Subgroups — C₁₃₇×#21538 #46870924

Order 35072.

1170 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#469×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#442×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×M₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#121×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1970×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1941×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1934×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1895×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#1809×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1776×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#1762×C₁₃₇	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₃₇	—	2	2	no	—
17	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
19	#469×C₁₃₇	C₄	1	8	yes	complemented
20	#442×C₁₃₇	—	2	4	no	—
21	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
27	C₄×M₁₆×C₁₃₇	—	2	4	no	—
28	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
32	#390×C₁₃₇	C₂²	1	2	yes	—
33	#389×C₁₃₇	C₂²	1	2	yes	—
34	#388×C₁₃₇	—	2	2	no	—
35	#386×C₁₃₇	—	2	2	no	—
36	#380×C₁₃₇	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
38	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
39	#340×C₁₃₇	C₂²	1	2	yes	—
40	#340×C₁₃₇	C₂²	1	2	yes	—
41	#340×C₁₃₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₁₃₇	—	2	2	no	—
43	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₁₃₇	—	2	2	no	—
45	C₂³×C₄×C₁₃₇	D₄	1	2	yes	—
46	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×D₈×C₁₃₇	—	2	2	no	—
48	C₂×D₈×C₁₃₇	—	2	2	no	—
49	C₂×D₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
50	C₂×M₁₆×C₁₃₇	—	2	2	no	—
51	C₂×M₁₆×C₁₃₇	—	2	2	no	—
52	C₂×M₁₆×C₁₃₇	—	2	2	no	—
53	C₂×M₁₆×C₁₃₇	—	2	2	no	—
54	C₂×M₁₆×C₁₃₇	—	2	2	no	—
55	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
56	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
60	#121×C₁₃₇	—	2	4	no	—
61	#121×C₁₃₇	C₂×C₄	1	8	yes	complemented
62	C₄×D₄×C₁₃₇	—	2	4	no	—
63	C₄×D₄×C₁₃₇	—	2	4	no	—
64	C₄×D₄×C₁₃₇	—	4	8	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄²×C₁₃₇	—	2	2	no	—
75	C₂×C₄²×C₁₃₇	—	2	2	no	—
76	C₂×C₄²×C₁₃₇	—	2	2	no	—
77	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
78	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
80	#107×C₁₃₇	—	2	2	no	—
81	#106×C₁₃₇	—	2	2	no	—
82	#106×C₁₃₇	C₂×C₄	1	4	yes	complemented
83	#102×C₁₃₇	—	2	4	no	—
84	#102×C₁₃₇	—	2	4	no	—
85	#102×C₁₃₇	C₂×C₄	1	8	yes	complemented
86	#97×C₁₃₇	—	2	2	no	—
87	#97×C₁₃₇	D₄	1	4	yes	complemented
88	C₄×C₈×C₁₃₇	—	2	2	no	—
89	C₄×C₈×C₁₃₇	—	2	2	no	—
90	#95×C₁₃₇	—	2	2	no	—
91	#95×C₁₃₇	—	2	2	no	—
92	#95×C₁₃₇	—	2	2	no	—
93	C₂⁴×C₁₃₇	—	2	2	no	—
94	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₁₃₇	—	2	8	no	—
96	C₂×D₄×C₁₃₇	—	2	2	no	—
97	C₂×D₄×C₁₃₇	—	2	2	no	—
98	C₂×D₄×C₁₃₇	—	2	2	no	—
99	C₂×D₄×C₁₃₇	—	2	2	no	—
100	C₂×D₄×C₁₃₇	—	2	2	no	—
101	C₂×D₄×C₁₃₇	—	4	4	no	—
102	C₂×D₄×C₁₃₇	—	4	4	no	—
103	C₂×D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
104	C₂²×C₄×C₁₃₇	—	2	1	no	—
105	C₂²×C₄×C₁₃₇	—	2	1	no	—
106	C₂²×C₄×C₁₃₇	—	2	1	no	—
107	C₂²×C₄×C₁₃₇	—	2	1	no	—
108	C₂²×C₄×C₁₃₇	—	2	2	no	—
109	C₂²×C₄×C₁₃₇	—	2	4	no	—
110	C₂²×C₄×C₁₃₇	—	2	4	no	—
111	C₂²×C₄×C₁₃₇	—	2	2	no	—
112	C₂²×C₄×C₁₃₇	—	2	2	no	—
113	C₂²×C₄×C₁₃₇	—	4	2	no	—
114	C₂²×C₄×C₁₃₇	—	4	4	no	—
115	C₂²×C₄×C₁₃₇	—	4	4	no	—
116	C₂²×C₄×C₁₃₇	—	4	4	no	—
117	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
122	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
