Subgroups — C₁₃₇×#20658 #46870044

Order 35072.

1646 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2280×C₁₃₇	C₂	1	2	yes	maximal
4	#1808×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1807×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1735×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₇	—	2	8	no	—
9	C₂³×C₈×C₁₃₇	C₄	1	8	yes	complemented
10	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
12	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₁₃₇	C₂²	1	4	yes	—
15	#427×C₁₃₇	—	2	8	no	—
16	C₂×#107×C₁₃₇	C₂²	1	4	yes	complemented
17	C₂×#106×C₁₃₇	C₂²	1	4	yes	complemented
18	#381×C₁₃₇	—	2	2	no	—
19	#380×C₁₃₇	C₂²	1	8	yes	—
20	#379×C₁₃₇	—	2	2	no	—
21	C₄×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
22	#340×C₁₃₇	C₂²	1	8	yes	complemented
23	C₂⁵×C₁₃₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₇	—	2	4	no	—
25	C₂³×C₄×C₁₃₇	—	2	4	no	—
26	C₂³×C₄×C₁₃₇	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
29	C₂³×C₄×C₁₃₇	Q₈	1	4	yes	—
30	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₇	—	2	4	no	—
33	C₂²×C₈×C₁₃₇	—	2	32	no	—
34	C₂²×C₈×C₁₃₇	—	2	8	no	—
35	C₂²×C₈×C₁₃₇	—	2	8	no	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₁₃₇	D₄	1	8	yes	—
41	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	#117×C₁₃₇	—	2	8	no	—
45	#117×C₁₃₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	8	yes	—
47	C₂×C₂²⋊C₄×C₁₃₇	—	2	8	no	—
48	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
49	C₂×C₄²×C₁₃₇	—	2	4	no	—
50	#107×C₁₃₇	—	2	8	no	—
51	#106×C₁₃₇	—	2	8	no	—
52	#95×C₁₃₇	—	2	8	no	—
53	C₂⁴×C₁₃₇	—	2	4	no	—
54	C₂⁴×C₁₃₇	—	2	8	no	—
55	C₂⁴×C₁₃₇	C₄⋊C₄	1	4	yes	—
56	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃₇	—	2	4	no	—
59	C₂²×C₄×C₁₃₇	—	2	8	no	—
60	C₂²×C₄×C₁₃₇	—	2	8	no	—
61	C₂²×C₄×C₁₃₇	—	2	8	no	—
62	C₂²×C₄×C₁₃₇	—	2	16	no	—
63	C₂²×C₄×C₁₃₇	—	2	4	no	—
64	C₂²×C₄×C₁₃₇	—	2	4	no	—
65	C₂²×C₄×C₁₃₇	—	2	8	no	—
66	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	8	yes	—
68	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
76	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	—	2	8	no	—
78	C₂×C₈×C₁₃₇	—	2	4	no	—
79	C₂×C₈×C₁₃₇	—	2	16	no	—
80	C₂×C₈×C₁₃₇	—	2	16	no	—
81	C₂×C₈×C₁₃₇	C₂²⋊C₄	1	16	yes	complemented
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
87	C₄⋊C₄×C₁₃₇	—	2	8	no	—
88	C₄⋊C₄×C₁₃₇	—	2	8	no	—
89	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
90	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
91	C₄²×C₁₃₇	—	4	8	no	—
92	C₂³×C₁₃₇	—	2	32	no	—
93	C₂³×C₁₃₇	—	2	4	no	—
94	C₂³×C₁₃₇	—	2	4	no	—
95	C₂³×C₁₃₇	—	2	4	no	—
96	C₂³×C₁₃₇	—	2	8	no	—
97	C₂³×C₁₃₇	—	2	16	no	—
98	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
100	C₂³×C₁₃₇	#121	1	4	yes	—
101	C₂³×C₁₃₇	#135	1	8	yes	—
102	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃₇	—	2	4	no	—
104	C₂×C₄×C₁₃₇	—	2	8	no	—
105	C₂×C₄×C₁₃₇	—	2	4	no	—
106	C₂×C₄×C₁₃₇	—	2	4	no	—
107	C₂×C₄×C₁₃₇	—	2	8	no	—
108	C₂×C₄×C₁₃₇	—	2	16	no	—
109	C₂×C₄×C₁₃₇	—	4	8	no	—
110	C₂×C₄×C₁₃₇	—	4	16	no	—
111	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
113	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃₇	#117	1	4	yes	—
115	C₂×C₄×C₁₃₇	#121	1	4	yes	—
116	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	8	yes	—
