Subgroups — C₁₃₇×#24743 #46874129

Order 35072.

932 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#464×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#462×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#440×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2547×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2351×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2068×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2042×C₁₃₇	C₂	1	2	yes	maximal
8	#1960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1940×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1878×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1804×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	#464×C₁₃₇	—	2	8	no	—
20	#462×C₁₃₇	—	2	4	no	—
21	#440×C₁₃₇	—	2	4	no	—
22	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
32	#397×C₁₃₇	—	2	1	no	—
33	#397×C₁₃₇	—	2	1	no	—
34	#391×C₁₃₇	C₂²	1	2	yes	—
35	#389×C₁₃₇	—	2	2	no	—
36	#389×C₁₃₇	C₂²	1	2	yes	—
37	#382×C₁₃₇	—	2	1	no	—
38	#382×C₁₃₇	C₂²	1	1	yes	characteristic
39	#380×C₁₃₇	—	2	2	no	—
40	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#336×C₁₃₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₁₃₇	—	2	1	no	—
48	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
49	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
50	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
53	C₂×SD₁₆×C₁₃₇	D₄	1	4	yes	—
54	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
58	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
59	#124×C₁₃₇	—	2	4	no	—
60	#124×C₁₃₇	—	4	2	no	—
61	C₄×Q₈×C₁₃₇	—	2	2	no	—
62	C₄×Q₈×C₁₃₇	—	2	4	no	—
63	C₄×Q₈×C₁₃₇	—	4	2	no	—
64	C₄×Q₈×C₁₃₇	D₄	1	4	yes	—
65	C₄×D₄×C₁₃₇	—	2	2	no	—
66	C₄×D₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
79	C₂×C₄²×C₁₃₇	—	2	1	no	—
80	C₂×C₄²×C₁₃₇	—	2	2	no	—
81	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
83	#107×C₁₃₇	—	2	2	no	—
84	C₄⋊C₈×C₁₃₇	—	2	4	no	—
85	C₄⋊C₈×C₁₃₇	—	2	2	no	—
86	#103×C₁₃₇	—	2	2	no	—
87	#103×C₁₃₇	—	2	4	no	—
88	#103×C₁₃₇	—	2	4	no	—
89	#103×C₁₃₇	—	4	2	no	—
90	#102×C₁₃₇	—	2	4	no	—
91	#102×C₁₃₇	—	2	2	no	—
92	#97×C₁₃₇	—	2	2	no	—
93	#95×C₁₃₇	—	2	1	no	—
94	#95×C₁₃₇	—	2	2	no	—
95	#95×C₁₃₇	—	2	2	no	—
96	#95×C₁₃₇	—	2	2	no	—
97	#95×C₁₃₇	—	2	1	no	—
98	#95×C₁₃₇	D₄	1	1	yes	characteristic
99	#95×C₁₃₇	D₄	1	1	yes	characteristic
100	C₂⁴×C₁₃₇	—	2	1	no	—
101	C₂×Q₈×C₁₃₇	—	2	1	no	—
102	C₂×Q₈×C₁₃₇	—	2	1	no	—
103	C₂×Q₈×C₁₃₇	—	2	1	no	—
104	C₂×Q₈×C₁₃₇	—	2	2	no	—
105	C₂×Q₈×C₁₃₇	—	2	2	no	—
106	C₂×Q₈×C₁₃₇	—	2	2	no	—
107	C₂×Q₈×C₁₃₇	—	2	2	no	—
108	C₂×Q₈×C₁₃₇	—	2	2	no	—
109	C₂×Q₈×C₁₃₇	—	2	2	no	—
110	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
112	C₂×Q₈×C₁₃₇	#41	1	2	yes	—
113	C₂×D₄×C₁₃₇	—	2	1	no	—
114	C₂×D₄×C₁₃₇	—	2	1	no	—
115	C₂×D₄×C₁₃₇	—	2	1	no	—
116	C₂×D₄×C₁₃₇	—	2	1	no	—
117	C₂×D₄×C₁₃₇	—	2	1	no	—
118	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₁₃₇	—	2	1	no	—
120	C₂²×C₄×C₁₃₇	—	2	1	no	—
121	C₂²×C₄×C₁₃₇	—	2	1	no	—
122	C₂²×C₄×C₁₃₇	—	2	2	no	—
123	C₂²×C₄×C₁₃₇	—	2	2	no	—
124	C₂²×C₄×C₁₃₇	—	4	4	no	—
125	C₂²×C₄×C₁₃₇	—	4	2	no	—
