Subgroups — C₁₃₇×#20875 #46870261

Order 35072.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#417×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#389×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1865×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1864×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1863×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	#1862×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#1736×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#1728×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×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	C₄×#41×C₁₃₇	—	2	4	no	—
16	C₄×#41×C₁₃₇	—	2	4	no	—
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
20	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
21	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₁₃₇	—	2	1	no	—
24	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
26	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₁₃₇	—	2	1	no	—
29	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
30	#428×C₁₃₇	—	2	4	no	—
31	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
33	#417×C₁₃₇	—	2	4	no	—
34	#417×C₁₃₇	C₄	1	8	yes	complemented
35	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
36	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
37	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
38	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
39	#389×C₁₃₇	—	2	1	no	—
40	#389×C₁₃₇	—	2	1	no	—
41	#389×C₁₃₇	C₂²	1	2	yes	—
42	#389×C₁₃₇	C₂²	1	2	yes	—
43	C₂×#95×C₁₃₇	—	2	1	no	—
44	#340×C₁₃₇	C₂²	1	2	yes	—
45	#340×C₁₃₇	C₂²	1	2	yes	—
46	#336×C₁₃₇	C₂²	1	2	yes	—
47	#336×C₁₃₇	C₂²	1	2	yes	—
48	C₂×#41×C₁₃₇	—	2	2	no	—
49	C₂×#41×C₁₃₇	—	2	2	no	—
50	C₂×#41×C₁₃₇	—	2	2	no	—
51	C₂×#41×C₁₃₇	C₂×C₄	1	4	yes	—
52	C₂×#41×C₁₃₇	D₄	1	4	yes	—
53	C₂×#41×C₁₃₇	Q₈	1	4	yes	—
54	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₁₃₇	—	2	1	no	—
56	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₁₃₇	—	2	1	no	—
58	C₂³×C₄×C₁₃₇	—	2	1	no	—
59	C₂³×C₄×C₁₃₇	—	2	1	no	—
60	C₂³×C₄×C₁₃₇	—	2	1	no	—
61	C₂³×C₄×C₁₃₇	—	2	1	no	—
62	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₁₃₇	—	2	2	no	—
66	C₂×M₁₆×C₁₃₇	—	2	2	no	—
67	C₂×M₁₆×C₁₃₇	—	2	2	no	—
68	C₂×M₁₆×C₁₃₇	—	2	2	no	—
69	C₂×M₁₆×C₁₃₇	—	2	2	no	—
70	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	complemented
71	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
72	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
73	C₄×Q₈×C₁₃₇	—	2	2	no	—
74	C₄×Q₈×C₁₃₇	—	2	2	no	—
75	C₄×Q₈×C₁₃₇	—	2	2	no	—
76	C₄×Q₈×C₁₃₇	—	2	2	no	—
77	C₄×D₄×C₁₃₇	—	2	8	no	—
78	C₄×D₄×C₁₃₇	—	2	8	no	—
79	C₄×D₄×C₁₃₇	—	2	2	no	—
80	C₄×D₄×C₁₃₇	—	2	2	no	—
81	C₄×D₄×C₁₃₇	—	2	2	no	—
82	C₄×D₄×C₁₃₇	—	2	2	no	—
83	#117×C₁₃₇	—	2	8	no	—
84	#117×C₁₃₇	—	2	2	no	—
85	#117×C₁₃₇	D₄	1	4	yes	—
86	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₄⋊C₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	4	yes	complemented
97	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
104	C₂×C₄²×C₁₃₇	—	2	8	no	—
105	C₂×C₄²×C₁₃₇	—	2	1	no	—
106	C₂×C₄²×C₁₃₇	—	2	1	no	—
107	C₂×C₄²×C₁₃₇	—	2	1	no	—
108	C₂×C₄²×C₁₃₇	—	2	2	no	—
109	C₂×C₄²×C₁₃₇	—	2	2	no	—
110	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
114	#107×C₁₃₇	—	2	4	no	—
115	#106×C₁₃₇	—	2	4	no	—
116	#103×C₁₃₇	—	2	4	no	—
117	#103×C₁₃₇	C₂×C₄	1	8	yes	complemented
118	#102×C₁₃₇	—	2	4	no	—
119	#102×C₁₃₇	C₂×C₄	1	8	yes	complemented
120	#95×C₁₃₇	—	2	2	no	—
