Subgroups — C₁₃₇×#24731 #46874117

Order 35072.

1086 conjugacy classes of subgroups.

The subgroup lattice has 566 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, complemented
2	C₂×#459×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2548×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2065×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2042×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1957×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1939×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1879×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1803×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×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	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
20	#463×C₁₃₇	—	2	8	no	—
21	#459×C₁₃₇	—	2	4	no	—
22	C₄×SD₁₆×C₁₃₇	—	2	4	no	—
23	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
33	#394×C₁₃₇	—	2	2	no	—
34	#390×C₁₃₇	—	2	1	no	—
35	#390×C₁₃₇	—	2	1	no	—
36	#390×C₁₃₇	C₂²	1	2	yes	—
37	#389×C₁₃₇	C₂²	1	2	yes	—
38	#385×C₁₃₇	—	2	2	no	—
39	#384×C₁₃₇	C₂²	1	1	yes	characteristic
40	#380×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	1	yes	characteristic
45	C₂²×D₄×C₁₃₇	—	2	1	no	—
46	C₂²×D₄×C₁₃₇	—	2	1	no	—
47	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₁₃₇	—	2	1	no	—
50	C₂³×C₄×C₁₃₇	—	2	2	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
53	C₂×SD₁₆×C₁₃₇	D₄	1	4	yes	complemented
54	C₂×D₈×C₁₃₇	—	2	4	no	—
55	C₂×D₈×C₁₃₇	—	2	4	no	—
56	C₂×D₈×C₁₃₇	—	2	4	no	—
57	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
60	#127×C₁₃₇	—	2	2	no	—
61	#127×C₁₃₇	—	2	2	no	—
62	#124×C₁₃₇	—	2	4	no	—
63	#124×C₁₃₇	—	4	2	no	—
64	C₄×Q₈×C₁₃₇	—	2	2	no	—
65	C₄×Q₈×C₁₃₇	—	2	2	no	—
66	C₄×D₄×C₁₃₇	—	2	4	no	—
67	C₄×D₄×C₁₃₇	—	2	2	no	—
68	C₄×D₄×C₁₃₇	—	4	2	no	—
69	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	4	4	no	—
75	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄²×C₁₃₇	—	2	1	no	—
87	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
88	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
89	#106×C₁₃₇	—	2	2	no	—
90	C₄⋊C₈×C₁₃₇	—	2	4	no	—
91	C₄⋊C₈×C₁₃₇	—	2	2	no	—
92	#103×C₁₃₇	—	2	4	no	—
93	#103×C₁₃₇	—	2	2	no	—
94	#102×C₁₃₇	—	2	2	no	—
95	#102×C₁₃₇	—	2	4	no	—
96	#102×C₁₃₇	—	2	4	no	—
97	#102×C₁₃₇	—	4	2	no	—
98	C₄×C₈×C₁₃₇	—	2	2	no	—
99	#95×C₁₃₇	—	2	2	no	—
100	#95×C₁₃₇	D₄	1	2	yes	—
101	C₂⁴×C₁₃₇	—	2	1	no	—
102	C₂⁴×C₁₃₇	—	2	2	no	—
103	C₂×Q₈×C₁₃₇	—	2	1	no	—
104	C₂×Q₈×C₁₃₇	—	2	1	no	—
105	C₂×Q₈×C₁₃₇	—	2	1	no	—
106	C₂×Q₈×C₁₃₇	—	2	1	no	—
107	C₂×Q₈×C₁₃₇	—	2	1	no	—
108	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
109	C₂×D₄×C₁₃₇	—	2	2	no	—
110	C₂×D₄×C₁₃₇	—	2	2	no	—
111	C₂×D₄×C₁₃₇	—	2	1	no	—
112	C₂×D₄×C₁₃₇	—	2	2	no	—
113	C₂×D₄×C₁₃₇	—	2	2	no	—
114	C₂×D₄×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	—	4	2	no	—
119	C₂×D₄×C₁₃₇	—	4	2	no	—
120	C₂×D₄×C₁₃₇	—	4	2	no	—
121	C₂×D₄×C₁₃₇	—	4	2	no	—
122	C₂×D₄×C₁₃₇	—	8	1	no	—
123	C₂×D₄×C₁₃₇	—	8	1	no	—
124	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₁₃₇	#41	1	2	yes	—
127	C₂²×C₄×C₁₃₇	—	2	1	no	—
128	C₂²×C₄×C₁₃₇	—	2	1	no	—
129	C₂²×C₄×C₁₃₇	—	2	4	no	—
