Subgroups — C₁₃₇×#24747 #46874133

Order 35072.

968 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#464×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#441×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2548×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2351×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2065×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2042×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1961×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1940×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1880×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1805×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	#464×C₁₃₇	—	2	8	no	—
20	#460×C₁₃₇	—	2	4	no	—
21	#441×C₁₃₇	—	2	4	no	—
22	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
32	#394×C₁₃₇	—	2	1	no	—
33	#394×C₁₃₇	—	2	1	no	—
34	#390×C₁₃₇	—	2	2	no	—
35	#390×C₁₃₇	C₂²	1	2	yes	—
36	#389×C₁₃₇	—	2	2	no	—
37	#389×C₁₃₇	C₂²	1	2	yes	—
38	#382×C₁₃₇	—	2	1	no	—
39	#382×C₁₃₇	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#336×C₁₃₇	C₂²	1	2	yes	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₁₃₇	—	2	1	no	—
47	C₂²×D₄×C₁₃₇	—	2	1	no	—
48	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
50	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
51	C₂×Q₁₆×C₁₃₇	D₄	1	4	yes	complemented
52	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
53	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
54	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
55	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
58	#127×C₁₃₇	—	2	2	no	—
59	#127×C₁₃₇	—	2	2	no	—
60	#124×C₁₃₇	—	2	4	no	—
61	#124×C₁₃₇	—	4	2	no	—
62	C₄×Q₈×C₁₃₇	—	2	2	no	—
63	C₄×Q₈×C₁₃₇	—	2	4	no	—
64	C₄×Q₈×C₁₃₇	—	2	2	no	—
65	C₄×Q₈×C₁₃₇	—	2	2	no	—
66	C₄×Q₈×C₁₃₇	—	4	2	no	—
67	C₄×Q₈×C₁₃₇	D₄	1	4	yes	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₄²×C₁₃₇	—	2	1	no	—
81	C₂×C₄²×C₁₃₇	—	2	2	no	—
82	C₂×C₄²×C₁₃₇	—	2	1	no	—
83	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
84	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
85	#106×C₁₃₇	—	2	2	no	—
86	C₄⋊C₈×C₁₃₇	—	2	4	no	—
87	C₄⋊C₈×C₁₃₇	—	2	2	no	—
88	#103×C₁₃₇	—	2	2	no	—
89	#103×C₁₃₇	—	2	4	no	—
90	#103×C₁₃₇	—	2	2	no	—
91	#102×C₁₃₇	—	2	4	no	—
92	#102×C₁₃₇	—	2	4	no	—
93	#102×C₁₃₇	—	4	2	no	—
94	#97×C₁₃₇	—	2	2	no	—
95	#95×C₁₃₇	—	2	1	no	—
96	#95×C₁₃₇	—	2	2	no	—
97	#95×C₁₃₇	—	2	1	no	—
98	#95×C₁₃₇	D₄	1	1	yes	characteristic
99	#95×C₁₃₇	D₄	1	1	yes	characteristic
100	C₂⁴×C₁₃₇	—	2	2	no	—
101	C₂×Q₈×C₁₃₇	—	2	1	no	—
102	C₂×Q₈×C₁₃₇	—	2	1	no	—
103	C₂×Q₈×C₁₃₇	—	2	1	no	—
104	C₂×Q₈×C₁₃₇	—	2	1	no	—
105	C₂×Q₈×C₁₃₇	—	2	1	no	—
106	C₂×Q₈×C₁₃₇	—	2	1	no	—
107	C₂×Q₈×C₁₃₇	—	2	1	no	—
108	C₂×Q₈×C₁₃₇	—	2	1	no	—
109	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
112	C₂×Q₈×C₁₃₇	#41	1	2	yes	—
113	C₂×D₄×C₁₃₇	—	2	2	no	—
114	C₂×D₄×C₁₃₇	—	2	2	no	—
115	C₂×D₄×C₁₃₇	—	2	2	no	—
116	C₂×D₄×C₁₃₇	—	2	2	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	2	2	no	—
119	C₂×D₄×C₁₃₇	—	4	4	no	—
120	C₂×D₄×C₁₃₇	—	4	2	no	—
121	C₂×D₄×C₁₃₇	—	4	2	no	—
122	C₂×D₄×C₁₃₇	—	8	2	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	2	no	—
127	C₂²×C₄×C₁₃₇	—	2	1	no	—
128	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₁₃₇	—	2	8	no	—