123	D₈×C₁₃₇	—	2	8	no	—
124	M₁₆×C₁₃₇	—	2	8	no	—
125	M₁₆×C₁₃₇	—	2	8	no	—
126	C₂×C₈×C₁₃₇	—	2	1	no	—
127	C₂×C₈×C₁₃₇	—	2	1	no	—
128	C₂×C₈×C₁₃₇	—	2	1	no	—
129	C₂×C₈×C₁₃₇	—	2	1	no	—
130	C₂×C₈×C₁₃₇	—	2	1	no	—
131	C₂×C₈×C₁₃₇	—	2	1	no	—
132	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₁₃₇	#41	1	2	yes	—
137	C₂×C₈×C₁₃₇	#41	1	2	yes	—
138	C₄⋊C₄×C₁₃₇	—	2	2	no	—
139	C₄⋊C₄×C₁₃₇	—	2	2	no	—
140	C₄⋊C₄×C₁₃₇	—	2	2	no	—
141	C₄⋊C₄×C₁₃₇	—	2	2	no	—
142	C₄⋊C₄×C₁₃₇	—	2	2	no	—
143	C₄⋊C₄×C₁₃₇	—	4	4	no	—
144	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
145	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
146	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
147	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
148	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
149	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
150	C₄²×C₁₃₇	—	2	4	no	—
151	C₄²×C₁₃₇	—	2	1	no	—
152	C₄²×C₁₃₇	—	2	1	no	—
153	C₄²×C₁₃₇	—	4	4	no	—
154	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
155	C₄²×C₁₃₇	#41	1	2	yes	—
156	C₂³×C₁₃₇	—	2	1	no	—
157	C₂³×C₁₃₇	—	2	1	no	—
158	C₂³×C₁₃₇	—	2	2	no	—
159	C₂³×C₁₃₇	—	2	2	no	—
160	C₂³×C₁₃₇	—	2	4	no	—
161	C₂³×C₁₃₇	—	4	2	no	—
162	C₂³×C₁₃₇	—	4	2	no	—
163	C₂³×C₁₃₇	—	4	2	no	—
164	C₂³×C₁₃₇	—	4	2	no	—
165	C₂³×C₁₃₇	—	4	2	no	—
166	C₂³×C₁₃₇	—	4	2	no	—
167	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
168	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
169	D₄×C₁₃₇	—	2	4	no	—
170	D₄×C₁₃₇	—	2	4	no	—
171	D₄×C₁₃₇	—	4	16	no	—
172	D₄×C₁₃₇	—	4	2	no	—
173	D₄×C₁₃₇	—	4	2	no	—
174	C₂×C₄×C₁₃₇	—	2	2	no	—
175	C₂×C₄×C₁₃₇	—	2	1	no	—
176	C₂×C₄×C₁₃₇	—	2	1	no	—
177	C₂×C₄×C₁₃₇	—	2	2	no	—
178	C₂×C₄×C₁₃₇	—	2	2	no	—
179	C₂×C₄×C₁₃₇	—	2	2	no	—
180	C₂×C₄×C₁₃₇	—	2	2	no	—
181	C₂×C₄×C₁₃₇	—	2	2	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	2	no	—
184	C₂×C₄×C₁₃₇	—	2	4	no	—
185	C₂×C₄×C₁₃₇	—	4	8	no	—
186	C₂×C₄×C₁₃₇	—	4	2	no	—
187	C₂×C₄×C₁₃₇	—	4	2	no	—
188	C₂×C₄×C₁₃₇	—	4	2	no	—
189	C₂×C₄×C₁₃₇	—	4	2	no	—
190	C₂×C₄×C₁₃₇	—	4	2	no	—
191	C₂×C₄×C₁₃₇	—	4	2	no	—
192	C₂×C₄×C₁₃₇	—	4	2	no	—
193	C₂×C₄×C₁₃₇	—	4	2	no	—
194	C₂×C₄×C₁₃₇	—	8	8	no	—
195	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
196	C₂×C₄×C₁₃₇	#121	1	4	yes	—
197	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
198	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
204	C₈×C₁₃₇	—	2	2	no	—
205	C₈×C₁₃₇	—	2	2	no	—
206	C₈×C₁₃₇	—	2	2	no	—
207	C₈×C₁₃₇	C₄×D₄	1	4	yes	complemented
208	C₂²×C₁₃₇	—	2	4	no	—
209	C₂²×C₁₃₇	—	2	2	no	—
210	C₂²×C₁₃₇	—	4	1	no	—
211	C₂²×C₁₃₇	—	4	1	no	—
212	C₂²×C₁₃₇	—	4	2	no	—
213	C₂²×C₁₃₇	—	4	4	no	—
214	C₂²×C₁₃₇	—	4	4	no	—
215	C₂²×C₁₃₇	—	4	4	no	—
216	C₂²×C₁₃₇	—	8	2	no	—
217	C₂²×C₁₃₇	—	8	2	no	—
218	C₂²×C₁₃₇	—	8	2	no	—
219	C₂²×C₁₃₇	—	8	2	no	—
220	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
222	C₂²×C₁₃₇	#532	1	1	yes	characteristic
223	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
224	C₂²×C₁₃₇	#575	1	2	yes	—
225	C₄×C₁₃₇	—	2	4	no	—
226	C₄×C₁₃₇	—	2	4	no	—
227	C₄×C₁₃₇	—	2	2	no	—
228	C₄×C₁₃₇	—	4	4	no	—
229	C₄×C₁₃₇	—	8	2	no	—
230	C₄×C₁₃₇	—	8	2	no	—
231	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
232	C₄×C₁₃₇	C₄×#41	1	1	yes	characteristic
233	C₄×C₁₃₇	#525	1	1	yes	characteristic
234	C₄×C₁₃₇	#529	1	1	yes	characteristic
235	C₂×C₁₃₇	—	4	2	no	—
236	C₂×C₁₃₇	—	8	4	no	—
237	C₂×C₁₃₇	C₄×#121	1	1	yes	characteristic
238	C₂×C₁₃₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
239	C₂×C₁₃₇	#3055	1	2	yes	—
240	C₂×C₁₃₇	#3183	1	1	yes	characteristic