126	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₁₃₇	—	4	8	no	—
132	SD₁₆×C₁₃₇	—	2	8	no	—
133	C₂×C₈×C₁₃₇	—	2	1	no	—
134	C₂×C₈×C₁₃₇	—	2	1	no	—
135	C₂×C₈×C₁₃₇	—	2	1	no	—
136	C₂×C₈×C₁₃₇	—	2	2	no	—
137	C₂×C₈×C₁₃₇	—	2	2	no	—
138	C₂×C₈×C₁₃₇	—	2	2	no	—
139	C₂×C₈×C₁₃₇	—	2	1	no	—
140	C₂×C₈×C₁₃₇	—	2	1	no	—
141	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₁₃₇	—	2	4	no	—
143	C₄⋊C₄×C₁₃₇	—	2	1	no	—
144	C₄⋊C₄×C₁₃₇	—	2	2	no	—
145	C₄⋊C₄×C₁₃₇	—	2	1	no	—
146	C₄⋊C₄×C₁₃₇	—	2	1	no	—
147	C₄⋊C₄×C₁₃₇	—	2	2	no	—
148	C₄⋊C₄×C₁₃₇	—	4	2	no	—
149	C₄⋊C₄×C₁₃₇	—	4	4	no	—
150	C₄⋊C₄×C₁₃₇	—	4	4	no	—
151	C₄⋊C₄×C₁₃₇	—	4	2	no	—
152	C₄⋊C₄×C₁₃₇	—	4	2	no	—
153	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
154	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
155	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
156	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
157	C₄²×C₁₃₇	—	2	4	no	—
158	C₄²×C₁₃₇	—	2	2	no	—
159	C₄²×C₁₃₇	—	2	2	no	—
160	C₄²×C₁₃₇	—	4	4	no	—
161	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
162	C₄²×C₁₃₇	#41	1	2	yes	—
163	C₂³×C₁₃₇	—	2	2	no	—
164	C₂³×C₁₃₇	—	4	1	no	—
165	C₂³×C₁₃₇	—	4	1	no	—
166	C₂³×C₁₃₇	—	4	2	no	—
167	C₂³×C₁₃₇	—	4	2	no	—
168	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₁₃₇	—	2	2	no	—
170	Q₈×C₁₃₇	—	2	2	no	—
171	Q₈×C₁₃₇	—	2	2	no	—
172	Q₈×C₁₃₇	—	4	2	no	—
173	Q₈×C₁₃₇	—	4	2	no	—
174	Q₈×C₁₃₇	—	4	2	no	—
175	Q₈×C₁₃₇	—	4	2	no	—
176	Q₈×C₁₃₇	#121	1	4	yes	—
177	D₄×C₁₃₇	—	2	2	no	—
178	D₄×C₁₃₇	—	2	2	no	—
179	D₄×C₁₃₇	—	4	1	no	—
180	D₄×C₁₃₇	—	4	1	no	—
181	C₂×C₄×C₁₃₇	—	2	2	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	2	no	—
184	C₂×C₄×C₁₃₇	—	2	2	no	—
185	C₂×C₄×C₁₃₇	—	2	2	no	—
186	C₂×C₄×C₁₃₇	—	2	2	no	—
187	C₂×C₄×C₁₃₇	—	2	2	no	—
188	C₂×C₄×C₁₃₇	—	2	2	no	—
189	C₂×C₄×C₁₃₇	—	2	2	no	—
190	C₂×C₄×C₁₃₇	—	2	2	no	—
191	C₂×C₄×C₁₃₇	—	2	2	no	—
192	C₂×C₄×C₁₃₇	—	2	2	no	—
193	C₂×C₄×C₁₃₇	—	4	2	no	—
194	C₂×C₄×C₁₃₇	—	4	1	no	—
195	C₂×C₄×C₁₃₇	—	4	2	no	—
196	C₂×C₄×C₁₃₇	—	4	2	no	—
197	C₂×C₄×C₁₃₇	—	4	2	no	—
198	C₂×C₄×C₁₃₇	—	4	2	no	—
199	C₂×C₄×C₁₃₇	—	4	2	no	—
200	C₂×C₄×C₁₃₇	—	4	2	no	—
201	C₂×C₄×C₁₃₇	—	4	2	no	—
202	C₂×C₄×C₁₃₇	—	8	4	no	—
203	C₂×C₄×C₁₃₇	#121	1	2	yes	—
204	C₂×C₄×C₁₃₇	#121	1	2	yes	—
205	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₁₃₇	—	2	2	no	—
212	C₈×C₁₃₇	—	4	2	no	—
213	C₈×C₁₃₇	—	4	1	no	—
214	C₂²×C₁₃₇	—	4	2	no	—
215	C₂²×C₁₃₇	—	4	4	no	—
216	C₂²×C₁₃₇	—	8	2	no	—
217	C₂²×C₁₃₇	—	8	1	no	—
218	C₂²×C₁₃₇	—	8	1	no	—
219	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
220	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
221	C₂²×C₁₃₇	#548	1	1	yes	characteristic
222	C₂²×C₁₃₇	#575	1	2	yes	—
223	C₂²×C₁₃₇	#578	1	2	yes	—
224	C₄×C₁₃₇	—	2	4	no	—
225	C₄×C₁₃₇	—	2	2	no	—
226	C₄×C₁₃₇	—	2	2	no	—
227	C₄×C₁₃₇	—	4	2	no	—
228	C₄×C₁₃₇	—	4	2	no	—
229	C₄×C₁₃₇	—	4	2	no	—
230	C₄×C₁₃₇	—	8	2	no	—
231	C₄×C₁₃₇	—	8	2	no	—
232	C₄×C₁₃₇	—	8	1	no	—
233	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
234	C₄×C₁₃₇	#537	1	1	yes	characteristic
235	C₄×C₁₃₇	#547	1	1	yes	characteristic
236	C₄×C₁₃₇	#553	1	1	yes	characteristic
237	C₂×C₁₃₇	—	8	2	no	—
238	C₂×C₁₃₇	#2416	1	1	yes	characteristic