134	SD₁₆×C₁₃₇	—	4	8	no	—
135	C₂×C₈×C₁₃₇	—	2	1	no	—
136	C₂×C₈×C₁₃₇	—	2	1	no	—
137	C₂×C₈×C₁₃₇	—	2	1	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	1	no	—
142	C₂×C₈×C₁₃₇	—	2	1	no	—
143	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₁₃₇	—	2	4	no	—
145	C₄⋊C₄×C₁₃₇	—	2	1	no	—
146	C₄⋊C₄×C₁₃₇	—	2	2	no	—
147	C₄⋊C₄×C₁₃₇	—	2	1	no	—
148	C₄⋊C₄×C₁₃₇	—	2	1	no	—
149	C₄⋊C₄×C₁₃₇	—	2	2	no	—
150	C₄⋊C₄×C₁₃₇	—	4	2	no	—
151	C₄⋊C₄×C₁₃₇	—	4	4	no	—
152	C₄⋊C₄×C₁₃₇	—	4	4	no	—
153	C₄⋊C₄×C₁₃₇	—	4	2	no	—
154	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
155	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
156	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
157	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
158	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
159	C₄²×C₁₃₇	—	2	4	no	—
160	C₄²×C₁₃₇	—	2	2	no	—
161	C₄²×C₁₃₇	—	2	2	no	—
162	C₄²×C₁₃₇	—	4	4	no	—
163	C₄²×C₁₃₇	—	4	2	no	—
164	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₄²×C₁₃₇	#41	1	2	yes	—
166	C₂³×C₁₃₇	—	4	2	no	—
167	C₂³×C₁₃₇	—	4	2	no	—
168	C₂³×C₁₃₇	—	4	2	no	—
169	C₂³×C₁₃₇	—	4	2	no	—
170	C₂³×C₁₃₇	—	4	2	no	—
171	C₂³×C₁₃₇	—	4	2	no	—
172	C₂³×C₁₃₇	—	4	2	no	—
173	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₁₃₇	—	2	2	no	—
175	Q₈×C₁₃₇	—	2	2	no	—
176	Q₈×C₁₃₇	—	2	2	no	—
177	Q₈×C₁₃₇	—	2	2	no	—
178	Q₈×C₁₃₇	—	2	2	no	—
179	Q₈×C₁₃₇	—	4	1	no	—
180	Q₈×C₁₃₇	—	4	1	no	—
181	Q₈×C₁₃₇	#121	1	4	yes	—
182	D₄×C₁₃₇	—	4	2	no	—
183	D₄×C₁₃₇	—	4	2	no	—
184	D₄×C₁₃₇	—	4	2	no	—
185	D₄×C₁₃₇	—	4	2	no	—
186	D₄×C₁₃₇	—	8	4	no	—
187	D₄×C₁₃₇	—	8	4	no	—
188	C₂×C₄×C₁₃₇	—	2	2	no	—
189	C₂×C₄×C₁₃₇	—	2	2	no	—
190	C₂×C₄×C₁₃₇	—	2	2	no	—
191	C₂×C₄×C₁₃₇	—	2	2	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	—	2	2	no	—
198	C₂×C₄×C₁₃₇	—	2	2	no	—
199	C₂×C₄×C₁₃₇	—	2	2	no	—
200	C₂×C₄×C₁₃₇	—	2	2	no	—
201	C₂×C₄×C₁₃₇	—	4	1	no	—
202	C₂×C₄×C₁₃₇	—	4	1	no	—
203	C₂×C₄×C₁₃₇	—	4	2	no	—
204	C₂×C₄×C₁₃₇	—	4	1	no	—
205	C₂×C₄×C₁₃₇	—	4	2	no	—
206	C₂×C₄×C₁₃₇	—	4	2	no	—
207	C₂×C₄×C₁₃₇	—	4	2	no	—
208	C₂×C₄×C₁₃₇	—	4	2	no	—
209	C₂×C₄×C₁₃₇	#121	1	2	yes	—
210	C₂×C₄×C₁₃₇	#121	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	1	yes	characteristic
214	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
217	C₈×C₁₃₇	—	2	2	no	—
218	C₈×C₁₃₇	—	4	2	no	—
219	C₈×C₁₃₇	—	4	1	no	—
220	C₂²×C₁₃₇	—	8	2	no	—
221	C₂²×C₁₃₇	—	8	2	no	—
222	C₂²×C₁₃₇	—	8	2	no	—
223	C₂²×C₁₃₇	—	8	2	no	—
224	C₂²×C₁₃₇	—	8	2	no	—
225	C₂²×C₁₃₇	—	8	2	no	—
226	C₂²×C₁₃₇	—	8	2	no	—
227	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
228	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
229	C₂²×C₁₃₇	#548	1	1	yes	characteristic
230	C₂²×C₁₃₇	#575	1	2	yes	—
231	C₂²×C₁₃₇	#577	1	2	yes	—
232	C₄×C₁₃₇	—	2	4	no	—
233	C₄×C₁₃₇	—	2	2	no	—
234	C₄×C₁₃₇	—	2	2	no	—
235	C₄×C₁₃₇	—	4	2	no	—
236	C₄×C₁₃₇	—	4	2	no	—
237	C₄×C₁₃₇	—	4	2	no	—
238	C₄×C₁₃₇	—	4	2	no	—
239	C₄×C₁₃₇	—	8	2	no	—
240	C₄×C₁₃₇	—	8	1	no	—
241	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
242	C₄×C₁₃₇	#537	1	1	yes	characteristic
243	C₄×C₁₃₇	#547	1	1	yes	characteristic
244	C₄×C₁₃₇	#553	1	1	yes	characteristic
245	C₂×C₁₃₇	—	16	2	no	—
246	C₂×C₁₃₇	#2416	1	1	yes	characteristic
247	C₂×C₁₃₇	#3040	1	1	yes	characteristic
248	C₂×C₁₃₇	#3043	1	1	yes	characteristic
