Subgroups — C₁₃₇×#21355 #46870741

Order 35072.

938 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1940×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1850×C₁₃₇	C₂	1	4	yes	maximal
6	#1767×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₂×#117×C₁₃₇	C₄	1	2	yes	—
9	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
10	#432×C₁₃₇	—	2	4	no	—
11	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	8	yes	—
12	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#97×C₁₃₇	—	2	4	no	—
15	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₃₇	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₁₃₇	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	2	yes	—
23	#376×C₁₃₇	C₄	1	2	yes	—
24	C₄³×C₁₃₇	C₂²	1	2	yes	—
25	#336×C₁₃₇	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₇	—	2	4	no	—
30	C₂²×C₈×C₁₃₇	—	2	8	no	—
31	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
33	#117×C₁₃₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
43	C₂×C₄²×C₁₃₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₁₃₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₁₃₇	—	2	16	no	—
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
50	#97×C₁₃₇	—	2	4	no	—
51	#97×C₁₃₇	—	2	4	no	—
52	#97×C₁₃₇	—	2	4	no	—
53	#95×C₁₃₇	C₂×C₄	1	2	yes	—
54	#95×C₁₃₇	C₂×C₄	1	2	yes	—
55	#95×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂⁴×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₇	—	2	4	no	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	4	2	no	—
60	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃₇	—	2	16	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	—	2	8	no	—
77	C₂×C₈×C₁₃₇	#41	1	8	yes	—
78	C₄⋊C₄×C₁₃₇	—	2	8	no	—
79	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
80	C₂²⋊C₄×C₁₃₇	M₁₆	1	8	yes	—
81	C₄²×C₁₃₇	—	2	8	no	—
82	C₄²×C₁₃₇	—	2	8	no	—
83	C₄²×C₁₃₇	—	4	2	no	—
84	C₄²×C₁₃₇	—	4	2	no	—
85	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₁₃₇	C₄⋊C₄	1	4	yes	—
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
91	C₂³×C₁₃₇	—	2	2	no	—
92	C₂³×C₁₃₇	—	2	2	no	—
93	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₁₃₇	C₂×M₁₆	1	4	yes	—
95	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃₇	—	2	4	no	—
97	C₂×C₄×C₁₃₇	—	2	2	no	—
98	C₂×C₄×C₁₃₇	—	2	2	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	4	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	8	no	—
104	C₂×C₄×C₁₃₇	—	2	4	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	2	4	no	—
108	C₂×C₄×C₁₃₇	—	2	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₃₇	C₂×M₁₆	1	8	yes	—
116	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂×#41	1	4	yes	—
119	C₈×C₁₃₇	—	2	8	no	—
120	C₈×C₁₃₇	—	4	4	no	—
121	C₂²×C₁₃₇	—	2	2	no	—
122	C₂²×C₁₃₇	—	2	8	no	—
123	C₂²×C₁₃₇	—	2	2	no	—
124	C₂²×C₁₃₇	#423	1	4	yes	—
125	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₁₃₇	C₄×#41	1	2	yes	—
127	C₂²×C₁₃₇	C₂²×M₁₆	1	2	yes	—
128	C₂²×C₁₃₇	#568	1	2	yes	—
129	C₄×C₁₃₇	—	2	2	no	—
130	C₄×C₁₃₇	—	2	2	no	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	2	8	no	—
134	C₄×C₁₃₇	—	2	2	no	—
135	C₄×C₁₃₇	—	4	4	no	—
136	C₄×C₁₃₇	#525	1	2	yes	—
137	C₄×C₁₃₇	#533	1	2	yes	—
138	C₂×C₁₃₇	—	2	4	no	—
139	C₂×C₁₃₇	#2296	1	1	yes	characteristic
140	C₂×C₁₃₇	C₂×#423	1	1	yes	characteristic
141	C₂×C₁₃₇	#2905	1	1	yes	characteristic
142	C₂×C₁₃₇	#2970	1	4	yes	—
143	#21355	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₁₃₇	#21355	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×#432	C₂×C₁₃₇	1	1	yes	characteristic
146	C₂²×#97	C₂×C₁₃₇	1	1	yes	characteristic
147	C₄×#117	C₂×C₁₃₇	1	1	yes	characteristic
148	#1940	C₂×C₁₃₇	1	4	yes	complemented
149	#1850	C₂×C₁₃₇	1	4	yes	—
150	#1767	C₂×C₁₃₇	1	4	yes	complemented
151	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
152	C₂×#117	C₄×C₁₃₇	1	2	yes	—
