171	#390	C₂²×C₁₃₇	1	4	yes	—
172	#389	C₂²×C₁₃₇	1	4	yes	—
173	#388	—	2	4	no	—
174	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
175	C₄×C₂²⋊C₄	—	2	2	no	—
176	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
177	#340	C₂²×C₁₃₇	1	2	yes	—
178	#340	C₂²×C₁₃₇	1	4	yes	—
179	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
180	C₂³×C₄	—	2	2	no	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	C₂×C₄×C₁₃₇	1	4	yes	—
183	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
186	C₄×D₄	—	2	4	no	—
187	C₄×D₄	C₂×C₄×C₁₃₇	1	8	yes	—
188	#117	—	4	8	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
197	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
198	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
200	#107	C₂×C₄×C₁₃₇	1	4	yes	—
201	#106	—	2	4	no	—
202	#102	—	2	8	no	—
203	#102	C₂×C₄×C₁₃₇	1	8	yes	—
204	#97	—	2	4	no	—
205	#97	—	2	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
208	#95	—	2	4	no	—
209	#95	—	2	2	no	—
210	C₂⁴	—	2	2	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	C₂²×C₄×C₁₃₇	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
224	D₈	—	2	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
230	C₂×C₈	#41×C₁₃₇	1	4	yes	—
231	C₂×C₈	#41×C₁₃₇	1	4	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	8	no	—
236	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	4	8	no	—
239	C₂²⋊C₄	—	4	8	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	4	no	—
245	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
246	C₄²	#41×C₁₃₇	1	2	yes	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	—	4	4	no	—
252	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	4	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
269	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
270	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
272	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
273	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	#117×C₁₃₇	1	4	yes	complemented
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	8	4	no	—
281	C₂²	—	8	4	no	—
282	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
283	C₂²	#532×C₁₃₇	1	2	yes	—
284	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
285	C₂²	#575×C₁₃₇	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	4	4	no	—
288	C₄	—	4	4	no	—
289	C₄	—	8	4	no	—
290	C₄	#442×C₁₃₇	1	4	yes	—
291	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
292	C₄	C₄×#41×C₁₃₇	1	1	yes	characteristic
293	C₄	#529×C₁₃₇	1	2	yes	—
294	C₂	—	8	4	no	—
295	C₂	#2293×C₁₃₇	1	1	yes	characteristic
296	C₂	C₂×#442×C₁₃₇	1	1	yes	characteristic
297	C₂	#2943×C₁₃₇	1	1	yes	characteristic
298	C₂	#3171×C₁₃₇	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂²×D₄
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2293×C₁₃₇	C₂	semidirect
4. #1943×C₁₃₇	C₂	semidirect
5. #1936×C₁₃₇	C₂	semidirect
6. C₄×#106×C₁₃₇	C₂	semidirect
7. C₄×#102×C₁₃₇	C₂	semidirect
8. C₄×#97×C₁₃₇	C₂	semidirect
12. C₄×D₈×C₁₃₇	C₄	semidirect
126. C₈×C₁₃₇	#117	semidirect
149. #21277	C₁₃₇	direct (N × H)
150. C₁₃₇	#21277	direct (N × H)
152. #2293	C₂×C₁₃₇	semidirect
154. #1943	C₂×C₁₃₇	semidirect
155. #1936	C₂×C₁₃₇	semidirect
156. C₄×#106	C₂×C₁₃₇	semidirect
157. C₄×#102	C₂×C₁₃₇	semidirect
158. C₄×#97	C₂×C₁₃₇	semidirect
162. C₄×D₈	C₄×C₁₃₇	semidirect
276. C₈	#117×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₃₇	2
#2293×C₁₃₇	2
#1972×C₁₃₇	2
#1943×C₁₃₇	2
#1936×C₁₃₇	2
C₄×#106×C₁₃₇	2
C₄×#102×C₁₃₇	2
C₄×#97×C₁₃₇	2
#21277	137

Subgroups — C137×#21277 #46870663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21277 #46870663