Subgroups — C₁₃₇×#25049 #46874435

Order 35072.

776 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#470×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2421×C₁₃₇	C₂	1	2	yes	maximal
4	#2074×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2036×C₁₃₇	C₂	1	2	yes	maximal
6	#1969×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1943×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1863×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1809×C₁₃₇	C₂	1	1	yes	maximal, characteristic
11	#1753×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
16	#470×C₁₃₇	—	2	4	no	—
17	#441×C₁₃₇	—	2	4	no	—
18	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	#400×C₁₃₇	—	2	2	no	—
29	#397×C₁₃₇	—	2	2	no	—
30	#391×C₁₃₇	C₂²	1	2	yes	—
31	#389×C₁₃₇	C₂²	1	2	yes	—
32	#388×C₁₃₇	—	2	2	no	—
33	#387×C₁₃₇	—	2	2	no	—
34	#382×C₁₃₇	C₂²	1	2	yes	—
35	#380×C₁₃₇	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
37	#340×C₁₃₇	C₂²	1	2	yes	—
38	#340×C₁₃₇	C₂²	1	2	yes	—
39	#336×C₁₃₇	C₂²	1	1	yes	characteristic
40	#336×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
44	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
45	C₂×Q₁₆×C₁₃₇	D₄	1	4	yes	—
46	C₂×M₁₆×C₁₃₇	—	2	2	no	—
47	C₂×M₁₆×C₁₃₇	—	2	2	no	—
48	C₂×M₁₆×C₁₃₇	—	2	2	no	—
49	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
54	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
55	C₄×Q₈×C₁₃₇	—	2	4	no	—
56	C₄×Q₈×C₁₃₇	—	4	2	no	—
57	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
66	C₂×C₄²×C₁₃₇	—	2	2	no	—
67	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
69	#107×C₁₃₇	—	2	2	no	—
70	#107×C₁₃₇	—	4	2	no	—
71	#106×C₁₃₇	—	2	2	no	—
72	#106×C₁₃₇	—	2	2	no	—
73	#103×C₁₃₇	—	2	4	no	—
74	#103×C₁₃₇	—	2	4	no	—
75	#103×C₁₃₇	—	2	4	no	—
76	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
77	#97×C₁₃₇	—	2	2	no	—
78	C₄×C₈×C₁₃₇	—	2	2	no	—
79	#95×C₁₃₇	—	2	2	no	—
80	#95×C₁₃₇	—	2	2	no	—
81	#95×C₁₃₇	—	2	2	no	—
82	#95×C₁₃₇	—	2	2	no	—
83	#95×C₁₃₇	—	2	2	no	—
84	#95×C₁₃₇	—	2	2	no	—
85	#95×C₁₃₇	D₄	1	2	yes	—
86	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Q₈×C₁₃₇	—	2	2	no	—
88	C₂×Q₈×C₁₃₇	—	2	2	no	—
89	C₂×Q₈×C₁₃₇	—	2	2	no	—
90	C₂×Q₈×C₁₃₇	—	2	2	no	—
91	C₂×Q₈×C₁₃₇	—	2	2	no	—
92	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₁₃₇	—	2	2	no	—
94	C₂²×C₄×C₁₃₇	—	2	1	no	—
95	C₂²×C₄×C₁₃₇	—	2	1	no	—
96	C₂²×C₄×C₁₃₇	—	2	1	no	—
97	C₂²×C₄×C₁₃₇	—	2	1	no	—
98	C₂²×C₄×C₁₃₇	—	2	1	no	—
99	C₂²×C₄×C₁₃₇	—	2	1	no	—
100	C₂²×C₄×C₁₃₇	—	2	2	no	—
101	C₂²×C₄×C₁₃₇	—	2	2	no	—
102	C₂²×C₄×C₁₃₇	—	2	2	no	—
103	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
107	Q₁₆×C₁₃₇	—	2	8	no	—
