Subgroups — C₁₃₇×#21143 #46870529

Order 35072.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#424×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#423×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2285×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1939×C₁₃₇	C₂	1	8	yes	maximal, complemented
6	#1729×C₁₃₇	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	#424×C₁₃₇	—	2	8	no	—
13	#423×C₁₃₇	—	2	4	no	—
14	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
15	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃₇	—	2	4	no	—
17	C₂×C₄×C₈×C₁₃₇	C₂²	1	4	yes	—
18	#384×C₁₃₇	C₄	1	2	yes	—
19	#382×C₁₃₇	C₄	1	2	yes	—
20	#380×C₁₃₇	C₄	1	4	yes	—
21	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	4	yes	—
22	#336×C₁₃₇	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₁₃₇	—	2	4	no	—
26	C₂×M₁₆×C₁₃₇	—	2	4	no	—
27	C₂×M₁₆×C₁₃₇	—	2	4	no	—
28	C₂²×C₈×C₁₃₇	—	2	2	no	—
29	C₂²×C₈×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	—	2	2	no	—
31	C₂²×C₈×C₁₃₇	—	2	2	no	—
32	C₂²×C₈×C₁₃₇	D₄	1	4	yes	—
33	C₂²×C₈×C₁₃₇	Q₈	1	4	yes	—
34	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
37	#117×C₁₃₇	—	2	4	no	—
38	#117×C₁₃₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
47	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₁₃₇	—	2	16	no	—
49	C₄⋊C₈×C₁₃₇	—	2	8	no	—
50	C₄×C₈×C₁₃₇	—	2	8	no	—
51	#95×C₁₃₇	C₂×C₄	1	8	yes	—
52	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₇	—	2	8	no	—
54	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₁₃₇	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
64	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
65	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₁₃₇	—	4	8	no	—
67	C₂×C₈×C₁₃₇	—	2	8	no	—
68	C₂×C₈×C₁₃₇	—	2	8	no	—
69	C₂×C₈×C₁₃₇	—	2	2	no	—
70	C₂×C₈×C₁₃₇	—	2	2	no	—
71	C₂×C₈×C₁₃₇	—	2	2	no	—
72	C₂×C₈×C₁₃₇	—	2	2	no	—
73	C₂×C₈×C₁₃₇	—	2	4	no	—
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	4	no	—
76	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	8	no	—
79	C₄⋊C₄×C₁₃₇	—	2	16	no	—
80	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
81	C₄²×C₁₃₇	—	2	4	no	—
82	C₄²×C₁₃₇	—	2	4	no	—
83	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
84	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₁₃₇	C₂×D₄	1	4	yes	—
86	C₄²×C₁₃₇	C₂×Q₈	1	4	yes	—
87	C₂³×C₁₃₇	—	2	4	no	—
88	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
90	C₂³×C₁₃₇	C₄×Q₈	1	2	yes	—
91	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₇	—	2	4	no	—
93	C₂×C₄×C₁₃₇	—	2	16	no	—
94	C₂×C₄×C₁₃₇	—	2	8	no	—
95	C₂×C₄×C₁₃₇	—	2	16	no	—
96	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₃₇	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃₇	#121	1	4	yes	—
102	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₁₃₇	#125	1	4	yes	—
104	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₁₃₇	#131	1	8	yes	—
106	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
112	C₈×C₁₃₇	—	2	4	no	—
113	C₈×C₁₃₇	—	2	4	no	—
114	C₈×C₁₃₇	—	4	4	no	—
115	C₂²×C₁₃₇	—	2	4	no	—
116	C₂²×C₁₃₇	—	2	4	no	—
117	C₂²×C₁₃₇	—	2	4	no	—
118	C₂²×C₁₃₇	#389	1	4	yes	—
119	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₁₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₂²×C₁₃₇	C₂×#131	1	4	yes	—
123	C₄×C₁₃₇	—	2	4	no	—
124	C₄×C₁₃₇	—	4	8	no	—
125	C₄×C₁₃₇	#389	1	4	yes	—
126	C₄×C₁₃₇	#424	1	8	yes	complemented
127	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
128	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₃₇	C₂×#125	1	1	yes	characteristic
130	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	C₂×C₁₃₇	—	2	4	no	—
132	C₂×C₁₃₇	C₂×#389	1	1	yes	characteristic
133	C₂×C₁₃₇	C₂×#424	1	2	yes	—
134	C₂×C₁₃₇	#2931	1	2	yes	—
135	C₂×C₁₃₇	#2960	1	2	yes	—
136	#21143	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₇	#21143	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#424	C₂×C₁₃₇	1	2	yes	—
139	C₂×#423	C₂×C₁₃₇	1	1	yes	characteristic
140	C₂²×C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic
