Subgroups — C₁₃₇×#8642 #46858028

Order 35072.

542 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#1844×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#1802×C₁₃₇	C₂	1	2	yes	maximal
4	C₂×#334×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1296×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₁₃₇	—	2	1	no	—
7	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₃₇	—	2	2	no	—
9	C₈×D₄×C₁₃₇	C₄	1	4	yes	complemented
10	C₂×#106×C₁₃₇	—	2	1	no	—
11	C₂×#106×C₁₃₇	—	2	1	no	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#102×C₁₃₇	—	2	2	no	—
15	C₂×C₂²⋊C₈×C₁₃₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	#384×C₁₃₇	C₂²	1	2	yes	—
18	#340×C₁₃₇	—	2	2	no	—
19	#334×C₁₃₇	—	2	2	no	—
20	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₇	—	2	1	no	—
22	C₂²×C₈×C₁₃₇	—	2	1	no	—
23	C₂²×C₈×C₁₃₇	—	2	4	no	—
24	C₂²×C₈×C₁₃₇	—	2	2	no	—
25	C₂²×C₈×C₁₃₇	—	4	2	no	—
26	C₂²×C₈×C₁₃₇	—	4	2	no	—
27	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₃₇	—	2	1	no	—
30	C₄×D₄×C₁₃₇	—	2	1	no	—
31	C₄×D₄×C₁₃₇	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₁₃₇	Q₈	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
38	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
39	#106×C₁₃₇	—	4	2	no	—
40	#106×C₁₃₇	—	4	2	no	—
41	C₄⋊C₈×C₁₃₇	—	2	1	no	—
42	C₄⋊C₈×C₁₃₇	—	2	2	no	—
43	C₄⋊C₈×C₁₃₇	—	2	2	no	—
44	C₄⋊C₈×C₁₃₇	C₂×C₄	1	2	yes	—
45	#102×C₁₃₇	—	4	4	no	—
46	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
47	C₄×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
49	C₂⁴×C₁₃₇	—	2	1	no	—
50	C₂×D₄×C₁₃₇	—	2	1	no	—
51	C₂×D₄×C₁₃₇	—	2	2	no	—
52	C₂×D₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
53	C₂×D₄×C₁₃₇	SD₁₆	1	4	yes	—
54	C₂²×C₄×C₁₃₇	—	2	1	no	—
55	C₂²×C₄×C₁₃₇	—	2	2	no	—
56	C₂²×C₄×C₁₃₇	—	2	2	no	—
57	C₂²×C₄×C₁₃₇	—	4	1	no	—
58	C₂²×C₄×C₁₃₇	—	4	1	no	—
59	C₂²×C₄×C₁₃₇	—	4	1	no	—
60	C₂²×C₄×C₁₃₇	—	4	1	no	—
61	C₂²×C₄×C₁₃₇	—	4	2	no	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
65	C₂×C₈×C₁₃₇	—	2	1	no	—
66	C₂×C₈×C₁₃₇	—	2	1	no	—
67	C₂×C₈×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	4	1	no	—
69	C₂×C₈×C₁₃₇	—	4	1	no	—
70	C₂×C₈×C₁₃₇	—	4	8	no	—
71	C₂×C₈×C₁₃₇	—	4	2	no	—
72	C₂×C₈×C₁₃₇	—	4	2	no	—
73	C₂×C₈×C₁₃₇	—	4	2	no	—
74	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₁₃₇	D₈	1	2	yes	—
77	C₂×C₈×C₁₃₇	SD₁₆	1	2	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	1	no	—
79	C₄⋊C₄×C₁₃₇	—	4	2	no	—
80	C₄⋊C₄×C₁₃₇	—	4	2	no	—
81	C₄⋊C₄×C₁₃₇	—	4	2	no	—
82	C₄⋊C₄×C₁₃₇	—	4	2	no	—
83	C₄⋊C₄×C₁₃₇	—	8	2	no	—
84	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
86	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
87	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
90	C₄²×C₁₃₇	#41	1	1	yes	characteristic
91	C₂³×C₁₃₇	—	2	2	no	—
92	C₂³×C₁₃₇	—	2	4	no	—
93	C₂³×C₁₃₇	—	4	1	no	—
94	C₂³×C₁₃₇	—	4	1	no	—
95	C₂³×C₁₃₇	—	4	2	no	—
96	C₂³×C₁₃₇	#121	1	1	yes	characteristic