108	M₁₆×C₁₃₇	—	4	4	no	—
109	C₂×C₈×C₁₃₇	—	2	1	no	—
110	C₂×C₈×C₁₃₇	—	2	1	no	—
111	C₂×C₈×C₁₃₇	—	2	1	no	—
112	C₂×C₈×C₁₃₇	—	2	1	no	—
113	C₂×C₈×C₁₃₇	—	2	1	no	—
114	C₂×C₈×C₁₃₇	—	2	1	no	—
115	C₂×C₈×C₁₃₇	—	2	1	no	—
116	C₂×C₈×C₁₃₇	—	2	1	no	—
117	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₁₃₇	#41	1	2	yes	—
119	C₂×C₈×C₁₃₇	#41	1	2	yes	—
120	C₂×C₈×C₁₃₇	#41	1	2	yes	—
121	C₄⋊C₄×C₁₃₇	—	2	2	no	—
122	C₄⋊C₄×C₁₃₇	—	2	2	no	—
123	C₄⋊C₄×C₁₃₇	—	2	2	no	—
124	C₄⋊C₄×C₁₃₇	—	2	2	no	—
125	C₄⋊C₄×C₁₃₇	—	2	2	no	—
126	C₄⋊C₄×C₁₃₇	—	2	2	no	—
127	C₄⋊C₄×C₁₃₇	—	4	4	no	—
128	C₄⋊C₄×C₁₃₇	—	4	4	no	—
129	C₄⋊C₄×C₁₃₇	—	4	4	no	—
130	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
131	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
132	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
133	C₄²×C₁₃₇	—	2	1	no	—
134	C₄²×C₁₃₇	—	2	1	no	—
135	C₄²×C₁₃₇	—	2	1	no	—
136	C₄²×C₁₃₇	—	2	1	no	—
137	C₄²×C₁₃₇	—	4	4	no	—
138	C₂³×C₁₃₇	—	2	1	no	—
139	C₂³×C₁₃₇	—	2	1	no	—
140	C₂³×C₁₃₇	—	2	1	no	—
141	C₂³×C₁₃₇	—	2	2	no	—
142	C₂³×C₁₃₇	—	2	2	no	—
143	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₁₃₇	—	2	4	no	—
145	Q₈×C₁₃₇	—	2	4	no	—
146	Q₈×C₁₃₇	—	4	2	no	—
147	Q₈×C₁₃₇	—	4	2	no	—
148	C₂×C₄×C₁₃₇	—	2	1	no	—
149	C₂×C₄×C₁₃₇	—	2	2	no	—
150	C₂×C₄×C₁₃₇	—	2	4	no	—
151	C₂×C₄×C₁₃₇	—	2	1	no	—
152	C₂×C₄×C₁₃₇	—	2	2	no	—
153	C₂×C₄×C₁₃₇	—	2	1	no	—
154	C₂×C₄×C₁₃₇	—	2	2	no	—
155	C₂×C₄×C₁₃₇	—	2	2	no	—
156	C₂×C₄×C₁₃₇	—	2	2	no	—
157	C₂×C₄×C₁₃₇	—	2	2	no	—
158	C₂×C₄×C₁₃₇	—	4	2	no	—
159	C₂×C₄×C₁₃₇	—	4	2	no	—
160	C₂×C₄×C₁₃₇	—	4	1	no	—
161	C₂×C₄×C₁₃₇	—	4	1	no	—
162	C₂×C₄×C₁₃₇	—	4	1	no	—
163	C₂×C₄×C₁₃₇	—	4	1	no	—
164	C₂×C₄×C₁₃₇	—	4	2	no	—
165	C₂×C₄×C₁₃₇	—	4	2	no	—
166	C₂×C₄×C₁₃₇	—	4	2	no	—
167	C₂×C₄×C₁₃₇	—	4	2	no	—
168	C₂×C₄×C₁₃₇	—	4	2	no	—
169	C₂×C₄×C₁₃₇	—	4	2	no	—
170	C₂×C₄×C₁₃₇	—	4	2	no	—
171	C₂×C₄×C₁₃₇	—	4	2	no	—
172	C₂×C₄×C₁₃₇	—	4	2	no	—
173	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
179	C₈×C₁₃₇	—	2	2	no	—
180	C₈×C₁₃₇	—	4	1	no	—
181	C₈×C₁₃₇	—	4	1	no	—
182	C₈×C₁₃₇	#123	1	4	yes	—
183	C₂²×C₁₃₇	—	4	2	no	—
184	C₂²×C₁₃₇	—	4	1	no	—
185	C₂²×C₁₃₇	—	4	1	no	—
186	C₂²×C₁₃₇	—	4	1	no	—
187	C₂²×C₁₃₇	—	4	2	no	—
188	C₂²×C₁₃₇	#532	1	1	yes	characteristic
189	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
190	C₂²×C₁₃₇	#547	1	1	yes	characteristic
191	C₂²×C₁₃₇	#575	1	2	yes	—
192	C₂²×C₁₃₇	#578	1	2	yes	—
193	C₄×C₁₃₇	—	4	2	no	—
194	C₄×C₁₃₇	—	4	1	no	—
195	C₄×C₁₃₇	—	4	4	no	—
196	C₄×C₁₃₇	—	4	2	no	—
197	C₄×C₁₃₇	—	8	2	no	—
198	C₄×C₁₃₇	—	8	2	no	—
199	C₄×C₁₃₇	—	8	2	no	—
200	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
201	C₄×C₁₃₇	#542	1	1	yes	characteristic
202	C₄×C₁₃₇	#547	1	1	yes	characteristic