241	C₂×C₁₃₇	#3184	1	1	yes	characteristic
242	#21538	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₁₃₇	#21538	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#469	C₂×C₁₃₇	1	1	yes	characteristic
245	C₂×#442	C₂×C₁₃₇	1	1	yes	characteristic, complemented
246	C₂×C₄×M₁₆	C₂×C₁₃₇	1	1	yes	characteristic, complemented
247	C₄×#121	C₂×C₁₃₇	1	2	yes	complemented
248	#1970	C₂×C₁₃₇	1	1	yes	characteristic, complemented
249	#1941	C₂×C₁₃₇	1	1	yes	characteristic, complemented
250	#1934	C₂×C₁₃₇	1	2	yes	complemented
251	#1895	C₂×C₁₃₇	1	2	yes	complemented
252	#1809	C₂×C₁₃₇	1	1	yes	characteristic, complemented
253	#1776	C₂×C₁₃₇	1	1	yes	characteristic, complemented
254	#1762	C₂×C₁₃₇	1	2	yes	complemented
255	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
256	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
257	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
258	C₂×#121	C₂²×C₁₃₇	1	2	yes	—
259	C₂×C₄×D₄	—	2	2	no	—
260	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
261	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
262	#469	C₄×C₁₃₇	1	8	yes	complemented
263	#442	—	2	4	no	—
264	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
266	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
268	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
269	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
270	C₄×M₁₆	—	2	4	no	—
271	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
272	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
275	#390	C₂²×C₁₃₇	1	2	yes	—
276	#389	C₂²×C₁₃₇	1	2	yes	—
277	#388	—	2	2	no	—
278	#386	—	2	2	no	—
279	#380	C₂²×C₁₃₇	1	2	yes	—
280	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
281	C₄×C₂²⋊C₄	—	2	2	no	—
282	#340	C₂²×C₁₃₇	1	2	yes	—
283	#340	C₂²×C₁₃₇	1	2	yes	—
284	#340	C₂²×C₁₃₇	1	2	yes	—
285	C₂²×D₄	—	2	2	no	—
286	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
287	C₂³×C₄	—	2	2	no	—
288	C₂³×C₄	D₄×C₁₃₇	1	2	yes	—
289	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂×D₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	—	2	2	no	—
297	C₂×M₁₆	—	2	2	no	—
298	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	complemented
299	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
301	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
303	#121	—	2	4	no	—
304	#121	C₂×C₄×C₁₃₇	1	8	yes	complemented
305	C₄×D₄	—	2	4	no	—
306	C₄×D₄	—	2	4	no	—
307	C₄×D₄	—	4	8	no	—
308	C₂×C₄⋊C₄	—	2	2	no	—
309	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
321	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
323	#107	—	2	2	no	—
324	#106	—	2	2	no	—
325	#106	C₂×C₄×C₁₃₇	1	4	yes	complemented
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	C₂×C₄×C₁₃₇	1	8	yes	complemented
329	#97	—	2	2	no	—
330	#97	D₄×C₁₃₇	1	4	yes	complemented
331	C₄×C₈	—	2	2	no	—
332	C₄×C₈	—	2	2	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	#95	—	2	2	no	—
336	C₂⁴	—	2	2	no	—
337	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	4	4	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	C₂²×C₄×C₁₃₇	1	4	yes	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	4	no	—
358	C₂²×C₄	—	4	4	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
364	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
365	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
366	D₈	—	2	8	no	—
367	M₁₆	—	2	8	no	—
368	M₁₆	—	2	8	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
376	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
377	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