117	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
121	C₈×C₁₃₇	—	2	4	no	—
122	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	4	yes	—
124	C₂²×C₁₃₇	—	2	32	no	—
125	C₂²×C₁₃₇	—	2	4	no	—
126	C₂²×C₁₃₇	—	2	4	no	—
127	C₂²×C₁₃₇	—	2	4	no	—
128	C₂²×C₁₃₇	—	2	8	no	—
129	C₂²×C₁₃₇	—	2	16	no	—
130	C₂²×C₁₃₇	#380	1	4	yes	—
131	C₂²×C₁₃₇	#427	1	8	yes	complemented
132	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₁₃₇	C₂×#121	1	2	yes	—
134	C₂²×C₁₃₇	C₂×#135	1	4	yes	—
135	C₄×C₁₃₇	—	2	4	no	—
136	C₄×C₁₃₇	—	8	8	no	—
137	C₄×C₁₃₇	#380	1	4	yes	—
138	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₃₇	C₂×#117	1	1	yes	characteristic
140	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	2	yes	—
141	C₂×C₁₃₇	—	2	4	no	—
142	C₂×C₁₃₇	—	2	8	no	—
143	C₂×C₁₃₇	C₂×#380	1	1	yes	characteristic
144	C₂×C₁₃₇	C₂×#427	1	2	yes	—
145	C₂×C₁₃₇	#3050	1	4	yes	—
146	#20658	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₁₃₇	#20658	1	1	yes	characteristic, direct product factor, Sylow
148	C₂⁴×C₈	C₂×C₁₃₇	1	1	yes	characteristic
149	C₂×#427	C₂×C₁₃₇	1	2	yes	complemented
150	#2280	C₂×C₁₃₇	1	2	yes	—
151	#1808	C₂×C₁₃₇	1	4	yes	complemented
152	#1807	C₂×C₁₃₇	1	4	yes	complemented
153	#1735	C₂×C₁₃₇	1	2	yes	complemented
154	C₂⁴×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
155	C₂³×C₈	—	2	8	no	—
156	C₂³×C₈	C₄×C₁₃₇	1	8	yes	complemented
157	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
158	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
159	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
160	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
161	C₂×#117	C₂²×C₁₃₇	1	4	yes	—
162	#427	—	2	8	no	—
163	C₂×#107	C₂²×C₁₃₇	1	4	yes	complemented
164	C₂×#106	C₂²×C₁₃₇	1	4	yes	complemented
165	#381	—	2	2	no	—
166	#380	C₂²×C₁₃₇	1	8	yes	—
167	#379	—	2	2	no	—
168	C₄×C₂²⋊C₄	—	2	4	no	—
169	#340	C₂²×C₁₃₇	1	8	yes	complemented
170	C₂⁵	D₄×C₁₃₇	1	1	yes	characteristic
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	—	2	4	no	—
173	C₂³×C₄	C₂×C₄×C₁₃₇	1	4	yes	—
174	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
175	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
176	C₂³×C₄	Q₈×C₁₃₇	1	4	yes	—
177	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	32	no	—
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	—	2	8	no	—
183	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
184	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
185	C₂²×C₈	C₂×C₄×C₁₃₇	1	8	yes	—
186	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
187	C₂²×C₈	D₄×C₁₃₇	1	8	yes	—
188	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
191	#117	—	2	8	no	—
192	#117	—	2	8	no	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	8	yes	—
194	C₂×C₂²⋊C₄	—	2	4	no	—
195	C₂×C₂²⋊C₄	—	2	8	no	—
196	C₂×C₄²	—	2	4	no	—
197	#107	—	2	8	no	—
198	#106	—	2	8	no	—
199	#95	—	2	8	no	—
200	C₂⁴	—	2	4	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	C₄⋊C₄×C₁₃₇	1	4	yes	—
203	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
204	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	16	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
214	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