121	#95×C₁₃₇	—	2	2	no	—
122	C₂⁴×C₁₃₇	—	2	1	no	—
123	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
124	#41×C₁₃₇	—	2	8	no	—
125	#41×C₁₃₇	—	2	8	no	—
126	#41×C₁₃₇	—	2	8	no	—
127	#41×C₁₃₇	C₄⋊C₄	1	16	yes	complemented
128	C₂×Q₈×C₁₃₇	—	2	1	no	—
129	C₂×Q₈×C₁₃₇	—	2	1	no	—
130	C₂×Q₈×C₁₃₇	—	2	1	no	—
131	C₂×Q₈×C₁₃₇	C₂²×C₄	1	2	yes	—
132	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
133	C₂×Q₈×C₁₃₇	C₂×Q₈	1	2	yes	—
134	C₂×D₄×C₁₃₇	—	2	4	no	—
135	C₂×D₄×C₁₃₇	—	2	4	no	—
136	C₂×D₄×C₁₃₇	—	2	4	no	—
137	C₂×D₄×C₁₃₇	—	2	1	no	—
138	C₂×D₄×C₁₃₇	—	2	1	no	—
139	C₂×D₄×C₁₃₇	—	2	1	no	—
140	C₂×D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
141	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
142	C₂×D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
143	C₂²×C₄×C₁₃₇	—	2	1	no	—
144	C₂²×C₄×C₁₃₇	—	2	4	no	—
145	C₂²×C₄×C₁₃₇	—	2	4	no	—
146	C₂²×C₄×C₁₃₇	—	2	4	no	—
147	C₂²×C₄×C₁₃₇	—	2	4	no	—
148	C₂²×C₄×C₁₃₇	—	2	4	no	—
149	C₂²×C₄×C₁₃₇	—	2	4	no	—
150	C₂²×C₄×C₁₃₇	—	2	2	no	—
151	C₂²×C₄×C₁₃₇	—	2	2	no	—
152	C₂²×C₄×C₁₃₇	—	2	2	no	—
153	C₂²×C₄×C₁₃₇	—	2	1	no	—
154	C₂²×C₄×C₁₃₇	—	2	1	no	—
155	C₂²×C₄×C₁₃₇	—	2	2	no	—
156	C₂²×C₄×C₁₃₇	—	2	2	no	—
157	C₂²×C₄×C₁₃₇	—	2	2	no	—
158	C₂²×C₄×C₁₃₇	—	4	1	no	—
159	C₂²×C₄×C₁₃₇	—	4	1	no	—
160	C₂²×C₄×C₁₃₇	—	4	2	no	—
161	C₂²×C₄×C₁₃₇	—	4	2	no	—
162	C₂²×C₄×C₁₃₇	—	4	2	no	—
163	C₂²×C₄×C₁₃₇	—	4	2	no	—
164	C₂²×C₄×C₁₃₇	—	4	2	no	—
165	C₂²×C₄×C₁₃₇	—	4	2	no	—
166	C₂²×C₄×C₁₃₇	—	4	2	no	—
167	C₂²×C₄×C₁₃₇	—	4	2	no	—
168	C₂²×C₄×C₁₃₇	—	4	2	no	—
169	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
170	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₁₃₇	C₂×Q₈	1	2	yes	—
178	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
179	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
180	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
181	M₁₆×C₁₃₇	—	2	8	no	—
182	M₁₆×C₁₃₇	—	4	4	no	—
183	C₂×C₈×C₁₃₇	—	2	2	no	—
184	C₂×C₈×C₁₃₇	—	2	2	no	—
185	C₂×C₈×C₁₃₇	—	2	2	no	—
186	C₂×C₈×C₁₃₇	—	2	2	no	—
187	C₂×C₈×C₁₃₇	—	2	2	no	—
188	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
189	C₄⋊C₄×C₁₃₇	—	2	4	no	—
190	C₄⋊C₄×C₁₃₇	—	2	4	no	—
191	C₄⋊C₄×C₁₃₇	—	2	4	no	—
192	C₄⋊C₄×C₁₃₇	—	2	1	no	—
193	C₄⋊C₄×C₁₃₇	—	2	1	no	—
194	C₄⋊C₄×C₁₃₇	—	2	2	no	—
195	C₄⋊C₄×C₁₃₇	—	2	2	no	—
196	C₄⋊C₄×C₁₃₇	—	2	2	no	—
197	C₄⋊C₄×C₁₃₇	—	4	4	no	—
198	C₄⋊C₄×C₁₃₇	—	4	2	no	—
199	C₄⋊C₄×C₁₃₇	—	4	4	no	—
200	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
201	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
202	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
203	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
204	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
205	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
206	C₄²×C₁₃₇	—	2	4	no	—
207	C₄²×C₁₃₇	—	2	4	no	—
208	C₄²×C₁₃₇	—	2	2	no	—
209	C₄²×C₁₃₇	—	2	4	no	—
210	C₄²×C₁₃₇	—	2	2	no	—
211	C₄²×C₁₃₇	—	2	1	no	—
212	C₄²×C₁₃₇	—	2	1	no	—
213	C₄²×C₁₃₇	—	2	1	no	—
214	C₄²×C₁₃₇	—	2	1	no	—
215	C₂³×C₁₃₇	—	2	1	no	—
216	C₂³×C₁₃₇	—	2	1	no	—
217	C₂³×C₁₃₇	—	2	1	no	—
218	C₂³×C₁₃₇	—	2	1	no	—
219	C₂³×C₁₃₇	—	2	2	no	—
220	C₂³×C₁₃₇	—	2	2	no	—
221	C₂³×C₁₃₇	—	2	2	no	—
222	C₂³×C₁₃₇	—	4	1	no	—
223	C₂³×C₁₃₇	—	4	1	no	—
224	C₂³×C₁₃₇	—	4	1	no	—
225	C₂³×C₁₃₇	—	4	1	no	—
226	C₂³×C₁₃₇	C₄×D₄	1	2	yes	—
227	C₂³×C₁₃₇	#120	1	2	yes	—
228	C₂³×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
229	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
230	Q₈×C₁₃₇	—	2	2	no	—
231	Q₈×C₁₃₇	—	2	2	no	—
