Subgroups — C₁₃₇×#8264 #46857650

Order 35072.

462 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1744×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#322×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1273×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1273×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₃₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
17	#336×C₁₃₇	C₄	1	2	yes	—
18	#336×C₁₃₇	C₄	1	2	yes	—
19	#336×C₁₃₇	C₄	1	2	yes	—
20	#336×C₁₃₇	C₄	1	2	yes	—
21	#322×C₁₃₇	—	2	2	no	—
22	C₈²×C₁₃₇	—	2	2	no	—
23	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₁₃₇	—	2	4	no	—
37	C₄⋊C₈×C₁₃₇	C₈	1	8	yes	complemented
38	C₄×C₈×C₁₃₇	—	2	1	no	—
39	C₄×C₈×C₁₃₇	—	2	1	no	—
40	C₄×C₈×C₁₃₇	—	2	1	no	—
41	C₄×C₈×C₁₃₇	—	2	1	no	—
42	C₄×C₈×C₁₃₇	—	2	1	no	—
43	C₄×C₈×C₁₃₇	—	2	1	no	—
44	C₄×C₈×C₁₃₇	—	2	1	no	—
45	C₄×C₈×C₁₃₇	—	2	1	no	—
46	C₄×C₈×C₁₃₇	—	2	1	no	—
47	C₄×C₈×C₁₃₇	—	2	1	no	—
48	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
49	C₄×C₈×C₁₃₇	Q₈	1	2	yes	—
50	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₇	—	2	2	no	—
58	C₂×C₈×C₁₃₇	—	2	2	no	—
59	C₂×C₈×C₁₃₇	—	2	2	no	—
60	C₂×C₈×C₁₃₇	—	2	2	no	—
61	C₂×C₈×C₁₃₇	—	2	2	no	—
62	C₂×C₈×C₁₃₇	—	2	2	no	—
63	C₂×C₈×C₁₃₇	—	2	2	no	—
64	C₂×C₈×C₁₃₇	—	2	2	no	—
65	C₂×C₈×C₁₃₇	—	2	2	no	—
66	C₂×C₈×C₁₃₇	—	2	2	no	—
67	C₂×C₈×C₁₃₇	—	2	2	no	—
68	C₂×C₈×C₁₃₇	—	2	2	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₁₃₇	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₁₃₇	M₁₆	1	4	yes	—
75	C₄²×C₁₃₇	—	2	2	no	—
76	C₄²×C₁₃₇	—	2	2	no	—
77	C₄²×C₁₃₇	—	2	2	no	—
78	C₄²×C₁₃₇	—	2	2	no	—
79	C₄²×C₁₃₇	—	2	2	no	—
80	C₄²×C₁₃₇	C₂×C₈	1	4	yes	—
81	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
83	C₄²×C₁₃₇	#41	1	1	yes	characteristic
84	C₄²×C₁₃₇	#41	1	1	yes	characteristic
85	C₂³×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₄×C₁₃₇	—	2	1	no	—
88	C₂×C₄×C₁₃₇	—	2	1	no	—
89	C₂×C₄×C₁₃₇	—	2	1	no	—
90	C₂×C₄×C₁₃₇	—	2	1	no	—
91	C₂×C₄×C₁₃₇	—	2	1	no	—
92	C₂×C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₄×C₁₃₇	—	2	1	no	—
94	C₂×C₄×C₁₃₇	—	2	1	no	—
95	C₂×C₄×C₁₃₇	—	2	1	no	—
96	C₂×C₄×C₁₃₇	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₁₃₇	C₄×C₈	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₃₇	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₃₇	#131	1	2	yes	—
107	C₂×C₄×C₁₃₇	#131	1	2	yes	—
108	C₂×C₄×C₁₃₇	#131	1	2	yes	—
109	C₂×C₄×C₁₃₇	#131	1	2	yes	—
110	C₈×C₁₃₇	—	2	4	no	—
111	C₈×C₁₃₇	—	2	4	no	—
112	C₈×C₁₃₇	—	2	4	no	—
113	C₈×C₁₃₇	—	2	4	no	—
114	C₈×C₁₃₇	—	4	2	no	—
115	C₈×C₁₃₇	—	4	2	no	—
116	C₂²×C₁₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂²×C₁₃₇	C₄×M₁₆	1	1	yes	characteristic
119	C₂²×C₁₃₇	#405	1	1	yes	characteristic
120	C₂²×C₁₃₇	#405	1	1	yes	characteristic
121	C₂²×C₁₃₇	#405	1	1	yes	characteristic
122	C₂²×C₁₃₇	#405	1	1	yes	characteristic
123	C₄×C₁₃₇	—	2	2	no	—
124	C₄×C₁₃₇	—	2	2	no	—
125	C₄×C₁₃₇	—	4	1	no	—
126	C₄×C₁₃₇	—	4	1	no	—
127	C₄×C₁₃₇	#424	1	2	yes	—
128	C₄×C₁₃₇	#433	1	2	yes	—
129	C₄×C₁₃₇	C₈×D₄	1	2	yes	—
130	C₄×C₁₃₇	#435	1	2	yes	—
131	C₄×C₁₃₇	C₈×Q₈	1	2	yes	—
132	C₄×C₁₃₇	#446	1	2	yes	—
133	C₂×C₁₃₇	C₈×M₁₆	1	1	yes	characteristic
134	C₂×C₁₃₇	#1450	1	1	yes	characteristic
135	C₂×C₁₃₇	#1453	1	1	yes	characteristic
136	C₂×C₁₃₇	#1454	1	1	yes	characteristic
137	C₂×C₁₃₇	#1768	1	1	yes	characteristic
138	C₂×C₁₃₇	C₈×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₁₃₇	#1771	1	1	yes	characteristic
140	#8264	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₁₃₇	#8264	1	1	yes	characteristic, direct product factor, Sylow
142	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
143	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
