Subgroups — C₁₃₇×#19166 #46868552

Order 35072.

758 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3138×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3101×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#459×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	D₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1701×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1697×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1681×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1677×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#1565×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1550×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#1545×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₃₇	—	2	1	no	—
13	C₂²×M₁₆×C₁₃₇	—	2	1	no	—
14	#533×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	#490×C₁₃₇	—	2	2	no	—
18	#486×C₁₃₇	C₂²	1	2	yes	complemented
19	#486×C₁₃₇	C₂²	1	2	yes	complemented
20	#483×C₁₃₇	—	2	1	no	—
21	#476×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
22	#476×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
23	#461×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
24	#461×C₁₃₇	C₂²	1	1	yes	characteristic, complemented
25	#459×C₁₃₇	—	2	1	no	—
26	#459×C₁₃₇	—	2	1	no	—
27	#459×C₁₃₇	C₂²	1	2	yes	—
28	#459×C₁₃₇	C₂²	1	2	yes	—
29	#450×C₁₃₇	—	2	1	no	—
30	C₄×SD₁₆×C₁₃₇	—	2	1	no	—
31	C₄×SD₁₆×C₁₃₇	—	2	1	no	—
32	#436×C₁₃₇	C₂²	1	2	yes	complemented
33	#435×C₁₃₇	—	2	2	no	—
34	C₈×D₄×C₁₃₇	C₂²	1	2	yes	complemented
35	#431×C₁₃₇	C₂²	1	1	yes	characteristic
36	#423×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
38	#418×C₁₃₇	—	2	1	no	—
39	#418×C₁₃₇	—	2	1	no	—
40	C₂×#102×C₁₃₇	—	2	1	no	—
41	#407×C₁₃₇	—	2	1	no	—
42	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
43	#335×C₁₃₇	C₂²	1	2	yes	complemented
44	#334×C₁₃₇	C₂²	1	2	yes	complemented
45	#333×C₁₃₇	C₂²	1	2	yes	complemented
46	#332×C₁₃₇	C₂²	1	2	yes	complemented
47	#331×C₁₃₇	C₂²	1	2	yes	complemented
48	#331×C₁₃₇	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₁₃₇	—	2	1	no	—
50	C₂²×D₄×C₁₃₇	—	2	1	no	—
51	C₂²×D₄×C₁₃₇	—	2	1	no	—
52	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₁₃₇	—	2	1	no	—
54	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
55	C₂×SD₁₆×C₁₃₇	—	2	1	no	—
56	C₂×D₈×C₁₃₇	—	2	2	no	—
57	C₂×D₈×C₁₃₇	—	2	2	no	—
58	C₂×D₈×C₁₃₇	—	4	4	no	—
59	C₂×M₁₆×C₁₃₇	—	2	1	no	—
60	C₂×M₁₆×C₁₃₇	—	2	2	no	—
61	C₂×M₁₆×C₁₃₇	—	4	4	no	—
62	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
63	C₂×M₁₆×C₁₃₇	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₁₃₇	—	2	1	no	—
65	C₂²×C₈×C₁₃₇	—	2	2	no	—
66	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
68	#127×C₁₃₇	—	2	1	no	—
69	#127×C₁₃₇	—	2	1	no	—
70	#127×C₁₃₇	—	2	1	no	—
71	#127×C₁₃₇	C₂³	1	2	yes	—
72	#125×C₁₃₇	—	2	1	no	—
73	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
74	C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
75	C₄×Q₈×C₁₃₇	—	2	1	no	—
76	C₄×Q₈×C₁₃₇	—	2	1	no	—
77	C₄×D₄×C₁₃₇	—	2	1	no	—
78	C₄×D₄×C₁₃₇	—	2	1	no	—
79	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
80	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
81	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
83	#117×C₁₃₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
86	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