249	C₂×C₁₃₇	#3271	1	2	yes	—
250	C₂×C₁₃₇	#3371	1	2	yes	—
251	#24747	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₁₃₇	#24747	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#464	C₂×C₁₃₇	1	2	yes	complemented
254	C₂×#460	C₂×C₁₃₇	1	1	yes	characteristic
255	C₂×#441	C₂×C₁₃₇	1	1	yes	characteristic, complemented
256	#2548	C₂×C₁₃₇	1	1	yes	characteristic
257	#2351	C₂×C₁₃₇	1	1	yes	characteristic, complemented
258	#2065	C₂×C₁₃₇	1	2	yes	complemented
259	#2042	C₂×C₁₃₇	1	2	yes	complemented
260	#1961	C₂×C₁₃₇	1	1	yes	characteristic
261	#1940	C₂×C₁₃₇	1	1	yes	characteristic, complemented
262	#1880	C₂×C₁₃₇	1	2	yes	complemented
263	#1805	C₂×C₁₃₇	1	1	yes	characteristic, complemented
264	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
266	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂×#124	C₂²×C₁₃₇	1	2	yes	—
268	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
270	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
271	#464	—	2	8	no	—
272	#460	—	2	4	no	—
273	#441	—	2	4	no	—
274	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
277	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
278	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
279	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
280	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
281	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
282	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
283	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
284	#394	—	2	1	no	—
285	#394	—	2	1	no	—
286	#390	—	2	2	no	—
287	#390	C₂²×C₁₃₇	1	2	yes	—
288	#389	—	2	2	no	—
289	#389	C₂²×C₁₃₇	1	2	yes	—
290	#382	—	2	1	no	—
291	#382	C₂²×C₁₃₇	1	1	yes	characteristic
292	C₄×C₄⋊C₄	—	2	1	no	—
293	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
294	#340	C₂²×C₁₃₇	1	2	yes	—
295	#336	C₂²×C₁₃₇	1	2	yes	—
296	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
297	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
298	C₂²×D₄	—	2	1	no	—
299	C₂²×D₄	—	2	1	no	—
300	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
301	C₂×Q₁₆	—	2	2	no	—
302	C₂×Q₁₆	—	2	2	no	—
303	C₂×Q₁₆	D₄×C₁₃₇	1	4	yes	complemented
304	C₂×SD₁₆	—	2	4	no	—
305	C₂×SD₁₆	—	2	4	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
308	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
310	#127	—	2	2	no	—
311	#127	—	2	2	no	—
312	#124	—	2	4	no	—
313	#124	—	4	2	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	—	2	4	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	—	2	2	no	—
318	C₄×Q₈	—	4	2	no	—
319	C₄×Q₈	D₄×C₁₃₇	1	4	yes	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
336	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
337	#106	—	2	2	no	—
338	C₄⋊C₈	—	2	4	no	—
339	C₄⋊C₈	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	2	4	no	—
342	#103	—	2	2	no	—
343	#102	—	2	4	no	—
344	#102	—	2	4	no	—
345	#102	—	4	2	no	—
346	#97	—	2	2	no	—
347	#95	—	2	1	no	—
348	#95	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	D₄×C₁₃₇	1	1	yes	characteristic
351	#95	D₄×C₁₃₇	1	1	yes	characteristic
352	C₂⁴	—	2	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
362	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
