Subgroups — C₁₃₇×#25273 #46874659

Order 35072.

814 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×#487×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#476×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2547×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2369×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2066×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2042×C₁₃₇	C₂	1	2	yes	maximal
7	#1983×C₁₃₇	C₂	1	2	yes	maximal
8	#1981×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1939×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1927×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
16	#487×C₁₃₇	—	2	4	no	—
17	#476×C₁₃₇	—	2	4	no	—
18	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
21	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
22	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	#397×C₁₃₇	—	2	2	no	—
30	#391×C₁₃₇	—	2	2	no	—
31	#391×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	—	2	1	no	—
33	#389×C₁₃₇	C₂²	1	2	yes	—
34	#380×C₁₃₇	—	2	1	no	—
35	#380×C₁₃₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
37	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
40	#340×C₁₃₇	C₂²	1	1	yes	characteristic
41	#340×C₁₃₇	C₂²	1	1	yes	characteristic
42	#336×C₁₃₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₃₇	—	2	1	no	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₁₃₇	—	2	1	no	—
47	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
51	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
52	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
53	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
54	#124×C₁₃₇	—	2	4	no	—
55	#124×C₁₃₇	—	4	2	no	—
56	C₄×D₄×C₁₃₇	—	2	2	no	—
57	C₄×D₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄²×C₁₃₇	—	2	1	no	—
72	C₂×C₄²×C₁₃₇	—	2	2	no	—
73	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
75	#106×C₁₃₇	—	2	4	no	—
76	C₄⋊C₈×C₁₃₇	—	2	2	no	—
77	C₄⋊C₈×C₁₃₇	—	4	2	no	—
78	#103×C₁₃₇	—	2	4	no	—
79	#103×C₁₃₇	—	2	4	no	—
80	#103×C₁₃₇	—	2	4	no	—
81	#103×C₁₃₇	—	4	2	no	—
82	#102×C₁₃₇	—	2	2	no	—
83	#102×C₁₃₇	—	2	4	no	—
84	#102×C₁₃₇	—	2	2	no	—
85	C₄×C₈×C₁₃₇	—	2	2	no	—
86	#95×C₁₃₇	—	2	2	no	—
87	#95×C₁₃₇	—	2	2	no	—
88	#95×C₁₃₇	—	2	1	no	—
89	#95×C₁₃₇	—	2	2	no	—
90	#95×C₁₃₇	—	2	2	no	—
91	#95×C₁₃₇	D₄	1	2	yes	—
92	C₂⁴×C₁₃₇	—	2	1	no	—
93	C₂×Q₈×C₁₃₇	—	2	4	no	—
94	C₂×Q₈×C₁₃₇	—	2	2	no	—
95	C₂×Q₈×C₁₃₇	—	2	2	no	—
96	C₂×Q₈×C₁₃₇	—	2	2	no	—
97	C₂×Q₈×C₁₃₇	—	2	2	no	—
98	C₂×Q₈×C₁₃₇	—	2	2	no	—
99	C₂×Q₈×C₁₃₇	—	2	2	no	—
100	C₂×Q₈×C₁₃₇	—	8	1	no	—
101	C₂×Q₈×C₁₃₇	—	8	1	no	—
102	C₂×D₄×C₁₃₇	—	2	1	no	—
103	C₂×D₄×C₁₃₇	—	2	1	no	—
104	C₂×D₄×C₁₃₇	—	2	1	no	—
105	C₂×D₄×C₁₃₇	—	2	1	no	—
106	C₂×D₄×C₁₃₇	—	2	1	no	—
107	C₂×D₄×C₁₃₇	#41	1	2	yes	—
108	C₂²×C₄×C₁₃₇	—	2	1	no	—
109	C₂²×C₄×C₁₃₇	—	2	2	no	—
110	C₂²×C₄×C₁₃₇	—	2	1	no	—
111	C₂²×C₄×C₁₃₇	—	2	1	no	—
112	C₂²×C₄×C₁₃₇	—	2	2	no	—
113	C₂²×C₄×C₁₃₇	—	4	2	no	—
114	C₂²×C₄×C₁₃₇	—	4	2	no	—
115	C₂²×C₄×C₁₃₇	—	4	2	no	—
116	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₁₃₇	—	2	1	no	—
122	C₂×C₈×C₁₃₇	—	2	1	no	—
123	C₂×C₈×C₁₃₇	—	2	2	no	—
124	C₂×C₈×C₁₃₇	—	2	1	no	—
125	C₂×C₈×C₁₃₇	—	2	1	no	—
126	C₂×C₈×C₁₃₇	—	2	2	no	—
