Subgroups — C₁₃₇×#25914 #46875300

Order 35072.

1010 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#491×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#487×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2621×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2026×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2024×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1993×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1991×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1873×C₁₃₇	C₂	1	2	yes	maximal
10	#1872×C₁₃₇	C₂	1	2	yes	maximal
11	#1750×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	#544×C₁₃₇	—	2	4	no	—
13	#544×C₁₃₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₁₃₇	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₁₃₇	—	2	1	no	—
19	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
22	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	#491×C₁₃₇	—	2	4	no	—
28	#487×C₁₃₇	—	2	4	no	—
29	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
34	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
36	#398×C₁₃₇	—	2	1	no	—
37	#398×C₁₃₇	—	2	1	no	—
38	#397×C₁₃₇	—	2	1	no	—
39	#397×C₁₃₇	—	2	1	no	—
40	#391×C₁₃₇	C₂²	1	2	yes	—
41	#382×C₁₃₇	C₂²	1	1	yes	characteristic
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#340×C₁₃₇	C₂²	1	2	yes	—
44	#340×C₁₃₇	C₂²	1	2	yes	—
45	#340×C₁₃₇	C₂²	1	2	yes	—
46	#336×C₁₃₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₁₃₇	—	2	1	no	—
48	C₂²×Q₈×C₁₃₇	—	2	1	no	—
49	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₁₃₇	—	2	1	no	—
51	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
55	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
57	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
58	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
61	#125×C₁₃₇	—	2	8	no	—
62	#125×C₁₃₇	—	2	2	no	—
63	#125×C₁₃₇	—	2	2	no	—
64	#125×C₁₃₇	—	2	2	no	—
65	#125×C₁₃₇	D₄	1	4	yes	—
66	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
67	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
68	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
69	#117×C₁₃₇	—	2	2	no	—
70	#117×C₁₃₇	—	2	2	no	—
71	#117×C₁₃₇	—	2	2	no	—
72	#117×C₁₃₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	2	1	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	—	2	1	no	—
79	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₁₃₇	—	2	1	no	—
97	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
99	#103×C₁₃₇	—	2	4	no	—
100	#103×C₁₃₇	—	2	4	no	—
101	#103×C₁₃₇	—	2	4	no	—
102	#103×C₁₃₇	—	2	4	no	—
103	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
104	#97×C₁₃₇	—	2	2	no	—
105	C₄×C₈×C₁₃₇	—	2	2	no	—
106	#95×C₁₃₇	—	2	1	no	—
107	#95×C₁₃₇	—	2	1	no	—
108	#95×C₁₃₇	—	2	2	no	—
109	#95×C₁₃₇	—	2	2	no	—
110	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₁₃₇	—	2	4	no	—
112	C₂×Q₈×C₁₃₇	—	2	4	no	—
113	C₂×Q₈×C₁₃₇	—	2	2	no	—
114	C₂×Q₈×C₁₃₇	—	2	2	no	—
115	C₂×Q₈×C₁₃₇	—	2	4	no	—
116	C₂×Q₈×C₁₃₇	—	2	4	no	—
117	C₂×Q₈×C₁₃₇	—	8	2	no	—
118	C₂×Q₈×C₁₃₇	—	8	2	no	—
119	C₂²×C₄×C₁₃₇	—	2	1	no	—
120	C₂²×C₄×C₁₃₇	—	2	1	no	—