141	#2285	C₂×C₁₃₇	1	1	yes	characteristic
142	#1939	C₂×C₁₃₇	1	8	yes	complemented
143	#1729	C₂×C₁₃₇	1	2	yes	—
144	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
145	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
148	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
149	#424	—	2	8	no	—
150	#423	—	2	4	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
152	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	8	yes	—
153	C₂×C₄×C₈	—	2	4	no	—
154	C₂×C₄×C₈	C₂²×C₁₃₇	1	4	yes	—
155	#384	C₄×C₁₃₇	1	2	yes	—
156	#382	C₄×C₁₃₇	1	2	yes	—
157	#380	C₄×C₁₃₇	1	4	yes	—
158	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	4	yes	—
159	#336	C₂²×C₁₃₇	1	8	yes	—
160	C₂³×C₄	C₂×C₄×C₁₃₇	1	2	yes	—
161	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	D₄×C₁₃₇	1	4	yes	—
170	C₂²×C₈	Q₈×C₁₃₇	1	4	yes	—
171	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
172	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
173	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	8	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
182	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
183	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
184	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
185	C₄⋊C₈	—	2	16	no	—
186	C₄⋊C₈	—	2	8	no	—
187	C₄×C₈	—	2	8	no	—
188	#95	C₂×C₄×C₁₃₇	1	8	yes	—
189	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
192	C₂²×C₄	C₂²×C₄×C₁₃₇	1	8	yes	—
193	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
194	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
195	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
199	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
200	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
201	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
202	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	2	8	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	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	16	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
221	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
222	C₄²	C₂×D₄×C₁₃₇	1	4	yes	—
223	C₄²	C₂×Q₈×C₁₃₇	1	4	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
226	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
227	C₂³	C₄×Q₈×C₁₃₇	1	2	yes	—
228	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
236	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
237	C₂×C₄	C₄×Q₈×C₁₃₇	1	2	yes	—
238	C₂×C₄	#121×C₁₃₇	1	4	yes	—
239	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
240	C₂×C₄	#125×C₁₃₇	1	4	yes	—
241	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
242	C₂×C₄	#131×C₁₃₇	1	8	yes	—
243	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
247	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	2	4	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	#389×C₁₃₇	1	4	yes	—
256	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
257	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
258	C₂²	C₂×C₄×Q₈×C₁₃₇	1	1	yes	characteristic
259	C₂²	C₂×#131×C₁₃₇	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	#389×C₁₃₇	1	4	yes	—
263	C₄	#424×C₁₃₇	1	8	yes	complemented
264	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
265	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
266	C₄	C₂×#125×C₁₃₇	1	1	yes	characteristic
267	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×#389×C₁₃₇	1	1	yes	characteristic
270	C₂	C₂×#424×C₁₃₇	1	2	yes	—
271	C₂	#2931×C₁₃₇	1	2	yes	—
272	C₂	#2960×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. #1939×C₁₃₇	C₂	semidirect
126. C₄×C₁₃₇	#424	semidirect
136. #21143	C₁₃₇	direct (N × H)
137. C₁₃₇	#21143	direct (N × H)
142. #1939	C₂×C₁₃₇	semidirect
263. C₄	#424×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#424×C₁₃₇	2
C₂×#423×C₁₃₇	2
C₂²×C₄×C₈×C₁₃₇	2
#2285×C₁₃₇	2
#1939×C₁₃₇	2
#1729×C₁₃₇	2
#21143	137

Subgroups — C137×#21143 #46870529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21143 #46870529