87	#106×C₁₃₇	—	2	1	no	—
88	#106×C₁₃₇	—	2	1	no	—
89	C₄⋊C₈×C₁₃₇	—	2	1	no	—
90	C₄⋊C₈×C₁₃₇	—	2	1	no	—
91	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
92	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
93	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
95	#103×C₁₃₇	—	2	1	no	—
96	#103×C₁₃₇	—	2	1	no	—
97	#102×C₁₃₇	—	2	1	no	—
98	#102×C₁₃₇	—	2	1	no	—
99	#102×C₁₃₇	—	2	2	no	—
100	#102×C₁₃₇	—	2	2	no	—
101	#102×C₁₃₇	—	2	2	no	—
102	#102×C₁₃₇	—	2	2	no	—
103	#102×C₁₃₇	—	2	2	no	—
104	#102×C₁₃₇	—	2	2	no	—
105	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
106	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
107	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
108	#97×C₁₃₇	D₄	1	2	yes	—
109	C₄×C₈×C₁₃₇	—	2	1	no	—
110	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
111	C₂⁴×C₁₃₇	—	2	1	no	—
112	C₂⁴×C₁₃₇	—	4	1	no	—
113	C₂×Q₈×C₁₃₇	—	2	1	no	—
114	C₂×Q₈×C₁₃₇	—	2	1	no	—
115	C₂×D₄×C₁₃₇	—	2	2	no	—
116	C₂×D₄×C₁₃₇	—	2	1	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	2	2	no	—
119	C₂×D₄×C₁₃₇	—	2	2	no	—
120	C₂×D₄×C₁₃₇	—	4	1	no	—
121	C₂×D₄×C₁₃₇	—	4	2	no	—
122	C₂×D₄×C₁₃₇	—	4	2	no	—
123	C₂×D₄×C₁₃₇	—	4	1	no	—
124	C₂×D₄×C₁₃₇	—	4	1	no	—
125	C₂×D₄×C₁₃₇	—	4	1	no	—
126	C₂×D₄×C₁₃₇	—	4	2	no	—
127	C₂×D₄×C₁₃₇	—	4	2	no	—
128	C₂×D₄×C₁₃₇	—	8	2	no	—
129	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	1	no	—
134	C₂²×C₄×C₁₃₇	—	2	1	no	—
135	C₂²×C₄×C₁₃₇	—	2	1	no	—
136	C₂²×C₄×C₁₃₇	—	4	2	no	—
137	C₂²×C₄×C₁₃₇	—	4	2	no	—
138	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
141	SD₁₆×C₁₃₇	—	4	2	no	—
142	SD₁₆×C₁₃₇	—	4	2	no	—
143	D₈×C₁₃₇	—	4	4	no	—
144	D₈×C₁₃₇	—	4	4	no	—
145	M₁₆×C₁₃₇	—	2	2	no	—
146	M₁₆×C₁₃₇	—	2	2	no	—
147	M₁₆×C₁₃₇	—	4	4	no	—
148	M₁₆×C₁₃₇	—	4	2	no	—
149	C₂×C₈×C₁₃₇	—	2	1	no	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	2	no	—
152	C₂×C₈×C₁₃₇	—	2	2	no	—
153	C₂×C₈×C₁₃₇	—	4	4	no	—
154	C₂×C₈×C₁₃₇	—	4	2	no	—
155	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
156	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₁₃₇	—	2	1	no	—
158	C₄⋊C₄×C₁₃₇	—	2	1	no	—
159	C₄⋊C₄×C₁₃₇	—	2	1	no	—
160	C₄⋊C₄×C₁₃₇	—	2	1	no	—
161	C₄⋊C₄×C₁₃₇	—	2	1	no	—
162	C₄⋊C₄×C₁₃₇	—	4	1	no	—
163	C₄⋊C₄×C₁₃₇	—	4	1	no	—
164	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
166	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
167	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
168	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
169	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
170	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
171	C₄²×C₁₃₇	—	4	1	no	—
172	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₁₃₇	#41	1	1	yes	characteristic
175	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₁₃₇	—	2	1	no	—
177	C₂³×C₁₃₇	—	2	1	no	—
178	C₂³×C₁₃₇	—	4	2	no	—
179	C₂³×C₁₃₇	—	4	2	no	—
180	C₂³×C₁₃₇	—	4	2	no	—
181	C₂³×C₁₃₇	—	8	1	no	—
182	C₂³×C₁₃₇	—	8	2	no	—
183	C₂³×C₁₃₇	—	8	2	no	—
184	C₂³×C₁₃₇	—	8	2	no	—
185	C₂³×C₁₃₇	#121	1	1	yes	characteristic
186	Q₈×C₁₃₇	—	4	1	no	—
187	Q₈×C₁₃₇	—	4	1	no	—
188	Q₈×C₁₃₇	—	8	1	no	—
189	D₄×C₁₃₇	—	2	1	no	—
190	D₄×C₁₃₇	—	2	1	no	—
191	D₄×C₁₃₇	—	2	1	no	—
192	D₄×C₁₃₇	—	4	2	no	—
193	D₄×C₁₃₇	—	4	2	no	—
194	D₄×C₁₃₇	—	4	2	no	—
195	D₄×C₁₃₇	—	4	2	no	—
196	D₄×C₁₃₇	—	4	2	no	—
197	D₄×C₁₃₇	—	8	2	no	—
198	D₄×C₁₃₇	—	8	2	no	—
199	D₄×C₁₃₇	—	8	2	no	—
200	D₄×C₁₃₇	—	8	2	no	—