130	C₂²×C₄×C₁₃₇	—	2	2	no	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	4	4	no	—
133	C₂²×C₄×C₁₃₇	—	4	4	no	—
134	C₂²×C₄×C₁₃₇	—	4	2	no	—
135	C₂²×C₄×C₁₃₇	—	4	2	no	—
136	C₂²×C₄×C₁₃₇	—	8	2	no	—
137	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
142	SD₁₆×C₁₃₇	—	2	8	no	—
143	D₈×C₁₃₇	—	4	8	no	—
144	C₂×C₈×C₁₃₇	—	2	1	no	—
145	C₂×C₈×C₁₃₇	—	2	1	no	—
146	C₂×C₈×C₁₃₇	—	2	1	no	—
147	C₂×C₈×C₁₃₇	—	2	2	no	—
148	C₂×C₈×C₁₃₇	—	2	2	no	—
149	C₂×C₈×C₁₃₇	—	2	2	no	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	1	no	—
152	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₁₃₇	—	2	2	no	—
154	C₄⋊C₄×C₁₃₇	—	2	2	no	—
155	C₄⋊C₄×C₁₃₇	—	2	1	no	—
156	C₄⋊C₄×C₁₃₇	—	2	2	no	—
157	C₄⋊C₄×C₁₃₇	—	2	1	no	—
158	C₄⋊C₄×C₁₃₇	—	2	1	no	—
159	C₄⋊C₄×C₁₃₇	—	2	2	no	—
160	C₄⋊C₄×C₁₃₇	—	4	4	no	—
161	C₄⋊C₄×C₁₃₇	—	4	2	no	—
162	C₄⋊C₄×C₁₃₇	—	4	2	no	—
163	C₄⋊C₄×C₁₃₇	—	8	1	no	—
164	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
166	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
167	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
168	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
169	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
170	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
171	C₄²×C₁₃₇	—	2	2	no	—
172	C₄²×C₁₃₇	—	2	2	no	—
173	C₄²×C₁₃₇	—	4	2	no	—
174	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
175	C₄²×C₁₃₇	#41	1	2	yes	—
176	C₂³×C₁₃₇	—	2	2	no	—
177	C₂³×C₁₃₇	—	2	2	no	—
178	C₂³×C₁₃₇	—	2	2	no	—
179	C₂³×C₁₃₇	—	4	1	no	—
180	C₂³×C₁₃₇	—	4	1	no	—
181	C₂³×C₁₃₇	—	4	1	no	—
182	C₂³×C₁₃₇	—	4	1	no	—
183	C₂³×C₁₃₇	—	4	2	no	—
184	C₂³×C₁₃₇	—	4	2	no	—
185	C₂³×C₁₃₇	—	4	2	no	—
186	C₂³×C₁₃₇	—	4	2	no	—
187	C₂³×C₁₃₇	—	4	2	no	—
188	C₂³×C₁₃₇	—	4	2	no	—
189	C₂³×C₁₃₇	—	4	2	no	—
190	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₁₃₇	—	2	2	no	—
192	Q₈×C₁₃₇	—	2	2	no	—
193	Q₈×C₁₃₇	—	4	1	no	—
194	Q₈×C₁₃₇	—	4	1	no	—
195	D₄×C₁₃₇	—	2	2	no	—
196	D₄×C₁₃₇	—	2	2	no	—
197	D₄×C₁₃₇	—	2	2	no	—
198	D₄×C₁₃₇	—	4	2	no	—
199	D₄×C₁₃₇	—	4	2	no	—
200	D₄×C₁₃₇	—	4	2	no	—
201	D₄×C₁₃₇	—	4	2	no	—
202	D₄×C₁₃₇	—	8	4	no	—
203	D₄×C₁₃₇	—	8	4	no	—
204	D₄×C₁₃₇	#121	1	4	yes	—
205	C₂×C₄×C₁₃₇	—	2	1	no	—
206	C₂×C₄×C₁₃₇	—	2	1	no	—
207	C₂×C₄×C₁₃₇	—	2	2	no	—
208	C₂×C₄×C₁₃₇	—	2	2	no	—
209	C₂×C₄×C₁₃₇	—	2	2	no	—
210	C₂×C₄×C₁₃₇	—	2	2	no	—
211	C₂×C₄×C₁₃₇	—	2	2	no	—
212	C₂×C₄×C₁₃₇	—	2	2	no	—
213	C₂×C₄×C₁₃₇	—	2	2	no	—
214	C₂×C₄×C₁₃₇	—	2	2	no	—
215	C₂×C₄×C₁₃₇	—	4	1	no	—
216	C₂×C₄×C₁₃₇	—	4	1	no	—
217	C₂×C₄×C₁₃₇	—	4	1	no	—
218	C₂×C₄×C₁₃₇	—	4	1	no	—
219	C₂×C₄×C₁₃₇	—	4	8	no	—
220	C₂×C₄×C₁₃₇	—	4	1	no	—
221	C₂×C₄×C₁₃₇	—	4	2	no	—
222	C₂×C₄×C₁₃₇	—	4	2	no	—
223	C₂×C₄×C₁₃₇	—	4	1	no	—
224	C₂×C₄×C₁₃₇	—	4	2	no	—
225	C₂×C₄×C₁₃₇	—	4	1	no	—
226	C₂×C₄×C₁₃₇	—	8	8	no	—
227	C₂×C₄×C₁₃₇	#121	1	2	yes	—
228	C₂×C₄×C₁₃₇	#121	1	2	yes	—
229	C₂×C₄×C₁₃₇	#135	1	2	yes	—
230	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
231	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
237	C₈×C₁₃₇	—	2	2	no	—
238	C₈×C₁₃₇	—	4	2	no	—
239	C₈×C₁₃₇	—	4	1	no	—
240	C₂²×C₁₃₇	—	2	4	no	—
241	C₂²×C₁₃₇	—	4	2	no	—
242	C₂²×C₁₃₇	—	4	2	no	—
243	C₂²×C₁₃₇	—	4	2	no	—
244	C₂²×C₁₃₇	—	4	2	no	—