121	C₂²×C₄×C₁₃₇	—	2	2	no	—
122	C₂²×C₄×C₁₃₇	—	2	1	no	—
123	C₂²×C₄×C₁₃₇	—	2	1	no	—
124	C₂²×C₄×C₁₃₇	—	2	1	no	—
125	C₂²×C₄×C₁₃₇	—	2	1	no	—
126	C₂²×C₄×C₁₃₇	—	2	4	no	—
127	C₂²×C₄×C₁₃₇	—	2	2	no	—
128	C₂²×C₄×C₁₃₇	—	2	2	no	—
129	C₂²×C₄×C₁₃₇	—	8	2	no	—
130	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
136	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
137	C₂×C₈×C₁₃₇	—	2	2	no	—
138	C₂×C₈×C₁₃₇	—	2	2	no	—
139	C₂×C₈×C₁₃₇	—	2	2	no	—
140	C₂×C₈×C₁₃₇	—	2	2	no	—
141	C₂×C₈×C₁₃₇	—	2	2	no	—
142	C₂×C₈×C₁₃₇	—	2	2	no	—
143	C₄⋊C₄×C₁₃₇	—	2	4	no	—
144	C₄⋊C₄×C₁₃₇	—	2	4	no	—
145	C₄⋊C₄×C₁₃₇	—	2	4	no	—
146	C₄⋊C₄×C₁₃₇	—	2	4	no	—
147	C₄⋊C₄×C₁₃₇	—	2	4	no	—
148	C₄⋊C₄×C₁₃₇	—	2	4	no	—
149	C₄⋊C₄×C₁₃₇	—	2	4	no	—
150	C₄⋊C₄×C₁₃₇	—	2	2	no	—
151	C₄⋊C₄×C₁₃₇	—	2	2	no	—
152	C₄⋊C₄×C₁₃₇	—	2	1	no	—
153	C₄⋊C₄×C₁₃₇	—	2	1	no	—
154	C₄⋊C₄×C₁₃₇	—	2	1	no	—
155	C₄⋊C₄×C₁₃₇	—	2	1	no	—
156	C₄⋊C₄×C₁₃₇	—	2	4	no	—
157	C₄⋊C₄×C₁₃₇	—	4	2	no	—
158	C₄⋊C₄×C₁₃₇	—	4	2	no	—
159	C₄⋊C₄×C₁₃₇	—	4	4	no	—
160	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
164	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
165	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
166	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
167	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
168	C₄²×C₁₃₇	—	2	4	no	—
169	C₄²×C₁₃₇	—	2	1	no	—
170	C₄²×C₁₃₇	—	2	1	no	—
171	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
172	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
173	C₂³×C₁₃₇	—	2	1	no	—
174	C₂³×C₁₃₇	—	2	1	no	—
175	C₂³×C₁₃₇	—	2	2	no	—
176	C₂³×C₁₃₇	—	2	2	no	—
177	C₂³×C₁₃₇	#124	1	2	yes	—
178	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₁₃₇	—	4	4	no	—
180	Q₈×C₁₃₇	—	4	4	no	—
181	Q₈×C₁₃₇	—	8	4	no	—
182	Q₈×C₁₃₇	—	8	4	no	—
183	C₂×C₄×C₁₃₇	—	2	1	no	—
184	C₂×C₄×C₁₃₇	—	2	1	no	—
185	C₂×C₄×C₁₃₇	—	2	2	no	—
186	C₂×C₄×C₁₃₇	—	2	2	no	—
187	C₂×C₄×C₁₃₇	—	2	4	no	—
188	C₂×C₄×C₁₃₇	—	2	4	no	—
189	C₂×C₄×C₁₃₇	—	2	1	no	—
190	C₂×C₄×C₁₃₇	—	2	2	no	—
191	C₂×C₄×C₁₃₇	—	2	1	no	—
192	C₂×C₄×C₁₃₇	—	2	2	no	—
193	C₂×C₄×C₁₃₇	—	2	2	no	—
194	C₂×C₄×C₁₃₇	—	2	2	no	—
195	C₂×C₄×C₁₃₇	—	2	2	no	—
196	C₂×C₄×C₁₃₇	—	2	2	no	—
197	C₂×C₄×C₁₃₇	—	4	2	no	—
198	C₂×C₄×C₁₃₇	—	4	2	no	—
199	C₂×C₄×C₁₃₇	—	4	2	no	—
200	C₂×C₄×C₁₃₇	—	4	2	no	—
201	C₂×C₄×C₁₃₇	—	4	2	no	—
202	C₂×C₄×C₁₃₇	—	4	2	no	—
203	C₂×C₄×C₁₃₇	—	4	2	no	—
204	C₂×C₄×C₁₃₇	—	4	2	no	—
205	C₂×C₄×C₁₃₇	—	4	2	no	—
206	C₂×C₄×C₁₃₇	—	4	2	no	—
207	C₂×C₄×C₁₃₇	—	8	4	no	—
208	C₂×C₄×C₁₃₇	#120	1	2	yes	—
209	C₂×C₄×C₁₃₇	#120	1	2	yes	—
210	C₂×C₄×C₁₃₇	#124	1	2	yes	—
211	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
214	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
215	C₈×C₁₃₇	—	4	2	no	—
216	C₈×C₁₃₇	—	4	2	no	—
217	C₂²×C₁₃₇	—	2	4	no	—
218	C₂²×C₁₃₇	—	2	2	no	—
219	C₂²×C₁₃₇	—	4	2	no	—
220	C₂²×C₁₃₇	—	4	1	no	—
221	C₂²×C₁₃₇	—	4	1	no	—
222	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
223	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
224	C₂²×C₁₃₇	#536	1	1	yes	characteristic
225	C₂²×C₁₃₇	#577	1	2	yes	—
226	C₂²×C₁₃₇	#578	1	2	yes	—
227	C₄×C₁₃₇	—	2	2	no	—
228	C₄×C₁₃₇	—	2	2	no	—