379	C₂×C₈	#41×C₁₃₇	1	2	yes	—
380	C₂×C₈	#41×C₁₃₇	1	2	yes	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	4	4	no	—
387	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	—	4	4	no	—
390	C₂²⋊C₄	—	4	4	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₄²	—	2	4	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	4	4	no	—
397	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
398	C₄²	#41×C₁₃₇	1	2	yes	—
399	C₂³	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	4	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
411	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
412	D₄	—	2	4	no	—
413	D₄	—	2	4	no	—
414	D₄	—	4	16	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	4	8	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	8	8	no	—
438	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
439	C₂×C₄	#121×C₁₃₇	1	4	yes	—
440	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
441	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
446	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₈	—	2	2	no	—
450	C₈	C₄×D₄×C₁₃₇	1	4	yes	complemented
451	C₂²	—	2	2	no	—
452	C₂²	—	2	4	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
464	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
465	C₂²	#532×C₁₃₇	1	1	yes	characteristic
466	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
467	C₂²	#575×C₁₃₇	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	4	no	—
472	C₄	—	8	2	no	—
473	C₄	—	8	2	no	—
474	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
475	C₄	C₄×#41×C₁₃₇	1	1	yes	characteristic
476	C₄	#525×C₁₃₇	1	1	yes	characteristic
477	C₄	#529×C₁₃₇	1	1	yes	characteristic
478	C₂	—	4	2	no	—
479	C₂	—	8	4	no	—
480	C₂	C₄×#121×C₁₃₇	1	1	yes	characteristic
481	C₂	C₄×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
482	C₂	#3055×C₁₃₇	1	2	yes	—
483	C₂	#3183×C₁₃₇	1	1	yes	characteristic
484	C₂	#3184×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#469×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#442×C₁₃₇	C₂	semidirect
3. C₂×C₄×M₁₆×C₁₃₇	C₂	semidirect
4. C₄×#121×C₁₃₇	C₂	semidirect
5. #1970×C₁₃₇	C₂	semidirect
6. #1941×C₁₃₇	C₂	semidirect
7. #1934×C₁₃₇	C₂	semidirect
8. #1895×C₁₃₇	C₂	semidirect
9. #1809×C₁₃₇	C₂	semidirect
10. #1776×C₁₃₇	C₂	semidirect
11. #1762×C₁₃₇	C₂	semidirect
19. #469×C₁₃₇	C₄	semidirect
49. C₂×D₈×C₁₃₇	C₂×C₄	semidirect
55. C₂×M₁₆×C₁₃₇	C₂×C₄	semidirect
61. #121×C₁₃₇	C₂×C₄	semidirect
82. #106×C₁₃₇	C₂×C₄	semidirect
85. #102×C₁₃₇	C₂×C₄	semidirect
87. #97×C₁₃₇	D₄	semidirect
207. C₈×C₁₃₇	C₄×D₄	semidirect
242. #21538	C₁₃₇	direct (N × H)
243. C₁₃₇	#21538	direct (N × H)
245. C₂×#442	C₂×C₁₃₇	semidirect
246. C₂×C₄×M₁₆	C₂×C₁₃₇	semidirect
247. C₄×#121	C₂×C₁₃₇	semidirect
248. #1970	C₂×C₁₃₇	semidirect
249. #1941	C₂×C₁₃₇	semidirect
250. #1934	C₂×C₁₃₇	semidirect
251. #1895	C₂×C₁₃₇	semidirect
252. #1809	C₂×C₁₃₇	semidirect
253. #1776	C₂×C₁₃₇	semidirect
254. #1762	C₂×C₁₃₇	semidirect
262. #469	C₄×C₁₃₇	semidirect
292. C₂×D₈	C₂×C₄×C₁₃₇	semidirect
298. C₂×M₁₆	C₂×C₄×C₁₃₇	semidirect
304. #121	C₂×C₄×C₁₃₇	semidirect
325. #106	C₂×C₄×C₁₃₇	semidirect
328. #102	C₂×C₄×C₁₃₇	semidirect
330. #97	D₄×C₁₃₇	semidirect
450. C₈	C₄×D₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#469×C₁₃₇	2
C₂×#442×C₁₃₇	2
C₂×C₄×M₁₆×C₁₃₇	2
C₄×#121×C₁₃₇	2
#1970×C₁₃₇	2
#1941×C₁₃₇	2
#1934×C₁₃₇	2
#1895×C₁₃₇	2
#1809×C₁₃₇	2
#1776×C₁₃₇	2
#1762×C₁₃₇	2
#21538	137

Subgroups — C137×#21538 #46870924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21538 #46870924