216	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
219	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
221	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
222	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
223	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	16	no	—
227	C₂×C₈	—	2	16	no	—
228	C₂×C₈	C₂²⋊C₄×C₁₃₇	1	16	yes	complemented
229	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
230	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
232	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
233	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	4	8	no	—
237	C₂²⋊C₄	—	4	16	no	—
238	C₄²	—	4	8	no	—
239	C₂³	—	2	32	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	16	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
246	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
247	C₂³	#121×C₁₃₇	1	4	yes	—
248	C₂³	#135×C₁₃₇	1	8	yes	—
249	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	16	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
259	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
260	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
261	C₂×C₄	#117×C₁₃₇	1	4	yes	—
262	C₂×C₄	#121×C₁₃₇	1	4	yes	—
263	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	8	yes	—
264	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
266	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
270	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	4	yes	—
271	C₂²	—	2	32	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	16	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	#380×C₁₃₇	1	4	yes	—
278	C₂²	#427×C₁₃₇	1	8	yes	complemented
279	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
280	C₂²	C₂×#121×C₁₃₇	1	2	yes	—
281	C₂²	C₂×#135×C₁₃₇	1	4	yes	—
282	C₄	—	2	4	no	—
283	C₄	—	8	8	no	—
284	C₄	#380×C₁₃₇	1	4	yes	—
285	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
286	C₄	C₂×#117×C₁₃₇	1	1	yes	characteristic
287	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	2	yes	—
288	C₂	—	2	4	no	—
289	C₂	—	2	8	no	—
290	C₂	C₂×#380×C₁₃₇	1	1	yes	characteristic
291	C₂	C₂×#427×C₁₃₇	1	2	yes	—
292	C₂	#3050×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	C₂⁵×C₁₃₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₂⁴×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#427×C₁₃₇	C₂	semidirect
4. #1808×C₁₃₇	C₂	semidirect
5. #1807×C₁₃₇	C₂	semidirect
6. #1735×C₁₃₇	C₂	semidirect
9. C₂³×C₈×C₁₃₇	C₄	semidirect
16. C₂×#107×C₁₃₇	C₂²	semidirect
17. C₂×#106×C₁₃₇	C₂²	semidirect
22. #340×C₁₃₇	C₂²	semidirect
81. C₂×C₈×C₁₃₇	C₂²⋊C₄	semidirect
131. C₂²×C₁₃₇	#427	semidirect
146. #20658	C₁₃₇	direct (N × H)
147. C₁₃₇	#20658	direct (N × H)
149. C₂×#427	C₂×C₁₃₇	semidirect
151. #1808	C₂×C₁₃₇	semidirect
152. #1807	C₂×C₁₃₇	semidirect
153. #1735	C₂×C₁₃₇	semidirect
156. C₂³×C₈	C₄×C₁₃₇	semidirect
163. C₂×#107	C₂²×C₁₃₇	semidirect
164. C₂×#106	C₂²×C₁₃₇	semidirect
169. #340	C₂²×C₁₃₇	semidirect
228. C₂×C₈	C₂²⋊C₄×C₁₃₇	semidirect
278. C₂²	#427×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₃₇	2
C₂×#427×C₁₃₇	2
#2280×C₁₃₇	2
#1808×C₁₃₇	2
#1807×C₁₃₇	2
#1735×C₁₃₇	2
#20658	137

Subgroups — C137×#20658 #46870044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20658 #46870044