Subgroups — C₁₃₇×#24745 #46874131

Order 35072.

1008 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#461×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#442×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2547×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2349×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2066×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2042×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1960×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1940×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1877×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1804×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×D₈×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×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	#463×C₁₃₇	—	2	8	no	—
20	#461×C₁₃₇	—	2	4	no	—
21	#442×C₁₃₇	—	2	4	no	—
22	C₂×#106×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	2	yes	—
28	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
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	C₂²	1	2	yes	—
36	#385×C₁₃₇	—	2	2	no	—
37	#382×C₁₃₇	C₂²	1	1	yes	characteristic
38	#380×C₁₃₇	—	2	2	no	—
39	#380×C₁₃₇	—	2	1	no	—
40	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂×#95×C₁₃₇	—	2	1	no	—
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#336×C₁₃₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₁₃₇	—	2	1	no	—
48	C₂³×C₄×C₁₃₇	—	2	1	no	—
49	C₂³×C₄×C₁₃₇	—	2	2	no	—
50	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
53	C₂×D₈×C₁₃₇	—	2	2	no	—
54	C₂×D₈×C₁₃₇	—	2	2	no	—
55	C₂×D₈×C₁₃₇	D₄	1	4	yes	—
56	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
61	#124×C₁₃₇	—	2	4	no	—
62	#124×C₁₃₇	—	4	2	no	—
63	C₄×D₄×C₁₃₇	—	2	4	no	—
64	C₄×D₄×C₁₃₇	—	2	2	no	—
65	C₄×D₄×C₁₃₇	—	2	2	no	—
66	C₄×D₄×C₁₃₇	—	2	2	no	—
67	C₄×D₄×C₁₃₇	—	4	2	no	—
68	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	2	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	1	no	—
77	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
83	#106×C₁₃₇	—	2	2	no	—
84	C₄⋊C₈×C₁₃₇	—	2	4	no	—
85	C₄⋊C₈×C₁₃₇	—	2	2	no	—
86	#103×C₁₃₇	—	2	4	no	—
87	#103×C₁₃₇	—	2	4	no	—
88	#103×C₁₃₇	—	4	2	no	—
89	#102×C₁₃₇	—	2	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	C₂⁴×C₁₃₇	—	2	1	no	—
102	C₂×Q₈×C₁₃₇	—	2	2	no	—
103	C₂×Q₈×C₁₃₇	—	2	2	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×D₄×C₁₃₇	—	2	1	no	—
109	C₂×D₄×C₁₃₇	—	2	1	no	—
110	C₂×D₄×C₁₃₇	—	2	1	no	—
111	C₂×D₄×C₁₃₇	—	2	1	no	—
112	C₂×D₄×C₁₃₇	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
117	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₁₃₇	#41	1	2	yes	—
120	C₂²×C₄×C₁₃₇	—	2	1	no	—
121	C₂²×C₄×C₁₃₇	—	2	4	no	—
122	C₂²×C₄×C₁₃₇	—	2	1	no	—
123	C₂²×C₄×C₁₃₇	—	2	2	no	—
124	C₂²×C₄×C₁₃₇	—	2	2	no	—
125	C₂²×C₄×C₁₃₇	—	4	4	no	—
126	C₂²×C₄×C₁₃₇	—	4	4	no	—
127	C₂²×C₄×C₁₃₇	—	4	4	no	—
128	C₂²×C₄×C₁₃₇	—	4	2	no	—
129	C₂²×C₄×C₁₃₇	—	8	2	no	—
130	C₂²×C₄×C₁₃₇	—	8	2	no	—
131	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₁₃₇	—	4	8	no	—
137	D₈×C₁₃₇	—	2	8	no	—
138	C₂×C₈×C₁₃₇	—	2	1	no	—
139	C₂×C₈×C₁₃₇	—	2	1	no	—
140	C₂×C₈×C₁₃₇	—	2	1	no	—
141	C₂×C₈×C₁₃₇	—	2	2	no	—
142	C₂×C₈×C₁₃₇	—	2	2	no	—
