Subgroups — C₁₃₇×#24744 #46874130

Order 35072.

1180 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#459×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×#459×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#442×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2541×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2349×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2065×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2041×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1961×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1940×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1879×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1804×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#121×C₁₃₇	—	2	2	no	—
17	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
20	#459×C₁₃₇	—	2	8	no	—
21	#459×C₁₃₇	—	2	4	no	—
22	#442×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×#102×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#102×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×#97×C₁₃₇	C₂²	1	1	yes	characteristic
33	#392×C₁₃₇	—	2	1	no	—
34	#392×C₁₃₇	—	2	1	no	—
35	#390×C₁₃₇	C₂²	1	2	yes	—
36	#389×C₁₃₇	C₂²	1	2	yes	—
37	#385×C₁₃₇	—	2	2	no	—
38	#382×C₁₃₇	C₂²	1	1	yes	characteristic
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×D₄×C₁₃₇	—	2	1	no	—
45	C₂²×D₄×C₁₃₇	—	2	2	no	—
46	C₂²×D₄×C₁₃₇	—	2	2	no	—
47	C₂²×D₄×C₁₃₇	—	2	1	no	—
48	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₁₃₇	—	2	1	no	—
52	C₂³×C₄×C₁₃₇	—	2	1	no	—
53	C₂³×C₄×C₁₃₇	—	2	2	no	—
54	C₂×D₈×C₁₃₇	—	2	2	no	—
55	C₂×D₈×C₁₃₇	—	2	2	no	—
56	C₂×D₈×C₁₃₇	—	2	4	no	—
57	C₂×D₈×C₁₃₇	—	2	4	no	—
58	C₂×D₈×C₁₃₇	—	2	4	no	—
59	C₂×D₈×C₁₃₇	D₄	1	4	yes	complemented
60	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
63	#127×C₁₃₇	—	2	4	no	—
64	#127×C₁₃₇	—	2	2	no	—
65	#127×C₁₃₇	—	2	2	no	—
66	#127×C₁₃₇	—	4	2	no	—
67	#121×C₁₃₇	—	4	8	no	—
68	C₄×D₄×C₁₃₇	—	2	4	no	—
69	C₄×D₄×C₁₃₇	—	2	2	no	—
70	C₄×D₄×C₁₃₇	—	2	2	no	—
71	C₄×D₄×C₁₃₇	—	2	2	no	—
72	C₄×D₄×C₁₃₇	—	4	2	no	—
73	C₄×D₄×C₁₃₇	D₄	1	4	yes	complemented
74	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
86	#106×C₁₃₇	—	2	2	no	—
87	C₄⋊C₈×C₁₃₇	—	2	4	no	—
88	C₄⋊C₈×C₁₃₇	—	2	2	no	—
89	#102×C₁₃₇	—	2	2	no	—
90	#102×C₁₃₇	—	2	4	no	—
91	#102×C₁₃₇	—	2	4	no	—
92	#102×C₁₃₇	—	2	2	no	—
93	#102×C₁₃₇	—	2	4	no	—
94	#102×C₁₃₇	—	4	2	no	—
95	#97×C₁₃₇	—	2	2	no	—
96	#95×C₁₃₇	—	2	1	no	—
97	#95×C₁₃₇	—	2	2	no	—
98	#95×C₁₃₇	—	2	1	no	—
99	#95×C₁₃₇	D₄	1	1	yes	characteristic
100	#95×C₁₃₇	D₄	1	1	yes	characteristic
101	C₂⁴×C₁₃₇	—	2	1	no	—
102	C₂⁴×C₁₃₇	—	2	1	no	—
103	C₂⁴×C₁₃₇	—	2	2	no	—
104	C₂×D₄×C₁₃₇	—	2	1	no	—
105	C₂×D₄×C₁₃₇	—	2	2	no	—
106	C₂×D₄×C₁₃₇	—	2	2	no	—
107	C₂×D₄×C₁₃₇	—	2	1	no	—
108	C₂×D₄×C₁₃₇	—	2	1	no	—
109	C₂×D₄×C₁₃₇	—	2	1	no	—
110	C₂×D₄×C₁₃₇	—	2	2	no	—
111	C₂×D₄×C₁₃₇	—	2	2	no	—
112	C₂×D₄×C₁₃₇	—	2	1	no	—
113	C₂×D₄×C₁₃₇	—	2	1	no	—
114	C₂×D₄×C₁₃₇	—	2	2	no	—
115	C₂×D₄×C₁₃₇	—	2	1	no	—
116	C₂×D₄×C₁₃₇	—	2	1	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	4	4	no	—
119	C₂×D₄×C₁₃₇	—	4	4	no	—