245	C₂²×C₁₃₇	—	4	2	no	—
246	C₂²×C₁₃₇	—	4	2	no	—
247	C₂²×C₁₃₇	—	8	2	no	—
248	C₂²×C₁₃₇	—	8	2	no	—
249	C₂²×C₁₃₇	—	8	2	no	—
250	C₂²×C₁₃₇	—	8	2	no	—
251	C₂²×C₁₃₇	—	8	2	no	—
252	C₂²×C₁₃₇	—	8	2	no	—
253	C₂²×C₁₃₇	—	8	2	no	—
254	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
255	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
256	C₂²×C₁₃₇	#548	1	1	yes	characteristic
257	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
258	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂²×C₁₃₇	#576	1	1	yes	characteristic
260	C₂²×C₁₃₇	#577	1	1	yes	characteristic
261	C₄×C₁₃₇	—	2	2	no	—
262	C₄×C₁₃₇	—	4	2	no	—
263	C₄×C₁₃₇	—	4	2	no	—
264	C₄×C₁₃₇	—	4	2	no	—
265	C₄×C₁₃₇	—	4	2	no	—
266	C₄×C₁₃₇	—	8	2	no	—
267	C₄×C₁₃₇	—	8	1	no	—
268	C₄×C₁₃₇	#463	1	4	yes	complemented
269	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
270	C₄×C₁₃₇	#537	1	1	yes	characteristic
271	C₄×C₁₃₇	#547	1	1	yes	characteristic
272	C₄×C₁₃₇	#553	1	1	yes	characteristic
273	C₂×C₁₃₇	—	4	4	no	—
274	C₂×C₁₃₇	—	16	2	no	—
275	C₂×C₁₃₇	#2416	1	1	yes	characteristic
276	C₂×C₁₃₇	C₂×#463	1	1	yes	characteristic
277	C₂×C₁₃₇	#3045	1	1	yes	characteristic
278	C₂×C₁₃₇	#3274	1	1	yes	characteristic
279	C₂×C₁₃₇	#3284	1	1	yes	characteristic
280	C₂×C₁₃₇	#3361	1	1	yes	characteristic
281	C₂×C₁₃₇	#3373	1	1	yes	characteristic
282	#24731	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₁₃₇	#24731	1	1	yes	characteristic, direct product factor, Sylow
284	C₂×#463	C₂×C₁₃₇	1	2	yes	complemented
285	C₂×#459	C₂×C₁₃₇	1	1	yes	characteristic
286	C₂×C₄×SD₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
287	#2548	C₂×C₁₃₇	1	1	yes	characteristic, complemented
288	D₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
289	#2065	C₂×C₁₃₇	1	2	yes	complemented
290	#2042	C₂×C₁₃₇	1	2	yes	complemented
291	#1957	C₂×C₁₃₇	1	1	yes	characteristic, complemented
292	#1939	C₂×C₁₃₇	1	1	yes	characteristic, complemented
293	#1879	C₂×C₁₃₇	1	2	yes	complemented
294	#1803	C₂×C₁₃₇	1	1	yes	characteristic, complemented
295	C₂²×SD₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
296	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
297	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
298	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
299	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
300	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	—	2	1	no	—
303	#463	—	2	8	no	—
304	#459	—	2	4	no	—
305	C₄×SD₁₆	—	2	4	no	—
306	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
307	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
309	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
310	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
311	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
312	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
313	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
314	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
316	#394	—	2	2	no	—
317	#390	—	2	1	no	—
318	#390	—	2	1	no	—
319	#390	C₂²×C₁₃₇	1	2	yes	—
320	#389	C₂²×C₁₃₇	1	2	yes	—
321	#385	—	2	2	no	—
322	#384	C₂²×C₁₃₇	1	1	yes	characteristic
323	#380	—	2	1	no	—