203	C₄×C₁₃₇	#553	1	1	yes	characteristic
204	C₂×C₁₃₇	—	8	1	no	—
205	C₂×C₁₃₇	#2422	1	1	yes	characteristic
206	C₂×C₁₃₇	#3177	1	1	yes	characteristic
207	C₂×C₁₃₇	#3183	1	1	yes	characteristic
208	C₂×C₁₃₇	#3277	1	2	yes	—
209	C₂×C₁₃₇	#3349	1	2	yes	—
210	#25049	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₁₃₇	#25049	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#470	C₂×C₁₃₇	1	1	yes	characteristic
213	C₂×#441	C₂×C₁₃₇	1	1	yes	characteristic, complemented
214	#2421	C₂×C₁₃₇	1	2	yes	—
215	#2074	C₂×C₁₃₇	1	2	yes	complemented
216	#2036	C₂×C₁₃₇	1	2	yes	—
217	#1969	C₂×C₁₃₇	1	1	yes	characteristic, complemented
218	#1943	C₂×C₁₃₇	1	1	yes	characteristic, complemented
219	#1863	C₂×C₁₃₇	1	2	yes	complemented
220	#1846	C₂×C₁₃₇	1	1	yes	characteristic, complemented
221	#1809	C₂×C₁₃₇	1	1	yes	characteristic
222	#1753	C₂×C₁₃₇	1	1	yes	characteristic
223	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
224	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
225	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
226	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
227	#470	—	2	4	no	—
228	#441	—	2	4	no	—
229	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
230	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
231	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
232	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
233	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
234	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
235	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
236	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
237	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
239	#400	—	2	2	no	—
240	#397	—	2	2	no	—
241	#391	C₂²×C₁₃₇	1	2	yes	—
242	#389	C₂²×C₁₃₇	1	2	yes	—
243	#388	—	2	2	no	—
244	#387	—	2	2	no	—
245	#382	C₂²×C₁₃₇	1	2	yes	—
246	#380	C₂²×C₁₃₇	1	2	yes	—
247	C₄×C₂²⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
248	#340	C₂²×C₁₃₇	1	2	yes	—
249	#340	C₂²×C₁₃₇	1	2	yes	—
250	#336	C₂²×C₁₃₇	1	1	yes	characteristic
251	#336	C₂²×C₁₃₇	1	1	yes	characteristic
252	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
253	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
254	C₂×Q₁₆	—	2	2	no	—
255	C₂×Q₁₆	—	2	2	no	—
256	C₂×Q₁₆	D₄×C₁₃₇	1	4	yes	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
264	C₂²⋊Q₈	—	2	4	no	—
265	C₂²⋊Q₈	—	4	2	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
280	#107	—	2	2	no	—
281	#107	—	4	2	no	—
282	#106	—	2	2	no	—
283	#106	—	2	2	no	—
284	#103	—	2	4	no	—
285	#103	—	2	4	no	—
286	#103	—	2	4	no	—
287	C₂²⋊C₈	—	2	2	no	—
288	#97	—	2	2	no	—