120	C₂×D₄×C₁₃₇	—	4	4	no	—
121	C₂×D₄×C₁₃₇	—	4	4	no	—
122	C₂×D₄×C₁₃₇	—	4	2	no	—
123	C₂×D₄×C₁₃₇	—	4	2	no	—
124	C₂×D₄×C₁₃₇	—	4	4	no	—
125	C₂×D₄×C₁₃₇	—	8	2	no	—
126	C₂×D₄×C₁₃₇	—	8	2	no	—
127	C₂×D₄×C₁₃₇	—	8	2	no	—
128	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
129	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
130	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₁₃₇	#41	1	2	yes	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	4	no	—
134	C₂²×C₄×C₁₃₇	—	2	2	no	—
135	C₂²×C₄×C₁₃₇	—	2	1	no	—
136	C₂²×C₄×C₁₃₇	—	4	4	no	—
137	C₂²×C₄×C₁₃₇	—	4	4	no	—
138	C₂²×C₄×C₁₃₇	—	4	2	no	—
139	C₂²×C₄×C₁₃₇	—	4	4	no	—
140	C₂²×C₄×C₁₃₇	—	8	2	no	—
141	C₂²×C₄×C₁₃₇	—	8	2	no	—
142	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
147	D₈×C₁₃₇	—	2	8	no	—
148	D₈×C₁₃₇	—	4	8	no	—
149	C₂×C₈×C₁₃₇	—	2	1	no	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	1	no	—
152	C₂×C₈×C₁₃₇	—	2	2	no	—
153	C₂×C₈×C₁₃₇	—	2	2	no	—
154	C₂×C₈×C₁₃₇	—	2	2	no	—
155	C₂×C₈×C₁₃₇	—	2	1	no	—
156	C₂×C₈×C₁₃₇	—	2	1	no	—
157	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
158	C₄⋊C₄×C₁₃₇	—	2	4	no	—
159	C₄⋊C₄×C₁₃₇	—	2	1	no	—
160	C₄⋊C₄×C₁₃₇	—	2	2	no	—
161	C₄⋊C₄×C₁₃₇	—	2	1	no	—
162	C₄⋊C₄×C₁₃₇	—	2	1	no	—
163	C₄⋊C₄×C₁₃₇	—	2	2	no	—
164	C₄⋊C₄×C₁₃₇	—	8	2	no	—
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
167	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
168	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
169	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
170	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
171	C₄²×C₁₃₇	—	2	2	no	—
172	C₄²×C₁₃₇	—	2	2	no	—
173	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
174	C₄²×C₁₃₇	#41	1	2	yes	—
175	C₂³×C₁₃₇	—	2	2	no	—
176	C₂³×C₁₃₇	—	2	2	no	—
177	C₂³×C₁₃₇	—	2	2	no	—
178	C₂³×C₁₃₇	—	2	2	no	—
179	C₂³×C₁₃₇	—	4	2	no	—
180	C₂³×C₁₃₇	—	4	2	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	—	4	2	no	—
191	C₂³×C₁₃₇	—	4	2	no	—
192	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
193	D₄×C₁₃₇	—	2	2	no	—
194	D₄×C₁₃₇	—	2	2	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	1	no	—
200	D₄×C₁₃₇	—	4	2	no	—
201	D₄×C₁₃₇	—	4	2	no	—
202	D₄×C₁₃₇	—	4	1	no	—
203	D₄×C₁₃₇	—	4	2	no	—
204	D₄×C₁₃₇	—	8	8	no	—
205	D₄×C₁₃₇	—	8	4	no	—
206	D₄×C₁₃₇	—	8	4	no	—
207	D₄×C₁₃₇	—	8	8	no	—
208	D₄×C₁₃₇	#121	1	4	yes	complemented
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	—	2	2	no	—
216	C₂×C₄×C₁₃₇	—	2	2	no	—
217	C₂×C₄×C₁₃₇	—	4	2	no	—
218	C₂×C₄×C₁₃₇	—	4	1	no	—
219	C₂×C₄×C₁₃₇	—	4	2	no	—
220	C₂×C₄×C₁₃₇	—	4	2	no	—
221	C₂×C₄×C₁₃₇	—	4	2	no	—
222	C₂×C₄×C₁₃₇	—	4	2	no	—
223	C₂×C₄×C₁₃₇	—	8	4	no	—
224	C₂×C₄×C₁₃₇	—	8	8	no	—
225	C₂×C₄×C₁₃₇	—	8	4	no	—
226	C₂×C₄×C₁₃₇	#121	1	2	yes	—
227	C₂×C₄×C₁₃₇	#121	1	2	yes	—
228	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
229	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
234	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
235	C₈×C₁₃₇	—	2	2	no	—
236	C₈×C₁₃₇	—	4	2	no	—
237	C₈×C₁₃₇	—	4	1	no	—
238	C₂²×C₁₃₇	—	2	4	no	—
239	C₂²×C₁₃₇	—	4	4	no	—