201	D₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
202	C₂×C₄×C₁₃₇	—	2	1	no	—
203	C₂×C₄×C₁₃₇	—	2	1	no	—
204	C₂×C₄×C₁₃₇	—	2	1	no	—
205	C₂×C₄×C₁₃₇	—	2	1	no	—
206	C₂×C₄×C₁₃₇	—	4	1	no	—
207	C₂×C₄×C₁₃₇	—	4	2	no	—
208	C₂×C₄×C₁₃₇	—	4	1	no	—
209	C₂×C₄×C₁₃₇	—	4	1	no	—
210	C₂×C₄×C₁₃₇	—	4	1	no	—
211	C₂×C₄×C₁₃₇	—	4	1	no	—
212	C₂×C₄×C₁₃₇	—	4	2	no	—
213	C₂×C₄×C₁₃₇	—	4	2	no	—
214	C₂×C₄×C₁₃₇	—	4	1	no	—
215	C₂×C₄×C₁₃₇	—	4	1	no	—
216	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
217	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
218	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
219	C₂×C₄×C₁₃₇	#135	1	2	yes	—
220	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
221	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
224	C₈×C₁₃₇	—	2	2	no	—
225	C₈×C₁₃₇	—	2	2	no	—
226	C₈×C₁₃₇	—	4	2	no	—
227	C₈×C₁₃₇	—	4	2	no	—
228	C₈×C₁₃₇	—	4	1	no	—
229	C₈×C₁₃₇	—	4	1	no	—
230	C₂²×C₁₃₇	—	2	1	no	—
231	C₂²×C₁₃₇	—	2	1	no	—
232	C₂²×C₁₃₇	—	2	2	no	—
233	C₂²×C₁₃₇	—	4	1	no	—
234	C₂²×C₁₃₇	—	4	1	no	—
235	C₂²×C₁₃₇	—	4	1	no	—
236	C₂²×C₁₃₇	—	4	1	no	—
237	C₂²×C₁₃₇	—	4	1	no	—
238	C₂²×C₁₃₇	—	8	4	no	—
239	C₂²×C₁₃₇	—	8	2	no	—
240	C₂²×C₁₃₇	—	8	2	no	—
241	C₂²×C₁₃₇	—	8	2	no	—
242	C₂²×C₁₃₇	—	16	4	no	—
243	C₂²×C₁₃₇	#463	1	2	yes	—
244	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
245	C₄×C₁₃₇	—	2	1	no	—
246	C₄×C₁₃₇	—	2	1	no	—
247	C₄×C₁₃₇	—	2	1	no	—
248	C₄×C₁₃₇	—	4	1	no	—
249	C₄×C₁₃₇	—	4	1	no	—
250	C₄×C₁₃₇	—	8	1	no	—
251	C₄×C₁₃₇	—	8	1	no	—
252	C₄×C₁₃₇	#463	1	2	yes	—
253	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
254	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₄×C₁₃₇	#575	1	1	yes	characteristic
257	C₂×C₁₃₇	—	2	2	no	—
258	C₂×C₁₃₇	—	4	2	no	—
259	C₂×C₁₃₇	—	8	1	no	—
260	C₂×C₁₃₇	—	16	2	no	—
261	C₂×C₁₃₇	C₂×#463	1	1	yes	characteristic
262	C₂×C₁₃₇	#3055	1	1	yes	characteristic
263	C₂×C₁₃₇	#3068	1	1	yes	characteristic
264	#19166	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
265	C₁₃₇	#19166	1	1	yes	characteristic, direct product factor, Sylow
266	#3138	C₂×C₁₃₇	1	1	yes	characteristic, complemented
267	#3101	C₂×C₁₃₇	1	1	yes	characteristic, complemented
268	C₂×#459	C₂×C₁₃₇	1	1	yes	characteristic
269	D₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
270	#1701	C₂×C₁₃₇	1	2	yes	complemented
271	#1697	C₂×C₁₃₇	1	2	yes	complemented
272	#1681	C₂×C₁₃₇	1	2	yes	complemented
273	#1677	C₂×C₁₃₇	1	2	yes	complemented
274	#1565	C₂×C₁₃₇	1	1	yes	characteristic, complemented
275	#1550	C₂×C₁₃₇	1	1	yes	characteristic, complemented
276	#1545	C₂×C₁₃₇	1	1	yes	characteristic, complemented
277	C₂²×D₈	—	2	1	no	—
278	C₂²×M₁₆	—	2	1	no	—
279	#533	C₂²×C₁₃₇	1	1	yes	characteristic
280	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
282	#490	—	2	2	no	—
283	#486	C₂²×C₁₃₇	1	2	yes	complemented
284	#486	C₂²×C₁₃₇	1	2	yes	complemented
285	#483	—	2	1	no	—
286	#476	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
287	#476	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
288	#461	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
289	#461	C₂²×C₁₃₇	1	1	yes	characteristic, complemented
290	#459	—	2	1	no	—
291	#459	—	2	1	no	—
292	#459	C₂²×C₁₃₇	1	2	yes	—
293	#459	C₂²×C₁₃₇	1	2	yes	—
294	#450	—	2	1	no	—
295	C₄×SD₁₆	—	2	1	no	—
296	C₄×SD₁₆	—	2	1	no	—
297	#436	C₂²×C₁₃₇	1	2	yes	complemented
298	#435	—	2	2	no	—