97	D₄×C₁₃₇	—	2	2	no	—
98	D₄×C₁₃₇	—	2	4	no	—
99	D₄×C₁₃₇	#106	1	4	yes	complemented
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	4	1	no	—
103	C₂×C₄×C₁₃₇	—	4	1	no	—
104	C₂×C₄×C₁₃₇	—	4	2	no	—
105	C₂×C₄×C₁₃₇	—	4	2	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	2	no	—
108	C₂×C₄×C₁₃₇	—	8	2	no	—
109	C₂×C₄×C₁₃₇	—	8	2	no	—
110	C₂×C₄×C₁₃₇	—	8	2	no	—
111	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₁₃₇	#135	1	2	yes	—
121	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₁₃₇	#137	1	1	yes	characteristic
123	C₈×C₁₃₇	—	2	2	no	—
124	C₈×C₁₃₇	—	4	2	no	—
125	C₈×C₁₃₇	—	8	2	no	—
126	C₈×C₁₃₇	#102	1	4	yes	complemented
127	C₂²×C₁₃₇	—	2	4	no	—
128	C₂²×C₁₃₇	—	4	2	no	—
129	C₂²×C₁₃₇	—	4	2	no	—
130	C₂²×C₁₃₇	—	4	4	no	—
131	C₂²×C₁₃₇	—	4	4	no	—
132	C₂²×C₁₃₇	#380	1	1	yes	characteristic
133	C₂²×C₁₃₇	#459	1	1	yes	characteristic
134	C₂²×C₁₃₇	#461	1	1	yes	characteristic
135	C₂²×C₁₃₇	#465	1	2	yes	—
136	C₂²×C₁₃₇	#471	1	2	yes	—
137	C₄×C₁₃₇	—	2	1	no	—
138	C₄×C₁₃₇	—	2	1	no	—
139	C₄×C₁₃₇	—	4	2	no	—
140	C₄×C₁₃₇	—	16	2	no	—
141	C₄×C₁₃₇	C₂×#102	1	1	yes	characteristic
142	C₄×C₁₃₇	#417	1	1	yes	characteristic
143	C₄×C₁₃₇	C₂×#106	1	1	yes	characteristic
144	C₄×C₁₃₇	#427	1	1	yes	characteristic
145	C₄×C₁₃₇	#465	1	2	yes	—
146	C₄×C₁₃₇	#474	1	1	yes	characteristic
147	C₄×C₁₃₇	#476	1	1	yes	characteristic
148	C₂×C₁₃₇	—	4	4	no	—
149	C₂×C₁₃₇	#1665	1	2	yes	—
150	C₂×C₁₃₇	#1671	1	2	yes	—
151	C₂×C₁₃₇	#1802	1	1	yes	characteristic
152	C₂×C₁₃₇	#1807	1	1	yes	characteristic
153	C₂×C₁₃₇	#1815	1	1	yes	characteristic
154	#8642	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₁₃₇	#8642	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×C₈×D₄	C₂×C₁₃₇	1	1	yes	characteristic
157	#1844	C₂×C₁₃₇	1	1	yes	characteristic, complemented
158	#1802	C₂×C₁₃₇	1	2	yes	—
159	C₂×#334	C₂×C₁₃₇	1	1	yes	characteristic, complemented
160	#1296	C₂×C₁₃₇	1	2	yes	complemented
161	C₂³×C₈	—	2	1	no	—
162	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
163	C₈×D₄	—	2	2	no	—
164	C₈×D₄	C₄×C₁₃₇	1	4	yes	complemented
165	C₂×#106	—	2	1	no	—
166	C₂×#106	—	2	1	no	—
167	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
169	C₂×#102	—	2	2	no	—
170	C₂×C₂²⋊C₈	—	2	1	no	—
171	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
172	#384	C₂²×C₁₃₇	1	2	yes	—
173	#340	—	2	2	no	—
174	#334	—	2	2	no	—
175	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
176	C₂³×C₄	—	2	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	—	4	2	no	—
182	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
184	C₄×D₄	—	2	1	no	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×C₄×C₁₃₇	1	2	yes	—
187	C₄×D₄	Q₈×C₁₃₇	1	2	yes	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
194	#106	—	4	2	no	—
195	#106	—	4	2	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	C₂×C₄×C₁₃₇	1	2	yes	—
200	#102	—	4	4	no	—