324	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
325	#340	C₂²×C₁₃₇	1	2	yes	—
326	#336	C₂²×C₁₃₇	1	2	yes	—
327	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
331	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	—	2	2	no	—
336	C₂×SD₁₆	D₄×C₁₃₇	1	4	yes	complemented
337	C₂×D₈	—	2	4	no	—
338	C₂×D₈	—	2	4	no	—
339	C₂×D₈	—	2	4	no	—
340	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
341	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#127	—	2	2	no	—
345	#124	—	2	4	no	—
346	#124	—	4	2	no	—
347	C₄×Q₈	—	2	2	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×D₄	—	2	4	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	4	2	no	—
352	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	4	4	no	—
358	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
359	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
361	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
363	C₂×C₂²⋊C₄	—	2	1	no	—
364	C₂×C₂²⋊C₄	—	2	2	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₂²⋊C₄	—	2	2	no	—
367	C₂×C₂²⋊C₄	—	2	2	no	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
371	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
372	#106	—	2	2	no	—
373	C₄⋊C₈	—	2	4	no	—
374	C₄⋊C₈	—	2	2	no	—
375	#103	—	2	4	no	—
376	#103	—	2	2	no	—
377	#102	—	2	2	no	—
378	#102	—	2	4	no	—
379	#102	—	2	4	no	—
380	#102	—	4	2	no	—
381	C₄×C₈	—	2	2	no	—
382	#95	—	2	2	no	—
383	#95	D₄×C₁₃₇	1	2	yes	—
384	C₂⁴	—	2	1	no	—
385	C₂⁴	—	2	2	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	1	no	—
391	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	4	2	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	4	2	no	—
405	C₂×D₄	—	8	1	no	—
406	C₂×D₄	—	8	1	no	—
407	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
408	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
409	C₂×D₄	#41×C₁₃₇	1	2	yes	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	—	4	4	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	—	4	2	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	8	2	no	—
420	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
422	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
425	SD₁₆	—	2	8	no	—
426	D₈	—	4	8	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	4	4	no	—
444	C₄⋊C₄	—	4	2	no	—
445	C₄⋊C₄	—	4	2	no	—
446	C₄⋊C₄	—	8	1	no	—
447	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	—	4	4	no	—
451	C₂²⋊C₄	—	4	4	no	—
452	C₂²⋊C₄	—	8	2	no	—
453	C₂²⋊C₄	—	8	2	no	—
454	C₄²	—	2	2	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	4	2	no	—
457	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
458	C₄²	#41×C₁₃₇	1	2	yes	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	4	1	no	—
463	C₂³	—	4	1	no	—
464	C₂³	—	4	1	no	—
465	C₂³	—	4	1	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	2	no	—
468	C₂³	—	4	2	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	2	no	—
471	C₂³	—	4	2	no	—
472	C₂³	—	4	2	no	—
473	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
474	Q₈	—	2	2	no	—
475	Q₈	—	2	2	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	4	1	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	2	no	—