153	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
154	#432	—	2	4	no	—
155	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	8	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#97	—	2	4	no	—
159	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
161	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
162	C₄×C₄⋊C₄	C₄×C₁₃₇	1	2	yes	—
163	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
164	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
165	C₄×C₂²⋊C₄	C₄×C₁₃₇	1	1	yes	characteristic
166	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	2	yes	—
167	#376	C₄×C₁₃₇	1	2	yes	—
168	C₄³	C₂²×C₁₃₇	1	2	yes	—
169	#336	C₂²×C₁₃₇	1	8	yes	—
170	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₃₇	1	1	yes	characteristic
172	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	4	no	—
174	C₂²×C₈	—	2	8	no	—
175	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
176	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
177	#117	—	2	8	no	—
178	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	2	yes	—
181	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
183	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
184	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
185	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
187	C₂×C₄²	Q₈×C₁₃₇	1	4	yes	—
188	C₂×C₄²	C₂³×C₁₃₇	1	4	yes	—
189	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
191	C₄⋊C₈	—	2	16	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₂²⋊C₈	—	2	4	no	—
194	#97	—	2	4	no	—
195	#97	—	2	4	no	—
196	#97	—	2	4	no	—
197	#95	C₂×C₄×C₁₃₇	1	2	yes	—
198	#95	C₂×C₄×C₁₃₇	1	2	yes	—
199	#95	C₂×C₄×C₁₃₇	1	4	yes	—
200	C₂⁴	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
205	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
214	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
215	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
216	C₂×C₈	—	2	16	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	#41×C₁₃₇	1	8	yes	—
222	C₄⋊C₄	—	2	8	no	—
223	C₂²⋊C₄	—	2	4	no	—
224	C₂²⋊C₄	M₁₆×C₁₃₇	1	8	yes	—
225	C₄²	—	2	8	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	4	2	no	—
228	C₄²	—	4	2	no	—
229	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
230	C₄²	C₄⋊C₄×C₁₃₇	1	4	yes	—
231	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
232	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
233	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
234	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
238	C₂³	C₂×M₁₆×C₁₃₇	1	4	yes	—
239	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
256	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
259	C₂×C₄	C₂×M₁₆×C₁₃₇	1	8	yes	—
260	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄	C₂×#41×C₁₃₇	1	4	yes	—
263	C₈	—	2	8	no	—
264	C₈	—	4	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	8	no	—
268	C₂²	#423×C₁₃₇	1	4	yes	—
269	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
270	C₂²	C₄×#41×C₁₃₇	1	2	yes	—
271	C₂²	C₂²×M₁₆×C₁₃₇	1	2	yes	—
272	C₂²	#568×C₁₃₇	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	8	no	—
279	C₄	—	4	4	no	—
280	C₄	#525×C₁₃₇	1	2	yes	—
281	C₄	#533×C₁₃₇	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	#2296×C₁₃₇	1	1	yes	characteristic
284	C₂	C₂×#423×C₁₃₇	1	1	yes	characteristic
285	C₂	#2905×C₁₃₇	1	1	yes	characteristic
286	C₂	#2970×C₁₃₇	1	4	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
4. #1940×C₁₃₇	C₂	semidirect
6. #1767×C₁₃₇	C₂	semidirect
143. #21355	C₁₃₇	direct (N × H)
144. C₁₃₇	#21355	direct (N × H)
148. #1940	C₂×C₁₃₇	semidirect
150. #1767	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×#432×C₁₃₇	2
C₂²×#97×C₁₃₇	2
C₄×#117×C₁₃₇	2
#1940×C₁₃₇	2
#1850×C₁₃₇	2
#1767×C₁₃₇	2
#21355	137

Subgroups — C137×#21355 #46870741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21355 #46870741