240	C₂²×C₁₃₇	—	4	4	no	—
241	C₂²×C₁₃₇	—	4	2	no	—
242	C₂²×C₁₃₇	—	4	2	no	—
243	C₂²×C₁₃₇	—	4	2	no	—
244	C₂²×C₁₃₇	—	4	4	no	—
245	C₂²×C₁₃₇	—	8	2	no	—
246	C₂²×C₁₃₇	—	8	1	no	—
247	C₂²×C₁₃₇	—	8	1	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	—	8	2	no	—
255	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
256	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
257	C₂²×C₁₃₇	#548	1	1	yes	characteristic
258	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
259	C₂²×C₁₃₇	#576	1	2	yes	—
260	C₄×C₁₃₇	—	2	4	no	—
261	C₄×C₁₃₇	—	2	2	no	—
262	C₄×C₁₃₇	—	2	2	no	—
263	C₄×C₁₃₇	—	4	2	no	—
264	C₄×C₁₃₇	—	4	2	no	—
265	C₄×C₁₃₇	—	8	2	no	—
266	C₄×C₁₃₇	—	8	1	no	—
267	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
268	C₄×C₁₃₇	#537	1	1	yes	characteristic
269	C₄×C₁₃₇	#547	1	1	yes	characteristic
270	C₄×C₁₃₇	#553	1	1	yes	characteristic
271	C₂×C₁₃₇	—	4	4	no	—
272	C₂×C₁₃₇	—	8	2	no	—
273	C₂×C₁₃₇	—	16	2	no	—
274	C₂×C₁₃₇	#2416	1	1	yes	characteristic
275	C₂×C₁₃₇	#3038	1	1	yes	characteristic
276	C₂×C₁₃₇	#3041	1	1	yes	characteristic
277	C₂×C₁₃₇	#3264	1	2	yes	—
278	C₂×C₁₃₇	#3366	1	2	yes	—
279	#24744	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₁₃₇	#24744	1	1	yes	characteristic, direct product factor, Sylow
281	C₂×#459	C₂×C₁₃₇	1	2	yes	complemented
282	C₂×#459	C₂×C₁₃₇	1	1	yes	characteristic, complemented
283	C₂×#442	C₂×C₁₃₇	1	1	yes	characteristic, complemented
284	#2541	C₂×C₁₃₇	1	1	yes	characteristic, complemented
285	#2349	C₂×C₁₃₇	1	1	yes	characteristic, complemented
286	#2065	C₂×C₁₃₇	1	2	yes	complemented
287	#2041	C₂×C₁₃₇	1	2	yes	complemented
288	#1961	C₂×C₁₃₇	1	1	yes	characteristic, complemented
289	#1940	C₂×C₁₃₇	1	1	yes	characteristic, complemented
290	#1879	C₂×C₁₃₇	1	2	yes	complemented
291	#1804	C₂×C₁₃₇	1	1	yes	characteristic, complemented
292	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
293	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
294	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
295	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
296	C₂×#121	—	2	2	no	—
297	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
299	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
300	#459	—	2	8	no	—
301	#459	—	2	4	no	—
302	#442	—	2	4	no	—
303	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
305	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
306	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
307	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
308	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
309	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
310	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
311	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
312	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
313	#392	—	2	1	no	—
314	#392	—	2	1	no	—
315	#390	C₂²×C₁₃₇	1	2	yes	—
316	#389	C₂²×C₁₃₇	1	2	yes	—
317	#385	—	2	2	no	—
318	#382	C₂²×C₁₃₇	1	1	yes	characteristic
319	#380	—	2	1	no	—
320	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
321	C₂×#95	—	2	1	no	—