143	C₂×C₈×C₁₃₇	—	2	2	no	—
144	C₂×C₈×C₁₃₇	—	2	1	no	—
145	C₂×C₈×C₁₃₇	—	2	1	no	—
146	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₁₃₇	—	2	1	no	—
148	C₄⋊C₄×C₁₃₇	—	2	2	no	—
149	C₄⋊C₄×C₁₃₇	—	2	1	no	—
150	C₄⋊C₄×C₁₃₇	—	2	1	no	—
151	C₄⋊C₄×C₁₃₇	—	2	2	no	—
152	C₄⋊C₄×C₁₃₇	—	4	2	no	—
153	C₄⋊C₄×C₁₃₇	—	4	2	no	—
154	C₄⋊C₄×C₁₃₇	—	4	2	no	—
155	C₄⋊C₄×C₁₃₇	—	8	2	no	—
156	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
158	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
159	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
160	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
161	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
162	C₄²×C₁₃₇	—	2	2	no	—
163	C₄²×C₁₃₇	—	2	2	no	—
164	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₄²×C₁₃₇	#41	1	2	yes	—
166	C₂³×C₁₃₇	—	2	2	no	—
167	C₂³×C₁₃₇	—	2	2	no	—
168	C₂³×C₁₃₇	—	2	2	no	—
169	C₂³×C₁₃₇	—	2	2	no	—
170	C₂³×C₁₃₇	—	4	2	no	—
171	C₂³×C₁₃₇	—	4	2	no	—
172	C₂³×C₁₃₇	—	4	1	no	—
173	C₂³×C₁₃₇	—	4	1	no	—
174	C₂³×C₁₃₇	—	4	2	no	—
175	C₂³×C₁₃₇	—	4	2	no	—
176	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₁₃₇	—	4	2	no	—
178	Q₈×C₁₃₇	—	4	2	no	—
179	Q₈×C₁₃₇	—	4	2	no	—
180	Q₈×C₁₃₇	—	4	2	no	—
181	D₄×C₁₃₇	—	2	2	no	—
182	D₄×C₁₃₇	—	2	2	no	—
183	D₄×C₁₃₇	—	2	2	no	—
184	D₄×C₁₃₇	—	2	2	no	—
185	D₄×C₁₃₇	—	2	2	no	—
186	D₄×C₁₃₇	—	4	1	no	—
187	D₄×C₁₃₇	—	4	1	no	—
188	D₄×C₁₃₇	#121	1	4	yes	—
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	—	2	2	no	—
194	C₂×C₄×C₁₃₇	—	2	2	no	—
195	C₂×C₄×C₁₃₇	—	2	2	no	—
196	C₂×C₄×C₁₃₇	—	2	2	no	—
197	C₂×C₄×C₁₃₇	—	2	2	no	—
198	C₂×C₄×C₁₃₇	—	4	2	no	—
199	C₂×C₄×C₁₃₇	—	4	1	no	—
200	C₂×C₄×C₁₃₇	—	4	2	no	—
201	C₂×C₄×C₁₃₇	—	4	2	no	—
202	C₂×C₄×C₁₃₇	—	4	2	no	—
203	C₂×C₄×C₁₃₇	—	4	2	no	—
204	C₂×C₄×C₁₃₇	—	4	2	no	—
205	C₂×C₄×C₁₃₇	—	4	2	no	—
206	C₂×C₄×C₁₃₇	—	4	2	no	—
207	C₂×C₄×C₁₃₇	—	8	4	no	—
208	C₂×C₄×C₁₃₇	—	8	4	no	—
209	C₂×C₄×C₁₃₇	—	8	8	no	—
210	C₂×C₄×C₁₃₇	#121	1	2	yes	—
211	C₂×C₄×C₁₃₇	#121	1	2	yes	—
212	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
218	C₈×C₁₃₇	—	2	2	no	—
219	C₈×C₁₃₇	—	4	2	no	—
220	C₈×C₁₃₇	—	4	1	no	—
221	C₂²×C₁₃₇	—	2	4	no	—
222	C₂²×C₁₃₇	—	4	4	no	—
223	C₂²×C₁₃₇	—	4	4	no	—
224	C₂²×C₁₃₇	—	4	2	no	—
225	C₂²×C₁₃₇	—	4	2	no	—
226	C₂²×C₁₃₇	—	4	2	no	—
227	C₂²×C₁₃₇	—	4	4	no	—
228	C₂²×C₁₃₇	—	8	2	no	—
229	C₂²×C₁₃₇	—	8	1	no	—
230	C₂²×C₁₃₇	—	8	1	no	—
231	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
232	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
233	C₂²×C₁₃₇	#548	1	1	yes	characteristic
234	C₂²×C₁₃₇	#575	1	2	yes	—
235	C₂²×C₁₃₇	#577	1	2	yes	—
236	C₄×C₁₃₇	—	2	2	no	—
237	C₄×C₁₃₇	—	2	2	no	—
238	C₄×C₁₃₇	—	4	2	no	—
239	C₄×C₁₃₇	—	4	2	no	—
240	C₄×C₁₃₇	—	4	2	no	—
241	C₄×C₁₃₇	—	8	2	no	—
242	C₄×C₁₃₇	—	8	2	no	—
243	C₄×C₁₃₇	—	8	1	no	—
244	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
245	C₄×C₁₃₇	#537	1	1	yes	characteristic
246	C₄×C₁₃₇	#547	1	1	yes	characteristic
247	C₄×C₁₃₇	#553	1	1	yes	characteristic
248	C₂×C₁₃₇	—	4	4	no	—
249	C₂×C₁₃₇	—	8	2	no	—
250	C₂×C₁₃₇	#2416	1	1	yes	characteristic
251	C₂×C₁₃₇	#3040	1	1	yes	characteristic
252	C₂×C₁₃₇	#3042	1	1	yes	characteristic
253	C₂×C₁₃₇	#3267	1	2	yes	—
254	C₂×C₁₃₇	#3376	1	2	yes	—