232	Q₈×C₁₃₇	—	2	2	no	—
233	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
234	D₄×C₁₃₇	—	2	16	no	—
235	D₄×C₁₃₇	—	2	2	no	—
236	D₄×C₁₃₇	—	2	2	no	—
237	D₄×C₁₃₇	—	2	2	no	—
238	D₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
239	C₂×C₄×C₁₃₇	—	2	2	no	—
240	C₂×C₄×C₁₃₇	—	2	1	no	—
241	C₂×C₄×C₁₃₇	—	2	1	no	—
242	C₂×C₄×C₁₃₇	—	2	2	no	—
243	C₂×C₄×C₁₃₇	—	2	2	no	—
244	C₂×C₄×C₁₃₇	—	2	1	no	—
245	C₂×C₄×C₁₃₇	—	2	1	no	—
246	C₂×C₄×C₁₃₇	—	2	16	no	—
247	C₂×C₄×C₁₃₇	—	2	4	no	—
248	C₂×C₄×C₁₃₇	—	2	1	no	—
249	C₂×C₄×C₁₃₇	—	2	1	no	—
250	C₂×C₄×C₁₃₇	—	2	2	no	—
251	C₂×C₄×C₁₃₇	—	2	2	no	—
252	C₂×C₄×C₁₃₇	—	2	1	no	—
253	C₂×C₄×C₁₃₇	—	2	1	no	—
254	C₂×C₄×C₁₃₇	—	2	1	no	—
255	C₂×C₄×C₁₃₇	—	2	1	no	—
256	C₂×C₄×C₁₃₇	—	4	4	no	—
257	C₂×C₄×C₁₃₇	—	4	4	no	—
258	C₂×C₄×C₁₃₇	—	4	8	no	—
259	C₂×C₄×C₁₃₇	—	4	8	no	—
260	C₂×C₄×C₁₃₇	—	4	8	no	—
261	C₂×C₄×C₁₃₇	—	4	2	no	—
262	C₂×C₄×C₁₃₇	—	4	1	no	—
263	C₂×C₄×C₁₃₇	—	4	1	no	—
264	C₂×C₄×C₁₃₇	—	4	2	no	—
265	C₂×C₄×C₁₃₇	—	4	1	no	—
266	C₂×C₄×C₁₃₇	—	4	1	no	—
267	C₂×C₄×C₁₃₇	—	4	2	no	—
268	C₂×C₄×C₁₃₇	—	4	2	no	—
269	C₂×C₄×C₁₃₇	—	4	2	no	—
270	C₂×C₄×C₁₃₇	—	8	4	no	—
271	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
272	C₂×C₄×C₁₃₇	C₄×D₄	1	4	yes	—
273	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
274	C₂×C₄×C₁₃₇	#120	1	2	yes	—
275	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	4	yes	—
276	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
277	C₂×C₄×C₁₃₇	#123	1	4	yes	—
278	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
279	C₂×C₄×C₁₃₇	#137	1	2	yes	—
280	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
281	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
284	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
286	C₈×C₁₃₇	—	2	4	no	—
287	C₈×C₁₃₇	—	4	2	no	—
288	C₂²×C₁₃₇	—	2	2	no	—
289	C₂²×C₁₃₇	—	2	2	no	—
290	C₂²×C₁₃₇	—	2	2	no	—
291	C₂²×C₁₃₇	—	2	2	no	—
292	C₂²×C₁₃₇	—	2	2	no	—
293	C₂²×C₁₃₇	—	2	2	no	—
294	C₂²×C₁₃₇	—	2	4	no	—
295	C₂²×C₁₃₇	—	4	2	no	—
296	C₂²×C₁₃₇	—	4	2	no	—
297	C₂²×C₁₃₇	—	4	2	no	—
298	C₂²×C₁₃₇	—	4	2	no	—
299	C₂²×C₁₃₇	—	4	2	no	—
300	C₂²×C₁₃₇	—	4	2	no	—
301	C₂²×C₁₃₇	#385	1	4	yes	—
302	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
303	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
304	C₂²×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
307	C₂²×C₁₃₇	#575	1	1	yes	characteristic
308	C₂²×C₁₃₇	#575	1	1	yes	characteristic
309	C₄×C₁₃₇	—	2	2	no	—
310	C₄×C₁₃₇	—	2	2	no	—
311	C₄×C₁₃₇	—	2	4	no	—
312	C₄×C₁₃₇	—	4	4	no	—
313	C₄×C₁₃₇	—	4	1	no	—
314	C₄×C₁₃₇	—	4	2	no	—
315	C₄×C₁₃₇	—	4	1	no	—
316	C₄×C₁₃₇	—	4	2	no	—
317	C₄×C₁₃₇	—	8	2	no	—
318	C₄×C₁₃₇	#385	1	4	yes	—
319	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
320	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
321	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
322	C₄×C₁₃₇	C₂×#123	1	1	yes	characteristic
323	C₂×C₁₃₇	—	2	4	no	—
324	C₂×C₁₃₇	—	4	4	no	—
325	C₂×C₁₃₇	C₂×#385	1	1	yes	characteristic
326	C₂×C₁₃₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
327	C₂×C₁₃₇	C₄×#137	1	1	yes	characteristic
328	C₂×C₁₃₇	#3006	1	1	yes	characteristic
329	C₂×C₁₃₇	#3007	1	1	yes	characteristic
330	C₂×C₁₃₇	#3079	1	1	yes	characteristic
331	C₂×C₁₃₇	#3080	1	1	yes	characteristic
332	#20875	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
333	C₁₃₇	#20875	1	1	yes	characteristic, direct product factor, Sylow
334	C₂×C₄×#41	C₂×C₁₃₇	1	1	yes	characteristic
335	C₂×#428	C₂×C₁₃₇	1	1	yes	characteristic, complemented
