Subgroups — C₁₃₇×#24750 #46874136

Order 35072.

1216 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#466×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×#459×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2856×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#121×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2067×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2054×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1982×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1947×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1893×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1802×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#121×C₁₃₇	—	2	1	no	—
16	C₂×#121×C₁₃₇	—	2	1	no	—
17	C₂×#121×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#121×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
20	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
22	#466×C₁₃₇	—	2	8	no	—
23	#459×C₁₃₇	—	2	4	no	—
24	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₁₃₇	—	2	4	no	—
26	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
33	#390×C₁₃₇	—	2	1	no	—
34	#390×C₁₃₇	—	2	1	no	—
35	#390×C₁₃₇	C₂²	1	1	yes	characteristic
36	#390×C₁₃₇	C₂²	1	2	yes	—
37	#389×C₁₃₇	C₂²	1	1	yes	characteristic
38	#388×C₁₃₇	—	2	1	no	—
39	#386×C₁₃₇	—	2	1	no	—
40	#384×C₁₃₇	C₂²	1	1	yes	characteristic
41	#380×C₁₃₇	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
44	#340×C₁₃₇	C₂²	1	2	yes	—
45	#340×C₁₃₇	C₂²	1	2	yes	—
46	C₂²×D₄×C₁₃₇	—	2	1	no	—
47	C₂²×D₄×C₁₃₇	—	2	2	no	—
48	C₂²×D₄×C₁₃₇	—	2	1	no	—
49	C₂²×D₄×C₁₃₇	—	2	1	no	—
50	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
51	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₁₃₇	—	2	1	no	—
53	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂×D₈×C₁₃₇	—	2	4	no	—
57	C₂×D₈×C₁₃₇	—	2	4	no	—
58	C₂×D₈×C₁₃₇	—	4	2	no	—
59	C₂²×C₈×C₁₃₇	—	2	4	no	—
60	C₂²×C₈×C₁₃₇	—	2	4	no	—
61	C₂²×C₈×C₁₃₇	—	4	2	no	—
62	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
64	#127×C₁₃₇	—	2	2	no	—
65	#127×C₁₃₇	—	2	2	no	—
66	#121×C₁₃₇	—	2	2	no	—
67	#121×C₁₃₇	—	2	4	no	—
68	#121×C₁₃₇	—	2	4	no	—
69	#121×C₁₃₇	—	4	4	no	—
70	#121×C₁₃₇	—	4	4	no	—
71	#121×C₁₃₇	D₄	1	4	yes	complemented
72	C₄×D₄×C₁₃₇	—	2	2	no	—
73	C₄×D₄×C₁₃₇	—	2	2	no	—
74	C₄×D₄×C₁₃₇	—	4	4	no	—
75	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
90	C₂×C₄²×C₁₃₇	—	2	2	no	—
91	C₂×C₄²×C₁₃₇	—	2	2	no	—
92	C₂×C₄²×C₁₃₇	—	2	1	no	—
93	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
97	#107×C₁₃₇	—	2	4	no	—
98	C₄⋊C₈×C₁₃₇	—	2	4	no	—
99	C₄⋊C₈×C₁₃₇	—	2	2	no	—
100	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	complemented
101	#102×C₁₃₇	—	2	4	no	—
102	#102×C₁₃₇	—	2	4	no	—
103	#102×C₁₃₇	—	2	4	no	—
104	#102×C₁₃₇	—	4	2	no	—
105	#95×C₁₃₇	—	2	1	no	—
106	#95×C₁₃₇	—	2	1	no	—
107	#95×C₁₃₇	—	2	1	no	—
108	C₂⁴×C₁₃₇	—	2	1	no	—
109	C₂⁴×C₁₃₇	—	2	2	no	—
110	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂×D₄×C₁₃₇	—	2	4	no	—
112	C₂×D₄×C₁₃₇	—	2	1	no	—