255	#24745	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₁₃₇	#24745	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#463	C₂×C₁₃₇	1	2	yes	—
258	C₂×#461	C₂×C₁₃₇	1	1	yes	characteristic, complemented
259	C₂×#442	C₂×C₁₃₇	1	1	yes	characteristic
260	#2547	C₂×C₁₃₇	1	1	yes	characteristic, complemented
261	#2349	C₂×C₁₃₇	1	1	yes	characteristic, complemented
262	#2066	C₂×C₁₃₇	1	2	yes	complemented
263	#2042	C₂×C₁₃₇	1	2	yes	complemented
264	#1960	C₂×C₁₃₇	1	1	yes	characteristic, complemented
265	#1940	C₂×C₁₃₇	1	1	yes	characteristic, complemented
266	#1877	C₂×C₁₃₇	1	2	yes	complemented
267	#1804	C₂×C₁₃₇	1	1	yes	characteristic, complemented
268	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
269	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
271	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
272	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
275	#463	—	2	8	no	—
276	#461	—	2	4	no	—
277	#442	—	2	4	no	—
278	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
281	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
282	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
283	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
284	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
285	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
286	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
287	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
288	#397	—	2	1	no	—
289	#397	—	2	1	no	—
290	#391	C₂²×C₁₃₇	1	2	yes	—
291	#389	C₂²×C₁₃₇	1	2	yes	—
292	#385	—	2	2	no	—
293	#382	C₂²×C₁₃₇	1	1	yes	characteristic
294	#380	—	2	2	no	—
295	#380	—	2	1	no	—
296	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
297	C₂×#95	—	2	1	no	—
298	#340	C₂²×C₁₃₇	1	2	yes	—
299	#336	C₂²×C₁₃₇	1	2	yes	—
300	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
301	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
302	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
303	C₂³×C₄	—	2	1	no	—
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	—	2	2	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×SD₁₆	—	2	4	no	—
309	C₂×D₈	—	2	2	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×D₈	D₄×C₁₃₇	1	4	yes	—
312	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
313	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
314	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	—	2	2	no	—
317	#124	—	2	4	no	—
318	#124	—	4	2	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	2	no	—
321	C₄×D₄	—	2	2	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	—	4	2	no	—
324	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
338	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
339	#106	—	2	2	no	—
340	C₄⋊C₈	—	2	4	no	—
341	C₄⋊C₈	—	2	2	no	—
342	#103	—	2	4	no	—
343	#103	—	2	4	no	—
344	#103	—	4	2	no	—
345	#102	—	2	2	no	—
346	#102	—	2	4	no	—
347	#102	—	2	2	no	—
348	#97	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	—	2	2	no	—
351	#95	—	2	2	no	—
352	#95	—	2	2	no	—
353	#95	—	2	1	no	—
354	#95	D₄×C₁₃₇	1	1	yes	characteristic
355	#95	D₄×C₁₃₇	1	1	yes	characteristic
356	C₂⁴	—	2	1	no	—
357	C₂⁴	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