363	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
364	C₂×Q₈	#41×C₁₃₇	1	2	yes	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	8	2	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
381	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
385	Q₁₆	—	2	8	no	—
386	SD₁₆	—	4	8	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	4	2	no	—
403	C₄⋊C₄	—	4	4	no	—
404	C₄⋊C₄	—	4	4	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
407	C₂²⋊C₄	—	4	4	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	8	2	no	—
410	C₂²⋊C₄	—	8	2	no	—
411	C₄²	—	2	4	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	4	4	no	—
415	C₄²	—	4	2	no	—
416	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
417	C₄²	#41×C₁₃₇	1	2	yes	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
426	Q₈	—	2	2	no	—
427	Q₈	—	2	2	no	—
428	Q₈	—	2	2	no	—
429	Q₈	—	2	2	no	—
430	Q₈	—	2	2	no	—
431	Q₈	—	4	1	no	—
432	Q₈	—	4	1	no	—
433	Q₈	#121×C₁₃₇	1	4	yes	—
434	D₄	—	4	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	8	4	no	—
439	D₄	—	8	4	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	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	#121×C₁₃₇	1	2	yes	—
462	C₂×C₄	#121×C₁₃₇	1	2	yes	—
463	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
467	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
468	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
469	C₈	—	2	2	no	—
470	C₈	—	4	2	no	—
471	C₈	—	4	1	no	—
472	C₂²	—	8	2	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	8	2	no	—
478	C₂²	—	8	2	no	—
479	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
480	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
481	C₂²	#548×C₁₃₇	1	1	yes	characteristic
482	C₂²	#575×C₁₃₇	1	2	yes	—
483	C₂²	#577×C₁₃₇	1	2	yes	—
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	4	2	no	—
491	C₄	—	8	1	no	—
492	C₄	—	8	2	no	—
493	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
494	C₄	#537×C₁₃₇	1	1	yes	characteristic
495	C₄	#547×C₁₃₇	1	1	yes	characteristic
496	C₄	#553×C₁₃₇	1	1	yes	characteristic
497	C₂	—	16	2	no	—
498	C₂	#2416×C₁₃₇	1	1	yes	characteristic
499	C₂	#3040×C₁₃₇	1	1	yes	characteristic
500	C₂	#3043×C₁₃₇	1	1	yes	characteristic
501	C₂	#3271×C₁₃₇	1	2	yes	—
502	C₂	#3371×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
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×#464×C₁₃₇	C₂	semidirect
3. C₂×#441×C₁₃₇	C₂	semidirect
5. #2351×C₁₃₇	C₂	semidirect
6. #2065×C₁₃₇	C₂	semidirect
7. #2042×C₁₃₇	C₂	semidirect
9. #1940×C₁₃₇	C₂	semidirect
10. #1880×C₁₃₇	C₂	semidirect
11. #1805×C₁₃₇	C₂	semidirect
51. C₂×Q₁₆×C₁₃₇	D₄	semidirect
251. #24747	C₁₃₇	direct (N × H)
252. C₁₃₇	#24747	direct (N × H)
253. C₂×#464	C₂×C₁₃₇	semidirect
255. C₂×#441	C₂×C₁₃₇	semidirect
257. #2351	C₂×C₁₃₇	semidirect
258. #2065	C₂×C₁₃₇	semidirect
259. #2042	C₂×C₁₃₇	semidirect
261. #1940	C₂×C₁₃₇	semidirect
262. #1880	C₂×C₁₃₇	semidirect
263. #1805	C₂×C₁₃₇	semidirect
303. C₂×Q₁₆	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×#464×C₁₃₇	2
C₂×#460×C₁₃₇	2
C₂×#441×C₁₃₇	2
#2548×C₁₃₇	2
#2351×C₁₃₇	2
#2065×C₁₃₇	2
#2042×C₁₃₇	2
#1961×C₁₃₇	2
#1940×C₁₃₇	2
#1880×C₁₃₇	2
#1805×C₁₃₇	2
#24747	137

Subgroups — C137×#24747 #46874133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24747 #46874133