336	C₂×#417	C₂×C₁₃₇	1	1	yes	characteristic
337	C₂×#417	C₂×C₁₃₇	1	1	yes	characteristic
338	C₂×#389	C₂×C₁₃₇	1	1	yes	characteristic, complemented
339	#1865	C₂×C₁₃₇	1	2	yes	complemented
340	#1864	C₂×C₁₃₇	1	2	yes	complemented
341	#1863	C₂×C₁₃₇	1	2	yes	complemented
342	#1862	C₂×C₁₃₇	1	2	yes	complemented
343	#1736	C₂×C₁₃₇	1	1	yes	characteristic, complemented
344	#1728	C₂×C₁₃₇	1	1	yes	characteristic, complemented
345	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
346	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
347	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
348	C₄×#41	—	2	4	no	—
349	C₄×#41	—	2	4	no	—
350	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
351	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
352	C₂×C₄×D₄	—	2	1	no	—
353	C₂×C₄×D₄	—	2	1	no	—
354	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
355	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
356	C₂×#117	—	2	1	no	—
357	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
358	C₂²×C₄⋊C₄	—	2	1	no	—
359	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
361	C₂²×C₄²	—	2	1	no	—
362	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
363	#428	—	2	4	no	—
364	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
365	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
366	#417	—	2	4	no	—
367	#417	C₄×C₁₃₇	1	8	yes	complemented
368	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
369	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
370	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
371	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
372	#389	—	2	1	no	—
373	#389	—	2	1	no	—
374	#389	C₂²×C₁₃₇	1	2	yes	—
375	#389	C₂²×C₁₃₇	1	2	yes	—
376	C₂×#95	—	2	1	no	—
377	#340	C₂²×C₁₃₇	1	2	yes	—
378	#340	C₂²×C₁₃₇	1	2	yes	—
379	#336	C₂²×C₁₃₇	1	2	yes	—
380	#336	C₂²×C₁₃₇	1	2	yes	—
381	C₂×#41	—	2	2	no	—
382	C₂×#41	—	2	2	no	—
383	C₂×#41	—	2	2	no	—
384	C₂×#41	C₂×C₄×C₁₃₇	1	4	yes	—
385	C₂×#41	D₄×C₁₃₇	1	4	yes	—
386	C₂×#41	Q₈×C₁₃₇	1	4	yes	—
387	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
388	C₂²×D₄	—	2	1	no	—
389	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	1	no	—
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	—	2	1	no	—
395	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
396	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
398	C₂×M₁₆	—	2	2	no	—
399	C₂×M₁₆	—	2	2	no	—
400	C₂×M₁₆	—	2	2	no	—
401	C₂×M₁₆	—	2	2	no	—
402	C₂×M₁₆	—	2	2	no	—
403	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	complemented
404	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
405	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	2	no	—
408	C₄×Q₈	—	2	2	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×D₄	—	2	8	no	—
411	C₄×D₄	—	2	8	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	#117	—	2	8	no	—
417	#117	—	2	2	no	—
418	#117	D₄×C₁₃₇	1	4	yes	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	2	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	2	no	—
425	C₂×C₄⋊C₄	—	2	2	no	—
426	C₂×C₄⋊C₄	—	2	2	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	4	4	no	—
429	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	4	yes	complemented
430	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
431	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
432	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₄²	—	2	8	no	—
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	2	no	—