322	#340	C₂²×C₁₃₇	1	2	yes	—
323	#336	C₂²×C₁₃₇	1	2	yes	—
324	C₂²×D₄	—	2	1	no	—
325	C₂²×D₄	—	2	2	no	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	1	no	—
328	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
329	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
330	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
331	C₂³×C₄	—	2	1	no	—
332	C₂³×C₄	—	2	1	no	—
333	C₂³×C₄	—	2	2	no	—
334	C₂×D₈	—	2	2	no	—
335	C₂×D₈	—	2	2	no	—
336	C₂×D₈	—	2	4	no	—
337	C₂×D₈	—	2	4	no	—
338	C₂×D₈	—	2	4	no	—
339	C₂×D₈	D₄×C₁₃₇	1	4	yes	complemented
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	4	no	—
344	#127	—	2	2	no	—
345	#127	—	2	2	no	—
346	#127	—	4	2	no	—
347	#121	—	4	8	no	—
348	C₄×D₄	—	2	4	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	2	2	no	—
352	C₄×D₄	—	4	2	no	—
353	C₄×D₄	D₄×C₁₃₇	1	4	yes	complemented
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
358	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
365	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
366	#106	—	2	2	no	—
367	C₄⋊C₈	—	2	4	no	—
368	C₄⋊C₈	—	2	2	no	—
369	#102	—	2	2	no	—
370	#102	—	2	4	no	—
371	#102	—	2	4	no	—
372	#102	—	2	2	no	—
373	#102	—	2	4	no	—
374	#102	—	4	2	no	—
375	#97	—	2	2	no	—
376	#95	—	2	1	no	—
377	#95	—	2	2	no	—
378	#95	—	2	1	no	—
379	#95	D₄×C₁₃₇	1	1	yes	characteristic
380	#95	D₄×C₁₃₇	1	1	yes	characteristic
381	C₂⁴	—	2	1	no	—
382	C₂⁴	—	2	1	no	—
383	C₂⁴	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	1	no	—
396	C₂×D₄	—	2	1	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	4	4	no	—
399	C₂×D₄	—	4	4	no	—
400	C₂×D₄	—	4	4	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	—	4	2	no	—
403	C₂×D₄	—	4	2	no	—
404	C₂×D₄	—	4	4	no	—
405	C₂×D₄	—	8	2	no	—
406	C₂×D₄	—	8	2	no	—
407	C₂×D₄	—	8	2	no	—
408	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
409	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
410	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
411	C₂×D₄	#41×C₁₃₇	1	2	yes	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	—	4	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	8	2	no	—
421	C₂²×C₄	—	8	2	no	—
422	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
423	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
427	D₈	—	2	8	no	—
428	D₈	—	4	8	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	8	2	no	—
445	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	4	4	no	—
449	C₂²⋊C₄	—	4	2	no	—
450	C₂²⋊C₄	—	4	4	no	—
451	C₄²	—	2	2	no	—
452	C₄²	—	2	2	no	—
453	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
454	C₄²	#41×C₁₃₇	1	2	yes	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	1	no	—
462	C₂³	—	4	1	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	2	no	—
465	C₂³	—	4	2	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	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
473	D₄	—	2	2	no	—
474	D₄	—	2	2	no	—
475	D₄	—	2	2	no	—
476	D₄	—	2	2	no	—
477	D₄	—	2	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	4	1	no	—
483	D₄	—	4	2	no	—
484	D₄	—	8	8	no	—
485	D₄	—	8	4	no	—