373	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
374	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
375	C₂×D₄	#41×C₁₃₇	1	2	yes	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	—	4	4	no	—
384	C₂²×C₄	—	4	2	no	—
385	C₂²×C₄	—	8	2	no	—
386	C₂²×C₄	—	8	2	no	—
387	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
391	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
392	SD₁₆	—	4	8	no	—
393	D₈	—	2	8	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	4	2	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	—	8	2	no	—
412	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	4	4	no	—
416	C₂²⋊C₄	—	4	4	no	—
417	C₂²⋊C₄	—	4	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	2	no	—
420	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
421	C₄²	#41×C₁₃₇	1	2	yes	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
433	Q₈	—	4	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	2	no	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	4	1	no	—
443	D₄	—	4	1	no	—
444	D₄	#121×C₁₃₇	1	4	yes	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	8	4	no	—
464	C₂×C₄	—	8	4	no	—
465	C₂×C₄	—	8	8	no	—
466	C₂×C₄	#121×C₁₃₇	1	2	yes	—
467	C₂×C₄	#121×C₁₃₇	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
472	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
473	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	1	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	8	1	no	—
485	C₂²	—	8	1	no	—
486	C₂²	—	8	2	no	—
487	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
488	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
489	C₂²	#548×C₁₃₇	1	1	yes	characteristic
490	C₂²	#575×C₁₃₇	1	2	yes	—
491	C₂²	#577×C₁₃₇	1	2	yes	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	2	no	—
500	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
501	C₄	#537×C₁₃₇	1	1	yes	characteristic
502	C₄	#547×C₁₃₇	1	1	yes	characteristic
503	C₄	#553×C₁₃₇	1	1	yes	characteristic
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	#2416×C₁₃₇	1	1	yes	characteristic
507	C₂	#3040×C₁₃₇	1	1	yes	characteristic
508	C₂	#3042×C₁₃₇	1	1	yes	characteristic
509	C₂	#3267×C₁₃₇	1	2	yes	—
510	C₂	#3376×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
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×#461×C₁₃₇	C₂	semidirect
4. #2547×C₁₃₇	C₂	semidirect
5. #2349×C₁₃₇	C₂	semidirect
6. #2066×C₁₃₇	C₂	semidirect
7. #2042×C₁₃₇	C₂	semidirect
8. #1960×C₁₃₇	C₂	semidirect
9. #1940×C₁₃₇	C₂	semidirect
10. #1877×C₁₃₇	C₂	semidirect
11. #1804×C₁₃₇	C₂	semidirect
255. #24745	C₁₃₇	direct (N × H)
256. C₁₃₇	#24745	direct (N × H)
258. C₂×#461	C₂×C₁₃₇	semidirect
260. #2547	C₂×C₁₃₇	semidirect
261. #2349	C₂×C₁₃₇	semidirect
262. #2066	C₂×C₁₃₇	semidirect
263. #2042	C₂×C₁₃₇	semidirect
264. #1960	C₂×C₁₃₇	semidirect
265. #1940	C₂×C₁₃₇	semidirect
266. #1877	C₂×C₁₃₇	semidirect
267. #1804	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×#463×C₁₃₇	2
C₂×#461×C₁₃₇	2
C₂×#442×C₁₃₇	2
#2547×C₁₃₇	2
#2349×C₁₃₇	2
#2066×C₁₃₇	2
#2042×C₁₃₇	2
#1960×C₁₃₇	2
#1940×C₁₃₇	2
#1877×C₁₃₇	2
#1804×C₁₃₇	2
#24745	137

Subgroups — C137×#24745 #46874131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24745 #46874131