127	C₂×C₈×C₁₃₇	—	2	2	no	—
128	C₂×C₈×C₁₃₇	—	2	1	no	—
129	C₂×C₈×C₁₃₇	—	2	1	no	—
130	C₄⋊C₄×C₁₃₇	—	2	4	no	—
131	C₄⋊C₄×C₁₃₇	—	2	2	no	—
132	C₄⋊C₄×C₁₃₇	—	2	2	no	—
133	C₄⋊C₄×C₁₃₇	—	2	2	no	—
134	C₄⋊C₄×C₁₃₇	—	2	1	no	—
135	C₄⋊C₄×C₁₃₇	—	2	1	no	—
136	C₄⋊C₄×C₁₃₇	—	2	1	no	—
137	C₄⋊C₄×C₁₃₇	—	2	1	no	—
138	C₄⋊C₄×C₁₃₇	—	2	1	no	—
139	C₄⋊C₄×C₁₃₇	—	4	2	no	—
140	C₄⋊C₄×C₁₃₇	—	4	2	no	—
141	C₄⋊C₄×C₁₃₇	—	8	2	no	—
142	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
143	C₄⋊C₄×C₁₃₇	#41	1	2	yes	—
144	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
145	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
146	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
147	C₄²×C₁₃₇	—	2	2	no	—
148	C₄²×C₁₃₇	—	2	2	no	—
149	C₄²×C₁₃₇	—	4	2	no	—
150	C₄²×C₁₃₇	—	8	2	no	—
151	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
152	C₄²×C₁₃₇	#41	1	2	yes	—
153	C₂³×C₁₃₇	—	2	2	no	—
154	C₂³×C₁₃₇	—	4	1	no	—
155	C₂³×C₁₃₇	—	4	1	no	—
156	C₂³×C₁₃₇	—	4	1	no	—
157	C₂³×C₁₃₇	—	4	1	no	—
158	C₂³×C₁₃₇	—	4	1	no	—
159	C₂³×C₁₃₇	—	4	1	no	—
160	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₁₃₇	—	4	2	no	—
162	Q₈×C₁₃₇	—	4	2	no	—
163	Q₈×C₁₃₇	—	4	2	no	—
164	Q₈×C₁₃₇	—	4	2	no	—
165	Q₈×C₁₃₇	—	8	4	no	—
166	D₄×C₁₃₇	—	2	2	no	—
167	D₄×C₁₃₇	—	2	2	no	—
168	D₄×C₁₃₇	—	4	1	no	—
169	D₄×C₁₃₇	—	4	1	no	—
170	C₂×C₄×C₁₃₇	—	2	1	no	—
171	C₂×C₄×C₁₃₇	—	2	1	no	—
172	C₂×C₄×C₁₃₇	—	2	4	no	—
173	C₂×C₄×C₁₃₇	—	2	2	no	—
174	C₂×C₄×C₁₃₇	—	2	2	no	—
175	C₂×C₄×C₁₃₇	—	2	2	no	—
176	C₂×C₄×C₁₃₇	—	2	2	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	—	4	1	no	—
181	C₂×C₄×C₁₃₇	—	4	1	no	—
182	C₂×C₄×C₁₃₇	—	4	2	no	—
183	C₂×C₄×C₁₃₇	—	4	2	no	—
184	C₂×C₄×C₁₃₇	—	4	1	no	—
185	C₂×C₄×C₁₃₇	—	4	2	no	—
186	C₂×C₄×C₁₃₇	—	4	2	no	—
187	C₂×C₄×C₁₃₇	—	4	1	no	—
188	C₂×C₄×C₁₃₇	—	4	2	no	—
189	C₂×C₄×C₁₃₇	—	4	1	no	—
190	C₂×C₄×C₁₃₇	—	4	1	no	—
191	C₂×C₄×C₁₃₇	—	4	1	no	—
192	C₂×C₄×C₁₃₇	—	4	1	no	—
193	C₂×C₄×C₁₃₇	—	8	4	no	—
194	C₂×C₄×C₁₃₇	#121	1	2	yes	—
195	C₂×C₄×C₁₃₇	#121	1	2	yes	—
196	C₂×C₄×C₁₃₇	#135	1	2	yes	—
197	C₂×C₄×C₁₃₇	#137	1	2	yes	—
198	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₁₃₇	C₂×#41	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	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
204	C₈×C₁₃₇	—	4	1	no	—
205	C₈×C₁₃₇	—	4	1	no	—
206	C₈×C₁₃₇	—	4	2	no	—
207	C₂²×C₁₃₇	—	4	2	no	—
208	C₂²×C₁₃₇	—	4	2	no	—
209	C₂²×C₁₃₇	—	4	2	no	—
210	C₂²×C₁₃₇	—	8	1	no	—
211	C₂²×C₁₃₇	—	8	1	no	—
212	C₂²×C₁₃₇	—	8	1	no	—
213	C₂²×C₁₃₇	—	8	1	no	—
214	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
215	C₂²×C₁₃₇	#546	1	1	yes	characteristic
216	C₂²×C₁₃₇	#548	1	1	yes	characteristic
217	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
218	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
219	C₂²×C₁₃₇	#577	1	1	yes	characteristic
220	C₂²×C₁₃₇	#577	1	1	yes	characteristic
221	C₄×C₁₃₇	—	2	2	no	—
222	C₄×C₁₃₇	—	4	2	no	—
223	C₄×C₁₃₇	—	4	2	no	—
224	C₄×C₁₃₇	—	4	4	no	—
225	C₄×C₁₃₇	—	4	2	no	—
226	C₄×C₁₃₇	—	8	2	no	—
227	C₄×C₁₃₇	—	8	1	no	—
228	C₄×C₁₃₇	—	8	1	no	—
229	C₄×C₁₃₇	#464	1	4	yes	—
230	C₄×C₁₃₇	#529	1	1	yes	characteristic
231	C₄×C₁₃₇	#542	1	1	yes	characteristic
232	C₄×C₁₃₇	#548	1	1	yes	characteristic
233	C₄×C₁₃₇	#550	1	1	yes	characteristic
234	C₂×C₁₃₇	—	8	2	no	—