239	C₂×C₁₃₇	#3040	1	1	yes	characteristic
240	C₂×C₁₃₇	#3044	1	1	yes	characteristic
241	C₂×C₁₃₇	#3269	1	2	yes	—
242	C₂×C₁₃₇	#3377	1	2	yes	—
243	#24743	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₁₃₇	#24743	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#464	C₂×C₁₃₇	1	2	yes	—
246	C₂×#462	C₂×C₁₃₇	1	1	yes	characteristic, complemented
247	C₂×#440	C₂×C₁₃₇	1	1	yes	characteristic
248	#2547	C₂×C₁₃₇	1	1	yes	characteristic
249	#2351	C₂×C₁₃₇	1	1	yes	characteristic, complemented
250	#2068	C₂×C₁₃₇	1	2	yes	complemented
251	#2042	C₂×C₁₃₇	1	2	yes	—
252	#1960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
253	#1940	C₂×C₁₃₇	1	1	yes	characteristic, complemented
254	#1878	C₂×C₁₃₇	1	2	yes	complemented
255	#1804	C₂×C₁₃₇	1	1	yes	characteristic
256	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
257	C₂²×SD₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
259	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
260	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
261	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
263	#464	—	2	8	no	—
264	#462	—	2	4	no	—
265	#440	—	2	4	no	—
266	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
269	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
270	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
271	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
272	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
273	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
274	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
275	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
276	#397	—	2	1	no	—
277	#397	—	2	1	no	—
278	#391	C₂²×C₁₃₇	1	2	yes	—
279	#389	—	2	2	no	—
280	#389	C₂²×C₁₃₇	1	2	yes	—
281	#382	—	2	1	no	—
282	#382	C₂²×C₁₃₇	1	1	yes	characteristic
283	#380	—	2	2	no	—
284	C₄×C₄⋊C₄	—	2	1	no	—
285	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
286	#340	C₂²×C₁₃₇	1	2	yes	—
287	#336	C₂²×C₁₃₇	1	2	yes	—
288	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
289	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
290	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
291	C₂³×C₄	—	2	1	no	—
292	C₂×Q₁₆	—	2	4	no	—
293	C₂×Q₁₆	—	2	4	no	—
294	C₂×Q₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	2	2	no	—
296	C₂×SD₁₆	—	2	2	no	—
297	C₂×SD₁₆	D₄×C₁₃₇	1	4	yes	—
298	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
299	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
301	C₄⋊Q₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	#124	—	2	4	no	—
304	#124	—	4	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	—	2	4	no	—
307	C₄×Q₈	—	4	2	no	—
308	C₄×Q₈	D₄×C₁₃₇	1	4	yes	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	—	2	1	no	—
324	C₂×C₄²	—	2	2	no	—
325	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
326	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
327	#107	—	2	2	no	—
328	C₄⋊C₈	—	2	4	no	—
329	C₄⋊C₈	—	2	2	no	—
330	#103	—	2	2	no	—
331	#103	—	2	4	no	—
332	#103	—	2	4	no	—
333	#103	—	4	2	no	—
334	#102	—	2	4	no	—
335	#102	—	2	2	no	—
336	#97	—	2	2	no	—
337	#95	—	2	1	no	—
338	#95	—	2	2	no	—