113	C₂×D₄×C₁₃₇	—	2	2	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	—	2	2	no	—
119	C₂×D₄×C₁₃₇	—	2	1	no	—
120	C₂×D₄×C₁₃₇	—	2	2	no	—
121	C₂×D₄×C₁₃₇	—	2	1	no	—
122	C₂×D₄×C₁₃₇	—	2	2	no	—
123	C₂×D₄×C₁₃₇	—	4	2	no	—
124	C₂×D₄×C₁₃₇	—	4	2	no	—
125	C₂×D₄×C₁₃₇	—	4	2	no	—
126	C₂×D₄×C₁₃₇	—	4	2	no	—
127	C₂×D₄×C₁₃₇	—	4	4	no	—
128	C₂×D₄×C₁₃₇	—	4	4	no	—
129	C₂×D₄×C₁₃₇	—	4	4	no	—
130	C₂×D₄×C₁₃₇	—	4	2	no	—
131	C₂×D₄×C₁₃₇	—	4	2	no	—
132	C₂×D₄×C₁₃₇	—	4	2	no	—
133	C₂×D₄×C₁₃₇	—	4	2	no	—
134	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₁₃₇	—	2	2	no	—
136	C₂²×C₄×C₁₃₇	—	2	2	no	—
137	C₂²×C₄×C₁₃₇	—	2	1	no	—
138	C₂²×C₄×C₁₃₇	—	2	1	no	—
139	C₂²×C₄×C₁₃₇	—	2	1	no	—
140	C₂²×C₄×C₁₃₇	—	2	1	no	—
141	C₂²×C₄×C₁₃₇	—	2	2	no	—
142	C₂²×C₄×C₁₃₇	—	2	1	no	—
143	C₂²×C₄×C₁₃₇	—	2	2	no	—
144	C₂²×C₄×C₁₃₇	—	2	1	no	—
145	C₂²×C₄×C₁₃₇	—	2	1	no	—
146	C₂²×C₄×C₁₃₇	—	2	2	no	—
147	C₂²×C₄×C₁₃₇	—	4	1	no	—
148	C₂²×C₄×C₁₃₇	—	4	2	no	—
149	C₂²×C₄×C₁₃₇	—	4	2	no	—
150	C₂²×C₄×C₁₃₇	—	4	2	no	—
151	C₂²×C₄×C₁₃₇	D₈	1	4	yes	complemented
152	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
159	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
160	D₈×C₁₃₇	—	4	8	no	—
161	C₂×C₈×C₁₃₇	—	2	2	no	—
162	C₂×C₈×C₁₃₇	—	2	2	no	—
163	C₂×C₈×C₁₃₇	—	2	2	no	—
164	C₂×C₈×C₁₃₇	—	2	2	no	—
165	C₂×C₈×C₁₃₇	—	2	2	no	—
166	C₂×C₈×C₁₃₇	—	4	8	no	—
167	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
168	C₄⋊C₄×C₁₃₇	—	2	1	no	—
169	C₄⋊C₄×C₁₃₇	—	2	1	no	—
170	C₄⋊C₄×C₁₃₇	—	2	2	no	—
171	C₄⋊C₄×C₁₃₇	—	2	1	no	—
172	C₄⋊C₄×C₁₃₇	—	2	2	no	—
173	C₄⋊C₄×C₁₃₇	—	2	1	no	—
174	C₄⋊C₄×C₁₃₇	—	2	1	no	—
175	C₄⋊C₄×C₁₃₇	—	4	2	no	—
176	C₄⋊C₄×C₁₃₇	—	4	2	no	—
177	C₄⋊C₄×C₁₃₇	—	4	2	no	—
178	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
179	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
180	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
181	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
182	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
183	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
184	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
185	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
186	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
187	C₄²×C₁₃₇	—	2	2	no	—
188	C₄²×C₁₃₇	—	2	2	no	—
189	C₄²×C₁₃₇	—	2	1	no	—
190	C₄²×C₁₃₇	—	2	1	no	—
191	C₄²×C₁₃₇	—	4	2	no	—
192	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
193	C₄²×C₁₃₇	#41	1	2	yes	—
194	C₂³×C₁₃₇	—	2	1	no	—
195	C₂³×C₁₃₇	—	2	1	no	—
196	C₂³×C₁₃₇	—	2	1	no	—
197	C₂³×C₁₃₇	—	2	1	no	—
198	C₂³×C₁₃₇	—	2	2	no	—
199	C₂³×C₁₃₇	—	4	1	no	—
200	C₂³×C₁₃₇	—	4	1	no	—
201	C₂³×C₁₃₇	—	4	1	no	—
202	C₂³×C₁₃₇	—	4	1	no	—
203	C₂³×C₁₃₇	—	4	1	no	—
204	C₂³×C₁₃₇	—	4	1	no	—
205	C₂³×C₁₃₇	—	4	2	no	—
206	C₂³×C₁₃₇	—	4	2	no	—
207	C₂³×C₁₃₇	—	4	2	no	—
208	C₂³×C₁₃₇	—	4	2	no	—
209	C₂³×C₁₃₇	—	4	2	no	—
210	C₂³×C₁₃₇	—	4	2	no	—
211	C₂³×C₁₃₇	—	4	2	no	—
212	C₂³×C₁₃₇	#121	1	2	yes	—
213	C₂³×C₁₃₇	C₂×D₈	1	2	yes	—
214	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