480	D₄	—	2	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	2	no	—
483	D₄	—	4	2	no	—
484	D₄	—	4	2	no	—
485	D₄	—	8	4	no	—
486	D₄	—	8	4	no	—
487	D₄	#121×C₁₃₇	1	4	yes	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	8	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	1	no	—
509	C₂×C₄	—	8	8	no	—
510	C₂×C₄	#121×C₁₃₇	1	2	yes	—
511	C₂×C₄	#121×C₁₃₇	1	2	yes	—
512	C₂×C₄	#135×C₁₃₇	1	2	yes	—
513	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
514	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
517	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
518	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
519	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
520	C₈	—	2	2	no	—
521	C₈	—	4	2	no	—
522	C₈	—	4	1	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	2	no	—
532	C₂²	—	8	2	no	—
533	C₂²	—	8	2	no	—
534	C₂²	—	8	2	no	—
535	C₂²	—	8	2	no	—
536	C₂²	—	8	2	no	—
537	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
538	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
539	C₂²	#548×C₁₃₇	1	1	yes	characteristic
540	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
541	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
542	C₂²	#576×C₁₃₇	1	1	yes	characteristic
543	C₂²	#577×C₁₃₇	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	2	no	—
549	C₄	—	8	1	no	—
550	C₄	—	8	2	no	—
551	C₄	#463×C₁₃₇	1	4	yes	complemented
552	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
553	C₄	#537×C₁₃₇	1	1	yes	characteristic
554	C₄	#547×C₁₃₇	1	1	yes	characteristic
555	C₄	#553×C₁₃₇	1	1	yes	characteristic
556	C₂	—	4	4	no	—
557	C₂	—	16	2	no	—
558	C₂	#2416×C₁₃₇	1	1	yes	characteristic
559	C₂	C₂×#463×C₁₃₇	1	1	yes	characteristic
560	C₂	#3045×C₁₃₇	1	1	yes	characteristic
561	C₂	#3274×C₁₃₇	1	1	yes	characteristic
562	C₂	#3284×C₁₃₇	1	1	yes	characteristic
563	C₂	#3361×C₁₃₇	1	1	yes	characteristic
564	C₂	#3373×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#459×C₁₃₇	C₂
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
1. C₂×#463×C₁₃₇	C₂	semidirect
3. C₂×C₄×SD₁₆×C₁₃₇	C₂	semidirect
4. #2548×C₁₃₇	C₂	semidirect
5. D₄×C₄⋊C₄×C₁₃₇	C₂	semidirect
6. #2065×C₁₃₇	C₂	semidirect
7. #2042×C₁₃₇	C₂	semidirect
8. #1957×C₁₃₇	C₂	semidirect
9. #1939×C₁₃₇	C₂	semidirect
10. #1879×C₁₃₇	C₂	semidirect
11. #1803×C₁₃₇	C₂	semidirect
53. C₂×SD₁₆×C₁₃₇	D₄	semidirect
268. C₄×C₁₃₇	#463	semidirect
282. #24731	C₁₃₇	direct (N × H)
283. C₁₃₇	#24731	direct (N × H)
284. C₂×#463	C₂×C₁₃₇	semidirect
286. C₂×C₄×SD₁₆	C₂×C₁₃₇	semidirect
287. #2548	C₂×C₁₃₇	semidirect
288. D₄×C₄⋊C₄	C₂×C₁₃₇	semidirect
289. #2065	C₂×C₁₃₇	semidirect
290. #2042	C₂×C₁₃₇	semidirect
291. #1957	C₂×C₁₃₇	semidirect
292. #1939	C₂×C₁₃₇	semidirect
293. #1879	C₂×C₁₃₇	semidirect
294. #1803	C₂×C₁₃₇	semidirect
336. C₂×SD₁₆	D₄×C₁₃₇	semidirect
551. C₄	#463×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×#459×C₁₃₇	2
C₂×C₄×SD₁₆×C₁₃₇	2
#2548×C₁₃₇	2
D₄×C₄⋊C₄×C₁₃₇	2
#2065×C₁₃₇	2
#2042×C₁₃₇	2
#1957×C₁₃₇	2
#1939×C₁₃₇	2
#1879×C₁₃₇	2
#1803×C₁₃₇	2
#24731	137

Subgroups — C137×#24731 #46874117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24731 #46874117