201	C₂²⋊C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
203	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
204	C₂⁴	—	2	1	no	—
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
208	C₂×D₄	SD₁₆×C₁₃₇	1	4	yes	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
219	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	8	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
230	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	2	yes	—
231	C₂×C₈	D₈×C₁₃₇	1	2	yes	—
232	C₂×C₈	SD₁₆×C₁₃₇	1	2	yes	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	8	2	no	—
239	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
244	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
245	C₄²	#41×C₁₃₇	1	1	yes	characteristic
246	C₂³	—	2	2	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	1	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#121×C₁₃₇	1	1	yes	characteristic
252	D₄	—	2	2	no	—
253	D₄	—	2	4	no	—
254	D₄	#106×C₁₃₇	1	4	yes	complemented
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	8	2	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
272	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
275	C₂×C₄	#135×C₁₃₇	1	2	yes	—
276	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	#137×C₁₃₇	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₈	—	8	2	no	—
281	C₈	#102×C₁₃₇	1	4	yes	complemented
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	#380×C₁₃₇	1	1	yes	characteristic
288	C₂²	#459×C₁₃₇	1	1	yes	characteristic
289	C₂²	#461×C₁₃₇	1	1	yes	characteristic
290	C₂²	#465×C₁₃₇	1	2	yes	—
291	C₂²	#471×C₁₃₇	1	2	yes	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	16	2	no	—
296	C₄	C₂×#102×C₁₃₇	1	1	yes	characteristic
297	C₄	#417×C₁₃₇	1	1	yes	characteristic
298	C₄	C₂×#106×C₁₃₇	1	1	yes	characteristic
299	C₄	#427×C₁₃₇	1	1	yes	characteristic
300	C₄	#465×C₁₃₇	1	2	yes	—
301	C₄	#474×C₁₃₇	1	1	yes	characteristic
302	C₄	#476×C₁₃₇	1	1	yes	characteristic
303	C₂	—	4	4	no	—
304	C₂	#1665×C₁₃₇	1	2	yes	—
305	C₂	#1671×C₁₃₇	1	2	yes	—
306	C₂	#1802×C₁₃₇	1	1	yes	characteristic
307	C₂	#1807×C₁₃₇	1	1	yes	characteristic
308	C₂	#1815×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	#121
commutator	C₄²	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1844×C₁₃₇	C₂	semidirect
4. C₂×#334×C₁₃₇	C₂	semidirect
5. #1296×C₁₃₇	C₂	semidirect
9. C₈×D₄×C₁₃₇	C₄	semidirect
99. D₄×C₁₃₇	#106	semidirect
126. C₈×C₁₃₇	#102	semidirect
154. #8642	C₁₃₇	direct (N × H)
155. C₁₃₇	#8642	direct (N × H)
157. #1844	C₂×C₁₃₇	semidirect
159. C₂×#334	C₂×C₁₃₇	semidirect
160. #1296	C₂×C₁₃₇	semidirect
164. C₈×D₄	C₄×C₁₃₇	semidirect
254. D₄	#106×C₁₃₇	semidirect
281. C₈	#102×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃₇	2
#1844×C₁₃₇	2
#1802×C₁₃₇	2
C₂×#334×C₁₃₇	2
#1296×C₁₃₇	2
#8642	137

Subgroups — C137×#8642 #46858028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8642 #46858028