215	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
216	D₄×C₁₃₇	—	2	2	no	—
217	D₄×C₁₃₇	—	2	2	no	—
218	D₄×C₁₃₇	—	4	8	no	—
219	D₄×C₁₃₇	—	4	2	no	—
220	D₄×C₁₃₇	—	4	2	no	—
221	D₄×C₁₃₇	—	4	1	no	—
222	D₄×C₁₃₇	—	4	2	no	—
223	D₄×C₁₃₇	—	4	1	no	—
224	D₄×C₁₃₇	—	4	2	no	—
225	D₄×C₁₃₇	—	8	4	no	—
226	D₄×C₁₃₇	—	8	8	no	—
227	D₄×C₁₃₇	—	8	4	no	—
228	C₂×C₄×C₁₃₇	—	2	1	no	—
229	C₂×C₄×C₁₃₇	—	2	1	no	—
230	C₂×C₄×C₁₃₇	—	2	1	no	—
231	C₂×C₄×C₁₃₇	—	2	1	no	—
232	C₂×C₄×C₁₃₇	—	2	1	no	—
233	C₂×C₄×C₁₃₇	—	2	1	no	—
234	C₂×C₄×C₁₃₇	—	2	2	no	—
235	C₂×C₄×C₁₃₇	—	2	2	no	—
236	C₂×C₄×C₁₃₇	—	2	2	no	—
237	C₂×C₄×C₁₃₇	—	2	2	no	—
238	C₂×C₄×C₁₃₇	—	2	8	no	—
239	C₂×C₄×C₁₃₇	—	2	1	no	—
240	C₂×C₄×C₁₃₇	—	2	2	no	—
241	C₂×C₄×C₁₃₇	—	2	1	no	—
242	C₂×C₄×C₁₃₇	—	2	1	no	—
243	C₂×C₄×C₁₃₇	—	4	4	no	—
244	C₂×C₄×C₁₃₇	—	4	1	no	—
245	C₂×C₄×C₁₃₇	—	4	1	no	—
246	C₂×C₄×C₁₃₇	—	4	1	no	—
247	C₂×C₄×C₁₃₇	—	4	2	no	—
248	C₂×C₄×C₁₃₇	—	4	1	no	—
249	C₂×C₄×C₁₃₇	—	4	1	no	—
250	C₂×C₄×C₁₃₇	—	4	1	no	—
251	C₂×C₄×C₁₃₇	—	4	2	no	—
252	C₂×C₄×C₁₃₇	—	4	1	no	—
253	C₂×C₄×C₁₃₇	—	4	2	no	—
254	C₂×C₄×C₁₃₇	—	4	1	no	—
255	C₂×C₄×C₁₃₇	—	8	4	no	—
256	C₂×C₄×C₁₃₇	#121	1	2	yes	—
257	C₂×C₄×C₁₃₇	#121	1	2	yes	—
258	C₂×C₄×C₁₃₇	#121	1	2	yes	—
259	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
260	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
261	C₂×C₄×C₁₃₇	#135	1	2	yes	—
262	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
266	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
267	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
268	C₈×C₁₃₇	—	2	4	no	—
269	C₈×C₁₃₇	—	4	2	no	—
270	C₂²×C₁₃₇	—	2	2	no	—
271	C₂²×C₁₃₇	—	2	2	no	—
272	C₂²×C₁₃₇	—	2	2	no	—
273	C₂²×C₁₃₇	—	2	2	no	—
274	C₂²×C₁₃₇	—	2	2	no	—
275	C₂²×C₁₃₇	—	2	2	no	—
276	C₂²×C₁₃₇	—	4	2	no	—
277	C₂²×C₁₃₇	—	4	2	no	—
278	C₂²×C₁₃₇	—	4	2	no	—
279	C₂²×C₁₃₇	—	8	1	no	—
280	C₂²×C₁₃₇	—	8	1	no	—
281	C₂²×C₁₃₇	—	8	1	no	—
282	C₂²×C₁₃₇	—	8	1	no	—
283	C₂²×C₁₃₇	—	8	2	no	—
284	C₂²×C₁₃₇	—	8	2	no	—
285	C₂²×C₁₃₇	—	8	2	no	—
286	C₂²×C₁₃₇	—	8	2	no	—
287	C₂²×C₁₃₇	—	8	2	no	—
288	C₂²×C₁₃₇	—	8	2	no	—
289	C₂²×C₁₃₇	—	8	2	no	—
290	C₂²×C₁₃₇	#459	1	4	yes	complemented
291	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
292	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
293	C₂²×C₁₃₇	#537	1	1	yes	characteristic
294	C₂²×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
295	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
296	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
297	C₂²×C₁₃₇	#575	1	1	yes	characteristic
298	C₄×C₁₃₇	—	2	2	no	—
299	C₄×C₁₃₇	—	2	2	no	—
300	C₄×C₁₃₇	—	2	2	no	—
301	C₄×C₁₃₇	—	4	2	no	—
302	C₄×C₁₃₇	—	4	1	no	—
303	C₄×C₁₃₇	—	8	1	no	—
304	C₄×C₁₃₇	—	8	2	no	—
305	C₄×C₁₃₇	—	8	1	no	—
306	C₄×C₁₃₇	#466	1	4	yes	complemented
307	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
308	C₄×C₁₃₇	#547	1	1	yes	characteristic
309	C₄×C₁₃₇	#548	1	1	yes	characteristic
310	C₄×C₁₃₇	#553	1	1	yes	characteristic
311	C₂×C₁₃₇	—	2	4	no	—
312	C₂×C₁₃₇	—	8	2	no	—
313	C₂×C₁₃₇	—	16	2	no	—