299	C₈×D₄	C₂²×C₁₃₇	1	2	yes	complemented
300	#431	C₂²×C₁₃₇	1	1	yes	characteristic
301	#423	C₂²×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
303	#418	—	2	1	no	—
304	#418	—	2	1	no	—
305	C₂×#102	—	2	1	no	—
306	#407	—	2	1	no	—
307	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
308	#335	C₂²×C₁₃₇	1	2	yes	complemented
309	#334	C₂²×C₁₃₇	1	2	yes	complemented
310	#333	C₂²×C₁₃₇	1	2	yes	complemented
311	#332	C₂²×C₁₃₇	1	2	yes	complemented
312	#331	C₂²×C₁₃₇	1	2	yes	complemented
313	#331	C₂²×C₁₃₇	1	2	yes	complemented
314	C₂²×D₄	—	2	1	no	—
315	C₂²×D₄	—	2	1	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
318	C₂³×C₄	—	2	1	no	—
319	C₂×SD₁₆	—	2	1	no	—
320	C₂×SD₁₆	—	2	1	no	—
321	C₂×D₈	—	2	2	no	—
322	C₂×D₈	—	2	2	no	—
323	C₂×D₈	—	4	4	no	—
324	C₂×M₁₆	—	2	1	no	—
325	C₂×M₁₆	—	2	2	no	—
326	C₂×M₁₆	—	4	4	no	—
327	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
328	C₂×M₁₆	D₄×C₁₃₇	1	1	yes	characteristic
329	C₂²×C₈	—	2	1	no	—
330	C₂²×C₈	—	2	2	no	—
331	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
333	#127	—	2	1	no	—
334	#127	—	2	1	no	—
335	#127	—	2	1	no	—
336	#127	C₂³×C₁₃₇	1	2	yes	—
337	#125	—	2	1	no	—
338	C₂²⋊Q₈	—	2	1	no	—
339	C₂²⋊Q₈	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	2	1	no	—
344	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
345	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
346	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
347	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
348	#117	—	2	1	no	—
349	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
352	#106	—	2	1	no	—
353	#106	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
357	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
359	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
360	#103	—	2	1	no	—
361	#103	—	2	1	no	—
362	#102	—	2	1	no	—
363	#102	—	2	1	no	—
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	—	2	2	no	—
370	C₂²⋊C₈	—	2	1	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	#97	D₄×C₁₃₇	1	2	yes	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
376	C₂⁴	—	2	1	no	—
377	C₂⁴	—	4	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	4	2	no	—
388	C₂×D₄	—	4	1	no	—
389	C₂×D₄	—	4	1	no	—
390	C₂×D₄	—	4	1	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	8	2	no	—
394	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
395	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
405	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
406	SD₁₆	—	4	2	no	—
407	SD₁₆	—	4	2	no	—
408	D₈	—	4	4	no	—
409	D₈	—	4	4	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	4	4	no	—
413	M₁₆	—	4	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	4	4	no	—
419	C₂×C₈	—	4	2	no	—
420	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
421	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	4	1	no	—
428	C₄⋊C₄	—	4	1	no	—
429	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
430	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
431	C₂²⋊C₄	—	2	1	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₄²	—	4	1	no	—
437	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
439	C₄²	#41×C₁₃₇	1	1	yes	characteristic
440	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
441	C₂³	—	2	1	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	8	1	no	—
447	C₂³	—	8	2	no	—
448	C₂³	—	8	2	no	—