229	C₄×C₁₃₇	—	2	2	no	—
230	C₄×C₁₃₇	—	4	1	no	—
231	C₄×C₁₃₇	—	4	2	no	—
232	C₄×C₁₃₇	—	4	2	no	—
233	C₄×C₁₃₇	—	4	2	no	—
234	C₄×C₁₃₇	—	4	1	no	—
235	C₄×C₁₃₇	—	4	2	no	—
236	C₄×C₁₃₇	—	8	2	no	—
237	C₄×C₁₃₇	—	8	2	no	—
238	C₄×C₁₃₇	#548	1	2	yes	—
239	C₄×C₁₃₇	#548	1	2	yes	—
240	C₂×C₁₃₇	—	4	2	no	—
241	C₂×C₁₃₇	#2436	1	1	yes	characteristic
242	C₂×C₁₃₇	#3009	1	1	yes	characteristic
243	C₂×C₁₃₇	#3011	1	1	yes	characteristic
244	C₂×C₁₃₇	#3137	1	2	yes	—
245	C₂×C₁₃₇	#3242	1	2	yes	—
246	#25914	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₁₃₇	#25914	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×#544	C₂×C₁₃₇	1	1	yes	characteristic
249	C₂×#491	C₂×C₁₃₇	1	1	yes	characteristic, complemented
250	C₂×#487	C₂×C₁₃₇	1	1	yes	characteristic, complemented
251	#2621	C₂×C₁₃₇	1	1	yes	characteristic
252	#2026	C₂×C₁₃₇	1	2	yes	complemented
253	#2024	C₂×C₁₃₇	1	2	yes	complemented
254	#1993	C₂×C₁₃₇	1	1	yes	characteristic, complemented
255	#1991	C₂×C₁₃₇	1	1	yes	characteristic, complemented
256	#1873	C₂×C₁₃₇	1	2	yes	—
257	#1872	C₂×C₁₃₇	1	2	yes	—
258	#1750	C₂×C₁₃₇	1	1	yes	characteristic
259	#544	—	2	4	no	—
260	#544	—	2	4	no	—
261	C₂×C₄⋊Q₈	—	2	1	no	—
262	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂×#125	—	2	1	no	—
266	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₂×C₂²⋊Q₈	—	2	1	no	—
269	C₂×C₂²⋊Q₈	—	2	1	no	—
270	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
271	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
272	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
274	#491	—	2	4	no	—
275	#487	—	2	4	no	—
276	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
277	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
278	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
279	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
280	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
281	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
283	#398	—	2	1	no	—
284	#398	—	2	1	no	—
285	#397	—	2	1	no	—
286	#397	—	2	1	no	—
287	#391	C₂²×C₁₃₇	1	2	yes	—
288	#382	C₂²×C₁₃₇	1	1	yes	characteristic
289	#340	C₂²×C₁₃₇	1	2	yes	—
290	#340	C₂²×C₁₃₇	1	2	yes	—
291	#340	C₂²×C₁₃₇	1	2	yes	—
292	#340	C₂²×C₁₃₇	1	2	yes	—
293	#336	C₂²×C₁₃₇	1	2	yes	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	—	2	1	no	—
296	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
297	C₂³×C₄	—	2	1	no	—
298	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
300	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
301	C₄⋊Q₈	—	2	8	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	—	2	2	no	—
307	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
308	#125	—	2	8	no	—
309	#125	—	2	2	no	—
310	#125	—	2	2	no	—
311	#125	—	2	2	no	—
312	#125	D₄×C₁₃₇	1	4	yes	—
313	C₂²⋊Q₈	—	2	8	no	—
314	C₂²⋊Q₈	—	2	8	no	—
315	C₂²⋊Q₈	—	2	8	no	—
316	#117	—	2	2	no	—
317	#117	—	2	2	no	—
318	#117	—	2	2	no	—
319	#117	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	—	2	2	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
333	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