314	C₂×C₁₃₇	#2416	1	1	yes	characteristic
315	C₂×C₁₃₇	C₂×#459	1	1	yes	characteristic
316	C₂×C₁₃₇	#3035	1	1	yes	characteristic
317	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
318	C₂×C₁₃₇	#3055	1	1	yes	characteristic
319	C₂×C₁₃₇	#3245	1	1	yes	characteristic
320	C₂×C₁₃₇	#3247	1	1	yes	characteristic
321	#24750	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
322	C₁₃₇	#24750	1	1	yes	characteristic, direct product factor, Sylow
323	C₂×#466	C₂×C₁₃₇	1	2	yes	complemented
324	C₂×#459	C₂×C₁₃₇	1	1	yes	characteristic, complemented
325	C₂²×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
326	#2856	C₂×C₁₃₇	1	1	yes	characteristic, complemented
327	C₄×#121	C₂×C₁₃₇	1	1	yes	characteristic, complemented
328	#2067	C₂×C₁₃₇	1	2	yes	complemented
329	#2054	C₂×C₁₃₇	1	2	yes	complemented
330	#1982	C₂×C₁₃₇	1	1	yes	characteristic, complemented
331	#1947	C₂×C₁₃₇	1	1	yes	characteristic, complemented
332	#1893	C₂×C₁₃₇	1	2	yes	complemented
333	#1802	C₂×C₁₃₇	1	1	yes	characteristic, complemented
334	C₂²×D₈	C₂²×C₁₃₇	1	2	yes	—
335	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
336	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
337	C₂×#121	—	2	1	no	—
338	C₂×#121	—	2	1	no	—
339	C₂×#121	C₂²×C₁₃₇	1	2	yes	—
340	C₂×#121	C₂²×C₁₃₇	1	1	yes	characteristic
341	C₂×C₄×D₄	—	2	1	no	—
342	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
343	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
344	#466	—	2	8	no	—
345	#459	—	2	4	no	—
346	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
347	C₂×C₄⋊C₈	—	2	4	no	—
348	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
349	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
350	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
351	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
352	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
353	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
354	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
355	#390	—	2	1	no	—
356	#390	—	2	1	no	—
357	#390	C₂²×C₁₃₇	1	1	yes	characteristic
358	#390	C₂²×C₁₃₇	1	2	yes	—
359	#389	C₂²×C₁₃₇	1	1	yes	characteristic
360	#388	—	2	1	no	—
361	#386	—	2	1	no	—
362	#384	C₂²×C₁₃₇	1	1	yes	characteristic
363	#380	C₂²×C₁₃₇	1	1	yes	characteristic
364	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
365	C₄×C₂²⋊C₄	—	2	1	no	—
366	#340	C₂²×C₁₃₇	1	2	yes	—
367	#340	C₂²×C₁₃₇	1	2	yes	—
368	C₂²×D₄	—	2	1	no	—
369	C₂²×D₄	—	2	2	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	1	no	—
372	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
373	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
374	C₂³×C₄	—	2	1	no	—
375	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
376	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
378	C₂×D₈	—	2	4	no	—
379	C₂×D₈	—	2	4	no	—
380	C₂×D₈	—	4	2	no	—
381	C₂²×C₈	—	2	4	no	—
382	C₂²×C₈	—	2	4	no	—
383	C₂²×C₈	—	4	2	no	—
384	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