449	C₂³	—	8	2	no	—
450	C₂³	#121×C₁₃₇	1	1	yes	characteristic
451	Q₈	—	4	1	no	—
452	Q₈	—	4	1	no	—
453	Q₈	—	8	1	no	—
454	D₄	—	2	1	no	—
455	D₄	—	2	1	no	—
456	D₄	—	2	1	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	2	no	—
463	D₄	—	8	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	D₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
482	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
483	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
484	C₂×C₄	#135×C₁₃₇	1	2	yes	—
485	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	2	no	—
491	C₈	—	4	2	no	—
492	C₈	—	4	2	no	—
493	C₈	—	4	1	no	—
494	C₈	—	4	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	8	4	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	16	4	no	—
508	C₂²	#463×C₁₃₇	1	2	yes	—
509	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	2	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	1	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	1	no	—
517	C₄	#463×C₁₃₇	1	2	yes	—
518	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
519	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
520	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
521	C₄	#575×C₁₃₇	1	1	yes	characteristic
522	C₂	—	2	2	no	—
523	C₂	—	4	2	no	—
524	C₂	—	8	1	no	—
525	C₂	—	16	2	no	—
526	C₂	C₂×#463×C₁₃₇	1	1	yes	characteristic
527	C₂	#3055×C₁₃₇	1	1	yes	characteristic
528	C₂	#3068×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	C₂×#459×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3138×C₁₃₇	C₂	semidirect
2. #3101×C₁₃₇	C₂	semidirect
4. D₄×M₁₆×C₁₃₇	C₂	semidirect
5. #1701×C₁₃₇	C₂	semidirect
6. #1697×C₁₃₇	C₂	semidirect
7. #1681×C₁₃₇	C₂	semidirect
8. #1677×C₁₃₇	C₂	semidirect
9. #1565×C₁₃₇	C₂	semidirect
10. #1550×C₁₃₇	C₂	semidirect
11. #1545×C₁₃₇	C₂	semidirect
18. #486×C₁₃₇	C₂²	semidirect
19. #486×C₁₃₇	C₂²	semidirect
21. #476×C₁₃₇	C₂²	semidirect
22. #476×C₁₃₇	C₂²	semidirect
23. #461×C₁₃₇	C₂²	semidirect
24. #461×C₁₃₇	C₂²	semidirect
32. #436×C₁₃₇	C₂²	semidirect
34. C₈×D₄×C₁₃₇	C₂²	semidirect
43. #335×C₁₃₇	C₂²	semidirect
44. #334×C₁₃₇	C₂²	semidirect
45. #333×C₁₃₇	C₂²	semidirect
46. #332×C₁₃₇	C₂²	semidirect
47. #331×C₁₃₇	C₂²	semidirect
48. #331×C₁₃₇	C₂²	semidirect
264. #19166	C₁₃₇	direct (N × H)
265. C₁₃₇	#19166	direct (N × H)
266. #3138	C₂×C₁₃₇	semidirect
267. #3101	C₂×C₁₃₇	semidirect
269. D₄×M₁₆	C₂×C₁₃₇	semidirect
270. #1701	C₂×C₁₃₇	semidirect
271. #1697	C₂×C₁₃₇	semidirect
272. #1681	C₂×C₁₃₇	semidirect
273. #1677	C₂×C₁₃₇	semidirect
274. #1565	C₂×C₁₃₇	semidirect
275. #1550	C₂×C₁₃₇	semidirect
276. #1545	C₂×C₁₃₇	semidirect
283. #486	C₂²×C₁₃₇	semidirect
284. #486	C₂²×C₁₃₇	semidirect
286. #476	C₂²×C₁₃₇	semidirect
287. #476	C₂²×C₁₃₇	semidirect
288. #461	C₂²×C₁₃₇	semidirect
289. #461	C₂²×C₁₃₇	semidirect
297. #436	C₂²×C₁₃₇	semidirect
299. C₈×D₄	C₂²×C₁₃₇	semidirect
308. #335	C₂²×C₁₃₇	semidirect
309. #334	C₂²×C₁₃₇	semidirect
310. #333	C₂²×C₁₃₇	semidirect
311. #332	C₂²×C₁₃₇	semidirect
312. #331	C₂²×C₁₃₇	semidirect
313. #331	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3138×C₁₃₇	2
#3101×C₁₃₇	2
C₂×#459×C₁₃₇	2
D₄×M₁₆×C₁₃₇	2
#1701×C₁₃₇	2
#1697×C₁₃₇	2
#1681×C₁₃₇	2
#1677×C₁₃₇	2
#1565×C₁₃₇	2
#1550×C₁₃₇	2
#1545×C₁₃₇	2
#19166	137

Subgroups — C137×#19166 #46868552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19166 #46868552