442	C₂×C₄²	—	2	2	no	—
443	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
444	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
447	#107	—	2	4	no	—
448	#106	—	2	4	no	—
449	#103	—	2	4	no	—
450	#103	C₂×C₄×C₁₃₇	1	8	yes	complemented
451	#102	—	2	4	no	—
452	#102	C₂×C₄×C₁₃₇	1	8	yes	complemented
453	#95	—	2	2	no	—
454	#95	—	2	2	no	—
455	C₂⁴	—	2	1	no	—
456	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
457	#41	—	2	8	no	—
458	#41	—	2	8	no	—
459	#41	—	2	8	no	—
460	#41	C₄⋊C₄×C₁₃₇	1	16	yes	complemented
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	C₂²×C₄×C₁₃₇	1	2	yes	—
465	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
466	C₂×Q₈	C₂×Q₈×C₁₃₇	1	2	yes	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	C₂²×C₄×C₁₃₇	1	2	yes	—
474	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
475	C₂×D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	4	1	no	—
492	C₂²×C₄	—	4	1	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	—	4	2	no	—
496	C₂²×C₄	—	4	2	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	—	4	2	no	—
500	C₂²×C₄	—	4	2	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
503	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
508	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
509	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
510	C₂²×C₄	C₂×Q₈×C₁₃₇	1	2	yes	—
511	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
512	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
514	M₁₆	—	2	8	no	—
515	M₁₆	—	4	4	no	—
516	C₂×C₈	—	2	2	no	—
517	C₂×C₈	—	2	2	no	—
518	C₂×C₈	—	2	2	no	—
519	C₂×C₈	—	2	2	no	—
520	C₂×C₈	—	2	2	no	—
521	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	4	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	4	4	no	—
531	C₄⋊C₄	—	4	2	no	—
532	C₄⋊C₄	—	4	4	no	—
533	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
534	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
535	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
536	C₂²⋊C₄	—	2	4	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	—	2	4	no	—
539	C₄²	—	2	4	no	—
540	C₄²	—	2	4	no	—
541	C₄²	—	2	2	no	—
542	C₄²	—	2	4	no	—
543	C₄²	—	2	2	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	2	1	no	—
550	C₂³	—	2	1	no	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	2	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	4	1	no	—
559	C₂³	C₄×D₄×C₁₃₇	1	2	yes	—
560	C₂³	#120×C₁₃₇	1	2	yes	—
561	C₂³	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
562	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
563	Q₈	—	2	2	no	—
564	Q₈	—	2	2	no	—
565	Q₈	—	2	2	no	—
566	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	complemented
567	D₄	—	2	16	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	2	no	—
570	D₄	—	2	2	no	—
571	D₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	complemented
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	16	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	8	no	—
592	C₂×C₄	—	4	8	no	—
593	C₂×C₄	—	4	8	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	2	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	8	4	no	—
604	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
605	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
606	C₂×C₄	C₄×D₄×C₁₃₇	1	4	yes	—
607	C₂×C₄	#120×C₁₃₇	1	2	yes	—
608	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
609	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	4	yes	—