385	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
386	#127	—	2	2	no	—
387	#127	—	2	2	no	—
388	#121	—	2	2	no	—
389	#121	—	2	4	no	—
390	#121	—	2	4	no	—
391	#121	—	4	4	no	—
392	#121	—	4	4	no	—
393	#121	D₄×C₁₃₇	1	4	yes	complemented
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	4	4	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₂²⋊C₄	—	2	2	no	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
416	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
418	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
419	#107	—	2	4	no	—
420	C₄⋊C₈	—	2	4	no	—
421	C₄⋊C₈	—	2	2	no	—
422	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	complemented
423	#102	—	2	4	no	—
424	#102	—	2	4	no	—
425	#102	—	2	4	no	—
426	#102	—	4	2	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	C₂⁴	—	2	1	no	—
431	C₂⁴	—	2	2	no	—
432	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	—	4	4	no	—
452	C₂×D₄	—	4	2	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	—	4	2	no	—
456	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	4	1	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	D₈×C₁₃₇	1	4	yes	complemented
474	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
480	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
481	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
482	D₈	—	4	8	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	2	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	4	8	no	—
489	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	4	2	no	—
503	C₂²⋊C₄	—	4	4	no	—
504	C₂²⋊C₄	—	4	2	no	—
505	C₂²⋊C₄	—	4	2	no	—
506	C₂²⋊C₄	—	4	4	no	—
507	C₂²⋊C₄	—	4	2	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₄²	—	2	2	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	4	2	no	—
514	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
515	C₄²	#41×C₁₃₇	1	2	yes	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	4	1	no	—
522	C₂³	—	4	1	no	—
523	C₂³	—	4	1	no	—
524	C₂³	—	4	1	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	2	no	—
529	C₂³	—	4	2	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	4	2	no	—
534	C₂³	#121×C₁₃₇	1	2	yes	—
535	C₂³	C₂×D₈×C₁₃₇	1	2	yes	—
536	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
537	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
538	D₄	—	2	2	no	—
539	D₄	—	2	2	no	—
540	D₄	—	4	8	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	—	4	1	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	1	no	—
546	D₄	—	4	2	no	—
547	D₄	—	8	4	no	—
548	D₄	—	8	8	no	—
549	D₄	—	8	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	8	4	no	—
578	C₂×C₄	#121×C₁₃₇	1	2	yes	—
579	C₂×C₄	#121×C₁₃₇	1	2	yes	—
580	C₂×C₄	#121×C₁₃₇	1	2	yes	—
581	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
582	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
583	C₂×C₄	#135×C₁₃₇	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
587	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
588	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