144	#1744	C₂×C₁₃₇	1	1	yes	characteristic
145	C₂×#322	C₂×C₁₃₇	1	1	yes	characteristic
146	C₂×C₈²	C₂×C₁₃₇	1	1	yes	characteristic
147	#1273	C₂×C₁₃₇	1	1	yes	characteristic
148	#1273	C₂×C₁₃₇	1	1	yes	characteristic
149	C₂×C₄⋊C₈	C₄×C₁₃₇	1	2	yes	—
150	C₂×C₄⋊C₈	C₄×C₁₃₇	1	2	yes	—
151	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
157	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
158	#336	C₄×C₁₃₇	1	2	yes	—
159	#336	C₄×C₁₃₇	1	2	yes	—
160	#336	C₄×C₁₃₇	1	2	yes	—
161	#336	C₄×C₁₃₇	1	2	yes	—
162	#322	—	2	2	no	—
163	C₈²	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
165	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
168	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
169	C₂²×C₈	C₂×C₄×C₁₃₇	1	2	yes	—
170	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
172	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
177	C₄⋊C₈	—	2	4	no	—
178	C₄⋊C₈	C₈×C₁₃₇	1	8	yes	complemented
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
190	C₄×C₈	Q₈×C₁₃₇	1	2	yes	—
191	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
192	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
193	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
194	C₂²×C₄	C₄²×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
213	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
214	C₂×C₈	C₂×C₈×C₁₃₇	1	4	yes	—
215	C₂×C₈	M₁₆×C₁₃₇	1	4	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₂×C₈×C₁₃₇	1	4	yes	—
222	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
223	C₄²	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
224	C₄²	#41×C₁₃₇	1	1	yes	characteristic
225	C₄²	#41×C₁₃₇	1	1	yes	characteristic
226	C₂³	C₂×C₄²×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	C₄×C₈×C₁₃₇	1	2	yes	—
238	C₂×C₄	C₄×C₈×C₁₃₇	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
240	C₂×C₄	#117×C₁₃₇	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
242	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	C₄×Q₈×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₈×C₁₃₇	1	2	yes	—
246	C₂×C₄	C₂×M₁₆×C₁₃₇	1	2	yes	—
247	C₂×C₄	#131×C₁₃₇	1	2	yes	—
248	C₂×C₄	#131×C₁₃₇	1	2	yes	—
249	C₂×C₄	#131×C₁₃₇	1	2	yes	—
250	C₂×C₄	#131×C₁₃₇	1	2	yes	—
251	C₈	—	2	4	no	—
252	C₈	—	2	4	no	—
253	C₈	—	2	4	no	—
254	C₈	—	2	4	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₂²	C₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
258	C₂²	C₂×C₄×C₈×C₁₃₇	1	1	yes	characteristic
259	C₂²	C₄×M₁₆×C₁₃₇	1	1	yes	characteristic
260	C₂²	#405×C₁₃₇	1	1	yes	characteristic
261	C₂²	#405×C₁₃₇	1	1	yes	characteristic
262	C₂²	#405×C₁₃₇	1	1	yes	characteristic
263	C₂²	#405×C₁₃₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#424×C₁₃₇	1	2	yes	—
269	C₄	#433×C₁₃₇	1	2	yes	—
270	C₄	C₈×D₄×C₁₃₇	1	2	yes	—
271	C₄	#435×C₁₃₇	1	2	yes	—
272	C₄	C₈×Q₈×C₁₃₇	1	2	yes	—
273	C₄	#446×C₁₃₇	1	2	yes	—
274	C₂	C₈×M₁₆×C₁₃₇	1	1	yes	characteristic
275	C₂	#1450×C₁₃₇	1	1	yes	characteristic
276	C₂	#1453×C₁₃₇	1	1	yes	characteristic
277	C₂	#1454×C₁₃₇	1	1	yes	characteristic
278	C₂	#1768×C₁₃₇	1	1	yes	characteristic
279	C₂	C₈×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
280	C₂	#1771×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²×C₁₃₇	C₂³
commutator	C₂²	C₂×C₄×C₈×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
37. C₄⋊C₈×C₁₃₇	C₈	semidirect
140. #8264	C₁₃₇	direct (N × H)
141. C₁₃₇	#8264	direct (N × H)
178. C₄⋊C₈	C₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₁₃₇	2
#1744×C₁₃₇	2
C₂×#322×C₁₃₇	2
C₂×C₈²×C₁₃₇	2
#1273×C₁₃₇	2
#8264	137

Subgroups — C137×#8264 #46857650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8264 #46857650