610	C₂×C₄	#123×C₁₃₇	1	4	yes	—
611	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
612	C₂×C₄	#137×C₁₃₇	1	2	yes	—
613	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
614	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
615	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
616	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
617	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
618	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
619	C₈	—	2	4	no	—
620	C₈	—	4	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	4	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	4	2	no	—
633	C₂²	—	4	2	no	—
634	C₂²	#385×C₁₃₇	1	4	yes	—
635	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
636	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
637	C₂²	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
638	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
639	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
640	C₂²	#575×C₁₃₇	1	1	yes	characteristic
641	C₂²	#575×C₁₃₇	1	1	yes	characteristic
642	C₄	—	2	4	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	2	no	—
645	C₄	—	4	4	no	—
646	C₄	—	4	1	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	1	no	—
649	C₄	—	4	2	no	—
650	C₄	—	8	2	no	—
651	C₄	#385×C₁₃₇	1	4	yes	—
652	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
653	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
654	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
655	C₄	C₂×#123×C₁₃₇	1	1	yes	characteristic
656	C₂	—	2	4	no	—
657	C₂	—	4	4	no	—
658	C₂	C₂×#385×C₁₃₇	1	1	yes	characteristic
659	C₂	C₄×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
660	C₂	C₄×#137×C₁₃₇	1	1	yes	characteristic
661	C₂	#3006×C₁₃₇	1	1	yes	characteristic
662	C₂	#3007×C₁₃₇	1	1	yes	characteristic
663	C₂	#3079×C₁₃₇	1	1	yes	characteristic
664	C₂	#3080×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	C₂²×#41×C₁₃₇	C₂²
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×#428×C₁₃₇	C₂	semidirect
5. C₂×#389×C₁₃₇	C₂	semidirect
6. #1865×C₁₃₇	C₂	semidirect
7. #1864×C₁₃₇	C₂	semidirect
8. #1863×C₁₃₇	C₂	semidirect
9. #1862×C₁₃₇	C₂	semidirect
10. #1736×C₁₃₇	C₂	semidirect
11. #1728×C₁₃₇	C₂	semidirect
34. #417×C₁₃₇	C₄	semidirect
70. C₂×M₁₆×C₁₃₇	C₂×C₄	semidirect
96. C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	semidirect
117. #103×C₁₃₇	C₂×C₄	semidirect
119. #102×C₁₃₇	C₂×C₄	semidirect
127. #41×C₁₃₇	C₄⋊C₄	semidirect
233. Q₈×C₁₃₇	C₂×C₄⋊C₄	semidirect
238. D₄×C₁₃₇	C₂×C₄⋊C₄	semidirect
332. #20875	C₁₃₇	direct (N × H)
333. C₁₃₇	#20875	direct (N × H)
335. C₂×#428	C₂×C₁₃₇	semidirect
338. C₂×#389	C₂×C₁₃₇	semidirect
339. #1865	C₂×C₁₃₇	semidirect
340. #1864	C₂×C₁₃₇	semidirect
341. #1863	C₂×C₁₃₇	semidirect
342. #1862	C₂×C₁₃₇	semidirect
343. #1736	C₂×C₁₃₇	semidirect
344. #1728	C₂×C₁₃₇	semidirect
367. #417	C₄×C₁₃₇	semidirect
403. C₂×M₁₆	C₂×C₄×C₁₃₇	semidirect
429. C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	semidirect
450. #103	C₂×C₄×C₁₃₇	semidirect
452. #102	C₂×C₄×C₁₃₇	semidirect
460. #41	C₄⋊C₄×C₁₃₇	semidirect
566. Q₈	C₂×C₄⋊C₄×C₁₃₇	semidirect
571. D₄	C₂×C₄⋊C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₁₃₇	2
C₂×#428×C₁₃₇	2
C₂×#417×C₁₃₇	2
C₂×#389×C₁₃₇	2
#1865×C₁₃₇	2
#1864×C₁₃₇	2
#1863×C₁₃₇	2
#1862×C₁₃₇	2
#1736×C₁₃₇	2
#1728×C₁₃₇	2
#20875	137

Subgroups — C137×#20875 #46870261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20875 #46870261