235	C₂×C₁₃₇	#2426	1	1	yes	characteristic
236	C₂×C₁₃₇	C₂×#464	1	1	yes	characteristic
237	C₂×C₁₃₇	#3042	1	1	yes	characteristic
238	C₂×C₁₃₇	#3303	1	1	yes	characteristic
239	C₂×C₁₃₇	#3331	1	1	yes	characteristic
240	C₂×C₁₃₇	#3360	1	1	yes	characteristic
241	C₂×C₁₃₇	#3369	1	1	yes	characteristic
242	#25273	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₁₃₇	#25273	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×#487	C₂×C₁₃₇	1	1	yes	characteristic, complemented
245	C₂×#476	C₂×C₁₃₇	1	1	yes	characteristic
246	#2547	C₂×C₁₃₇	1	1	yes	characteristic
247	#2369	C₂×C₁₃₇	1	1	yes	characteristic, complemented
248	#2066	C₂×C₁₃₇	1	2	yes	complemented
249	#2042	C₂×C₁₃₇	1	2	yes	—
250	#1983	C₂×C₁₃₇	1	2	yes	—
251	#1981	C₂×C₁₃₇	1	1	yes	characteristic, complemented
252	#1939	C₂×C₁₃₇	1	1	yes	characteristic, complemented
253	#1927	C₂×C₁₃₇	1	2	yes	complemented
254	C₄×#102	C₂×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
257	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
258	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
259	#487	—	2	4	no	—
260	#476	—	2	4	no	—
261	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
262	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
264	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
265	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
266	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
267	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
268	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
269	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
270	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
272	#397	—	2	2	no	—
273	#391	—	2	2	no	—
274	#391	C₂²×C₁₃₇	1	2	yes	—
275	#389	—	2	1	no	—
276	#389	C₂²×C₁₃₇	1	2	yes	—
277	#380	—	2	1	no	—
278	#380	—	2	1	no	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
281	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
283	#340	C₂²×C₁₃₇	1	1	yes	characteristic
284	#340	C₂²×C₁₃₇	1	1	yes	characteristic
285	#336	C₂²×C₁₃₇	1	2	yes	—
286	C₂²×Q₈	—	2	1	no	—
287	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
288	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
289	C₂³×C₄	—	2	1	no	—
290	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
292	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
293	C₄⋊Q₈	—	2	2	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	2	no	—
296	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
297	#124	—	2	4	no	—
298	#124	—	4	2	no	—
299	C₄×D₄	—	2	2	no	—
300	C₄×D₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	2	no	—
316	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
317	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
318	#106	—	2	4	no	—
319	C₄⋊C₈	—	2	2	no	—
320	C₄⋊C₈	—	4	2	no	—
321	#103	—	2	4	no	—
322	#103	—	2	4	no	—
323	#103	—	2	4	no	—
324	#103	—	4	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	4	no	—
327	#102	—	2	2	no	—
328	C₄×C₈	—	2	2	no	—
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	#95	—	2	1	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	#95	D₄×C₁₃₇	1	2	yes	—
335	C₂⁴	—	2	1	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	8	1	no	—