289	C₄×C₈	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	D₄×C₁₃₇	1	2	yes	—
297	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
317	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
318	Q₁₆	—	2	8	no	—
319	M₁₆	—	4	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
329	C₂×C₈	#41×C₁₃₇	1	2	yes	—
330	C₂×C₈	#41×C₁₃₇	1	2	yes	—
331	C₂×C₈	#41×C₁₃₇	1	2	yes	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	4	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	4	4	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	2	1	no	—
351	C₂³	—	2	1	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
355	Q₈	—	2	4	no	—
356	Q₈	—	2	4	no	—
357	Q₈	—	4	2	no	—
358	Q₈	—	4	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
388	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
389	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
390	C₈	—	2	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	1	no	—
393	C₈	#123×C₁₃₇	1	4	yes	—
394	C₂²	—	4	2	no	—
395	C₂²	—	4	1	no	—
396	C₂²	—	4	1	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	2	no	—
399	C₂²	#532×C₁₃₇	1	1	yes	characteristic
400	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
401	C₂²	#547×C₁₃₇	1	1	yes	characteristic
402	C₂²	#575×C₁₃₇	1	2	yes	—
403	C₂²	#578×C₁₃₇	1	2	yes	—
404	C₄	—	4	2	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	4	no	—
407	C₄	—	4	2	no	—
408	C₄	—	8	2	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	2	no	—
411	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
412	C₄	#542×C₁₃₇	1	1	yes	characteristic
413	C₄	#547×C₁₃₇	1	1	yes	characteristic
414	C₄	#553×C₁₃₇	1	1	yes	characteristic
415	C₂	—	8	1	no	—
416	C₂	#2422×C₁₃₇	1	1	yes	characteristic
417	C₂	#3177×C₁₃₇	1	1	yes	characteristic
418	C₂	#3183×C₁₃₇	1	1	yes	characteristic
419	C₂	#3277×C₁₃₇	1	2	yes	—
420	C₂	#3349×C₁₃₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#441×C₁₃₇	C₂	semidirect
4. #2074×C₁₃₇	C₂	semidirect
6. #1969×C₁₃₇	C₂	semidirect
7. #1943×C₁₃₇	C₂	semidirect
8. #1863×C₁₃₇	C₂	semidirect
9. #1846×C₁₃₇	C₂	semidirect
210. #25049	C₁₃₇	direct (N × H)
211. C₁₃₇	#25049	direct (N × H)
213. C₂×#441	C₂×C₁₃₇	semidirect
215. #2074	C₂×C₁₃₇	semidirect
217. #1969	C₂×C₁₃₇	semidirect
218. #1943	C₂×C₁₃₇	semidirect
219. #1863	C₂×C₁₃₇	semidirect
220. #1846	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×#470×C₁₃₇	2
C₂×#441×C₁₃₇	2
#2421×C₁₃₇	2
#2074×C₁₃₇	2
#2036×C₁₃₇	2
#1969×C₁₃₇	2
#1943×C₁₃₇	2
#1863×C₁₃₇	2
#1846×C₁₃₇	2
#1809×C₁₃₇	2
#1753×C₁₃₇	2
#25049	137

Subgroups — C137×#25049 #46874435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25049 #46874435