339	#95	—	2	2	no	—
340	#95	—	2	2	no	—
341	#95	—	2	1	no	—
342	#95	D₄×C₁₃₇	1	1	yes	characteristic
343	#95	D₄×C₁₃₇	1	1	yes	characteristic
344	C₂⁴	—	2	1	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
355	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
356	C₂×Q₈	#41×C₁₃₇	1	2	yes	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	1	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
372	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
375	Q₁₆	—	4	8	no	—
376	SD₁₆	—	2	8	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	2	no	—
393	C₄⋊C₄	—	4	4	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	4	2	no	—
401	C₄²	—	2	4	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	4	4	no	—
405	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
406	C₄²	#41×C₁₃₇	1	2	yes	—
407	C₂³	—	2	2	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
413	Q₈	—	2	2	no	—
414	Q₈	—	2	2	no	—
415	Q₈	—	2	2	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	4	2	no	—
419	Q₈	—	4	2	no	—
420	Q₈	#121×C₁₃₇	1	4	yes	—
421	D₄	—	2	2	no	—
422	D₄	—	2	2	no	—
423	D₄	—	4	1	no	—
424	D₄	—	4	1	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	8	4	no	—
447	C₂×C₄	#121×C₁₃₇	1	2	yes	—
448	C₂×C₄	#121×C₁₃₇	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
451	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
453	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
454	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
455	C₈	—	2	2	no	—
456	C₈	—	4	2	no	—
457	C₈	—	4	1	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	8	1	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
464	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
465	C₂²	#548×C₁₃₇	1	1	yes	characteristic
466	C₂²	#575×C₁₃₇	1	2	yes	—
467	C₂²	#578×C₁₃₇	1	2	yes	—
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	4	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	8	1	no	—
475	C₄	—	8	2	no	—
476	C₄	—	8	2	no	—
477	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
478	C₄	#537×C₁₃₇	1	1	yes	characteristic
479	C₄	#547×C₁₃₇	1	1	yes	characteristic
480	C₄	#553×C₁₃₇	1	1	yes	characteristic
481	C₂	—	8	2	no	—
482	C₂	#2416×C₁₃₇	1	1	yes	characteristic
483	C₂	#3040×C₁₃₇	1	1	yes	characteristic
484	C₂	#3044×C₁₃₇	1	1	yes	characteristic
485	C₂	#3269×C₁₃₇	1	2	yes	—
486	C₂	#3377×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#462×C₁₃₇	C₂	semidirect
5. #2351×C₁₃₇	C₂	semidirect
6. #2068×C₁₃₇	C₂	semidirect
8. #1960×C₁₃₇	C₂	semidirect
9. #1940×C₁₃₇	C₂	semidirect
10. #1878×C₁₃₇	C₂	semidirect
243. #24743	C₁₃₇	direct (N × H)
244. C₁₃₇	#24743	direct (N × H)
246. C₂×#462	C₂×C₁₃₇	semidirect
249. #2351	C₂×C₁₃₇	semidirect
250. #2068	C₂×C₁₃₇	semidirect
252. #1960	C₂×C₁₃₇	semidirect
253. #1940	C₂×C₁₃₇	semidirect
254. #1878	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×#464×C₁₃₇	2
C₂×#462×C₁₃₇	2
C₂×#440×C₁₃₇	2
#2547×C₁₃₇	2
#2351×C₁₃₇	2
#2068×C₁₃₇	2
#2042×C₁₃₇	2
#1960×C₁₃₇	2
#1940×C₁₃₇	2
#1878×C₁₃₇	2
#1804×C₁₃₇	2
#24743	137

Subgroups — C137×#24743 #46874129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24743 #46874129