335	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
339	C₂×C₂²⋊C₄	—	2	1	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
342	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
346	#103	—	2	4	no	—
347	#103	—	2	4	no	—
348	#103	—	2	4	no	—
349	#103	—	2	4	no	—
350	C₂²⋊C₈	—	2	4	no	—
351	#97	—	2	2	no	—
352	C₄×C₈	—	2	2	no	—
353	#95	—	2	1	no	—
354	#95	—	2	1	no	—
355	#95	—	2	2	no	—
356	#95	—	2	2	no	—
357	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
358	C₂×Q₈	—	2	4	no	—
359	C₂×Q₈	—	2	4	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	8	2	no	—
365	C₂×Q₈	—	8	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	8	2	no	—
377	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
382	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
383	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	—	4	4	no	—
407	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
408	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
409	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
410	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
411	C₂²⋊C₄	—	2	4	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	4	4	no	—
415	C₄²	—	2	4	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
419	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	#124×C₁₃₇	1	2	yes	—
425	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
426	Q₈	—	4	4	no	—
427	Q₈	—	4	4	no	—
428	Q₈	—	8	4	no	—
429	Q₈	—	8	4	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	4	no	—
455	C₂×C₄	#120×C₁₃₇	1	2	yes	—
456	C₂×C₄	#120×C₁₃₇	1	2	yes	—
457	C₂×C₄	#124×C₁₃₇	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
461	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
462	C₈	—	4	2	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
470	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
471	C₂²	#536×C₁₃₇	1	1	yes	characteristic
472	C₂²	#577×C₁₃₇	1	2	yes	—
473	C₂²	#578×C₁₃₇	1	2	yes	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	1	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	8	2	no	—
484	C₄	—	8	2	no	—
485	C₄	#548×C₁₃₇	1	2	yes	—
486	C₄	#548×C₁₃₇	1	2	yes	—
487	C₂	—	4	2	no	—
488	C₂	#2436×C₁₃₇	1	1	yes	characteristic
489	C₂	#3009×C₁₃₇	1	1	yes	characteristic
490	C₂	#3011×C₁₃₇	1	1	yes	characteristic
491	C₂	#3137×C₁₃₇	1	2	yes	—
492	C₂	#3242×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#491×C₁₃₇	C₂	semidirect
3. C₂×#487×C₁₃₇	C₂	semidirect
5. #2026×C₁₃₇	C₂	semidirect
6. #2024×C₁₃₇	C₂	semidirect
7. #1993×C₁₃₇	C₂	semidirect
8. #1991×C₁₃₇	C₂	semidirect
246. #25914	C₁₃₇	direct (N × H)
247. C₁₃₇	#25914	direct (N × H)
249. C₂×#491	C₂×C₁₃₇	semidirect
250. C₂×#487	C₂×C₁₃₇	semidirect
252. #2026	C₂×C₁₃₇	semidirect
253. #2024	C₂×C₁₃₇	semidirect
254. #1993	C₂×C₁₃₇	semidirect
255. #1991	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×#544×C₁₃₇	2
C₂×#491×C₁₃₇	2
C₂×#487×C₁₃₇	2
#2621×C₁₃₇	2
#2026×C₁₃₇	2
#2024×C₁₃₇	2
#1993×C₁₃₇	2
#1991×C₁₃₇	2
#1873×C₁₃₇	2
#1872×C₁₃₇	2
#1750×C₁₃₇	2
#25914	137

Subgroups — C137×#25914 #46875300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25914 #46875300