486	D₄	—	8	4	no	—
487	D₄	—	8	8	no	—
488	D₄	#121×C₁₃₇	1	4	yes	complemented
489	C₂×C₄	—	2	2	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	—	4	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	8	4	no	—
504	C₂×C₄	—	8	8	no	—
505	C₂×C₄	—	8	4	no	—
506	C₂×C₄	#121×C₁₃₇	1	2	yes	—
507	C₂×C₄	#121×C₁₃₇	1	2	yes	—
508	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
511	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
512	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
513	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
514	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
515	C₈	—	2	2	no	—
516	C₈	—	4	2	no	—
517	C₈	—	4	1	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	2	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	4	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	8	1	no	—
527	C₂²	—	8	2	no	—
528	C₂²	—	8	2	no	—
529	C₂²	—	8	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	C₂×#121×C₁₃₇	1	1	yes	characteristic
536	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
537	C₂²	#548×C₁₃₇	1	1	yes	characteristic
538	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
539	C₂²	#576×C₁₃₇	1	2	yes	—
540	C₄	—	2	2	no	—
541	C₄	—	2	2	no	—
542	C₄	—	2	4	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	8	1	no	—
546	C₄	—	8	2	no	—
547	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
548	C₄	#537×C₁₃₇	1	1	yes	characteristic
549	C₄	#547×C₁₃₇	1	1	yes	characteristic
550	C₄	#553×C₁₃₇	1	1	yes	characteristic
551	C₂	—	4	4	no	—
552	C₂	—	8	2	no	—
553	C₂	—	16	2	no	—
554	C₂	#2416×C₁₃₇	1	1	yes	characteristic
555	C₂	#3038×C₁₃₇	1	1	yes	characteristic
556	C₂	#3041×C₁₃₇	1	1	yes	characteristic
557	C₂	#3264×C₁₃₇	1	2	yes	—
558	C₂	#3366×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
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×#459×C₁₃₇	C₂	semidirect
2. C₂×#459×C₁₃₇	C₂	semidirect
3. C₂×#442×C₁₃₇	C₂	semidirect
4. #2541×C₁₃₇	C₂	semidirect
5. #2349×C₁₃₇	C₂	semidirect
6. #2065×C₁₃₇	C₂	semidirect
7. #2041×C₁₃₇	C₂	semidirect
8. #1961×C₁₃₇	C₂	semidirect
9. #1940×C₁₃₇	C₂	semidirect
10. #1879×C₁₃₇	C₂	semidirect
11. #1804×C₁₃₇	C₂	semidirect
59. C₂×D₈×C₁₃₇	D₄	semidirect
73. C₄×D₄×C₁₃₇	D₄	semidirect
208. D₄×C₁₃₇	#121	semidirect
279. #24744	C₁₃₇	direct (N × H)
280. C₁₃₇	#24744	direct (N × H)
281. C₂×#459	C₂×C₁₃₇	semidirect
282. C₂×#459	C₂×C₁₃₇	semidirect
283. C₂×#442	C₂×C₁₃₇	semidirect
284. #2541	C₂×C₁₃₇	semidirect
285. #2349	C₂×C₁₃₇	semidirect
286. #2065	C₂×C₁₃₇	semidirect
287. #2041	C₂×C₁₃₇	semidirect
288. #1961	C₂×C₁₃₇	semidirect
289. #1940	C₂×C₁₃₇	semidirect
290. #1879	C₂×C₁₃₇	semidirect
291. #1804	C₂×C₁₃₇	semidirect
339. C₂×D₈	D₄×C₁₃₇	semidirect
353. C₄×D₄	D₄×C₁₃₇	semidirect
488. D₄	#121×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#459×C₁₃₇	2
C₂×#442×C₁₃₇	2
#2541×C₁₃₇	2
#2349×C₁₃₇	2
#2065×C₁₃₇	2
#2041×C₁₃₇	2
#1961×C₁₃₇	2
#1940×C₁₃₇	2
#1879×C₁₃₇	2
#1804×C₁₃₇	2
#24744	137

Subgroups — C137×#24744 #46874130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24744 #46874130