589	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
590	C₈	—	2	4	no	—
591	C₈	—	4	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	8	1	no	—
602	C₂²	—	8	1	no	—
603	C₂²	—	8	1	no	—
604	C₂²	—	8	1	no	—
605	C₂²	—	8	2	no	—
606	C₂²	—	8	2	no	—
607	C₂²	—	8	2	no	—
608	C₂²	—	8	2	no	—
609	C₂²	—	8	2	no	—
610	C₂²	—	8	2	no	—
611	C₂²	—	8	2	no	—
612	C₂²	#459×C₁₃₇	1	4	yes	complemented
613	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
614	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
615	C₂²	#537×C₁₃₇	1	1	yes	characteristic
616	C₂²	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
617	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
618	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
619	C₂²	#575×C₁₃₇	1	1	yes	characteristic
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	4	2	no	—
625	C₄	—	8	1	no	—
626	C₄	—	8	1	no	—
627	C₄	—	8	2	no	—
628	C₄	#466×C₁₃₇	1	4	yes	complemented
629	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
630	C₄	#547×C₁₃₇	1	1	yes	characteristic
631	C₄	#548×C₁₃₇	1	1	yes	characteristic
632	C₄	#553×C₁₃₇	1	1	yes	characteristic
633	C₂	—	2	4	no	—
634	C₂	—	8	2	no	—
635	C₂	—	16	2	no	—
636	C₂	#2416×C₁₃₇	1	1	yes	characteristic
637	C₂	C₂×#459×C₁₃₇	1	1	yes	characteristic
638	C₂	#3035×C₁₃₇	1	1	yes	characteristic
639	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
640	C₂	#3055×C₁₃₇	1	1	yes	characteristic
641	C₂	#3245×C₁₃₇	1	1	yes	characteristic
642	C₂	#3247×C₁₃₇	1	1	yes	characteristic

	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 ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#466×C₁₃₇	C₂	semidirect
2. C₂×#459×C₁₃₇	C₂	semidirect
3. C₂²×C₄⋊C₈×C₁₃₇	C₂	semidirect
4. #2856×C₁₃₇	C₂	semidirect
5. C₄×#121×C₁₃₇	C₂	semidirect
6. #2067×C₁₃₇	C₂	semidirect
7. #2054×C₁₃₇	C₂	semidirect
8. #1982×C₁₃₇	C₂	semidirect
9. #1947×C₁₃₇	C₂	semidirect
10. #1893×C₁₃₇	C₂	semidirect
11. #1802×C₁₃₇	C₂	semidirect
71. #121×C₁₃₇	D₄	semidirect
100. C₄⋊C₈×C₁₃₇	D₄	semidirect
151. C₂²×C₄×C₁₃₇	D₈	semidirect
290. C₂²×C₁₃₇	#459	semidirect
306. C₄×C₁₃₇	#466	semidirect
321. #24750	C₁₃₇	direct (N × H)
322. C₁₃₇	#24750	direct (N × H)
323. C₂×#466	C₂×C₁₃₇	semidirect
324. C₂×#459	C₂×C₁₃₇	semidirect
325. C₂²×C₄⋊C₈	C₂×C₁₃₇	semidirect
326. #2856	C₂×C₁₃₇	semidirect
327. C₄×#121	C₂×C₁₃₇	semidirect
328. #2067	C₂×C₁₃₇	semidirect
329. #2054	C₂×C₁₃₇	semidirect
330. #1982	C₂×C₁₃₇	semidirect
331. #1947	C₂×C₁₃₇	semidirect
332. #1893	C₂×C₁₃₇	semidirect
333. #1802	C₂×C₁₃₇	semidirect
393. #121	D₄×C₁₃₇	semidirect
422. C₄⋊C₈	D₄×C₁₃₇	semidirect
473. C₂²×C₄	D₈×C₁₃₇	semidirect
612. C₂²	#459×C₁₃₇	semidirect
628. C₄	#466×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#466×C₁₃₇	2
C₂×#459×C₁₃₇	2
C₂²×C₄⋊C₈×C₁₃₇	2
#2856×C₁₃₇	2
C₄×#121×C₁₃₇	2
#2067×C₁₃₇	2
#2054×C₁₃₇	2
#1982×C₁₃₇	2
#1947×C₁₃₇	2
#1893×C₁₃₇	2
#1802×C₁₃₇	2
#24750	137

Subgroups — C137×#24750 #46874136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24750 #46874136