344	C₂×Q₈	—	8	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	#41×C₁₃₇	1	2	yes	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
361	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
363	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	8	2	no	—
385	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
386	C₄⋊C₄	#41×C₁₃₇	1	2	yes	—
387	C₂²⋊C₄	—	4	4	no	—
388	C₂²⋊C₄	—	4	4	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	2	2	no	—
392	C₄²	—	4	2	no	—
393	C₄²	—	8	2	no	—
394	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
395	C₄²	#41×C₁₃₇	1	2	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	—	4	1	no	—
403	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
404	Q₈	—	4	2	no	—
405	Q₈	—	4	2	no	—
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	Q₈	—	8	4	no	—
409	D₄	—	2	2	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	1	no	—
412	D₄	—	4	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	8	4	no	—
437	C₂×C₄	#121×C₁₃₇	1	2	yes	—
438	C₂×C₄	#121×C₁₃₇	1	2	yes	—
439	C₂×C₄	#135×C₁₃₇	1	2	yes	—
440	C₂×C₄	#137×C₁₃₇	1	2	yes	—
441	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×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	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
446	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
447	C₈	—	4	1	no	—
448	C₈	—	4	1	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
458	C₂²	#546×C₁₃₇	1	1	yes	characteristic
459	C₂²	#548×C₁₃₇	1	1	yes	characteristic
460	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
461	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
462	C₂²	#577×C₁₃₇	1	1	yes	characteristic
463	C₂²	#577×C₁₃₇	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	4	4	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	8	1	no	—
470	C₄	—	8	2	no	—
471	C₄	—	8	1	no	—
472	C₄	#464×C₁₃₇	1	4	yes	—
473	C₄	#529×C₁₃₇	1	1	yes	characteristic
474	C₄	#542×C₁₃₇	1	1	yes	characteristic
475	C₄	#548×C₁₃₇	1	1	yes	characteristic
476	C₄	#550×C₁₃₇	1	1	yes	characteristic
477	C₂	—	8	2	no	—
478	C₂	#2426×C₁₃₇	1	1	yes	characteristic
479	C₂	C₂×#464×C₁₃₇	1	1	yes	characteristic
480	C₂	#3042×C₁₃₇	1	1	yes	characteristic
481	C₂	#3303×C₁₃₇	1	1	yes	characteristic
482	C₂	#3331×C₁₃₇	1	1	yes	characteristic
483	C₂	#3360×C₁₃₇	1	1	yes	characteristic
484	C₂	#3369×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
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×#487×C₁₃₇	C₂	semidirect
4. #2369×C₁₃₇	C₂	semidirect
5. #2066×C₁₃₇	C₂	semidirect
8. #1981×C₁₃₇	C₂	semidirect
9. #1939×C₁₃₇	C₂	semidirect
10. #1927×C₁₃₇	C₂	semidirect
242. #25273	C₁₃₇	direct (N × H)
243. C₁₃₇	#25273	direct (N × H)
244. C₂×#487	C₂×C₁₃₇	semidirect
247. #2369	C₂×C₁₃₇	semidirect
248. #2066	C₂×C₁₃₇	semidirect
251. #1981	C₂×C₁₃₇	semidirect
252. #1939	C₂×C₁₃₇	semidirect
253. #1927	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#487×C₁₃₇	2
C₂×#476×C₁₃₇	2
#2547×C₁₃₇	2
#2369×C₁₃₇	2
#2066×C₁₃₇	2
#2042×C₁₃₇	2
#1983×C₁₃₇	2
#1981×C₁₃₇	2
#1939×C₁₃₇	2
#1927×C₁₃₇	2
C₄×#102×C₁₃₇	2
#25273	137

Subgroups — C137×#25273 #46874659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25273 #46874659