Subgroups — C₁₃₇×#25322 #46874708

Order 35072.

946 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2567×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2366×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2071×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2055×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1987×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1939×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1877×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#1802×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#103×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
17	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
18	#486×C₁₃₇	—	2	4	no	—
19	#460×C₁₃₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
22	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	#395×C₁₃₇	—	2	2	no	—
29	#391×C₁₃₇	C₂²	1	2	yes	—
30	#390×C₁₃₇	—	2	1	no	—
31	#390×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	—	2	1	no	—
33	#389×C₁₃₇	—	2	1	no	—
34	#386×C₁₃₇	—	2	2	no	—
35	#384×C₁₃₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
38	#340×C₁₃₇	C₂²	1	1	yes	characteristic
39	#340×C₁₃₇	C₂²	1	1	yes	characteristic
40	#340×C₁₃₇	C₂²	1	2	yes	—
41	#340×C₁₃₇	C₂²	1	2	yes	—
42	#336×C₁₃₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₃₇	—	2	1	no	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₁₃₇	—	2	1	no	—
46	C₂²×D₄×C₁₃₇	—	2	1	no	—
47	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₁₃₇	—	2	2	no	—
49	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
50	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₃₇	—	4	2	no	—
52	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
57	#127×C₁₃₇	—	2	2	no	—
58	#127×C₁₃₇	D₄	1	4	yes	—
59	C₄×Q₈×C₁₃₇	—	2	2	no	—
60	C₄×Q₈×C₁₃₇	—	2	2	no	—
61	C₄×D₄×C₁₃₇	—	2	4	no	—
62	C₄×D₄×C₁₃₇	—	4	2	no	—
63	C₄×D₄×C₁₃₇	—	4	4	no	—
64	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₄²×C₁₃₇	—	2	1	no	—
78	C₂×C₄²×C₁₃₇	—	2	1	no	—
79	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
80	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₁₃₇	—	2	2	no	—
82	C₄⋊C₈×C₁₃₇	—	4	2	no	—
83	#103×C₁₃₇	—	2	2	no	—
84	#103×C₁₃₇	—	2	2	no	—
85	#102×C₁₃₇	—	2	4	no	—
86	#102×C₁₃₇	—	2	4	no	—
87	#102×C₁₃₇	—	2	4	no	—
88	C₄×C₈×C₁₃₇	—	2	2	no	—
89	#95×C₁₃₇	—	2	1	no	—
90	#95×C₁₃₇	—	2	2	no	—
91	#95×C₁₃₇	—	2	2	no	—
92	#95×C₁₃₇	D₄	1	2	yes	—
93	C₂⁴×C₁₃₇	—	2	2	no	—
94	C₂×Q₈×C₁₃₇	—	2	1	no	—
95	C₂×Q₈×C₁₃₇	—	2	1	no	—
96	C₂×Q₈×C₁₃₇	—	2	1	no	—
97	C₂×Q₈×C₁₃₇	—	2	1	no	—
98	C₂×Q₈×C₁₃₇	—	2	1	no	—
99	C₂×Q₈×C₁₃₇	—	4	2	no	—
100	C₂×Q₈×C₁₃₇	—	8	1	no	—
101	C₂×Q₈×C₁₃₇	—	8	1	no	—
102	C₂×Q₈×C₁₃₇	#41	1	2	yes	—
103	C₂×D₄×C₁₃₇	—	2	4	no	—
104	C₂×D₄×C₁₃₇	—	2	4	no	—
105	C₂×D₄×C₁₃₇	—	2	2	no	—
106	C₂×D₄×C₁₃₇	—	2	2	no	—
107	C₂×D₄×C₁₃₇	—	2	2	no	—
108	C₂×D₄×C₁₃₇	—	2	2	no	—
109	C₂×D₄×C₁₃₇	—	2	2	no	—
110	C₂×D₄×C₁₃₇	—	4	2	no	—
111	C₂×D₄×C₁₃₇	—	4	2	no	—
112	C₂×D₄×C₁₃₇	—	8	1	no	—
113	C₂×D₄×C₁₃₇	—	8	1	no	—
114	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₁₃₇	—	2	1	no	—
116	C₂²×C₄×C₁₃₇	—	2	1	no	—
117	C₂²×C₄×C₁₃₇	—	2	2	no	—
118	C₂²×C₄×C₁₃₇	—	2	1	no	—
119	C₂²×C₄×C₁₃₇	—	2	1	no	—
120	C₂²×C₄×C₁₃₇	—	4	4	no	—
121	C₂²×C₄×C₁₃₇	—	4	4	no	—
122	C₂²×C₄×C₁₃₇	—	8	2	no	—
123	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
128	SD₁₆×C₁₃₇	—	4	8	no	—
129	C₂×C₈×C₁₃₇	—	2	1	no	—
130	C₂×C₈×C₁₃₇	—	2	1	no	—
131	C₂×C₈×C₁₃₇	—	2	2	no	—
132	C₂×C₈×C₁₃₇	—	2	1	no	—
133	C₂×C₈×C₁₃₇	—	2	1	no	—
134	C₂×C₈×C₁₃₇	—	2	2	no	—
135	C₂×C₈×C₁₃₇	—	2	1	no	—
136	C₂×C₈×C₁₃₇	—	2	2	no	—
137	C₂×C₈×C₁₃₇	—	2	1	no	—
138	C₄⋊C₄×C₁₃₇	—	2	2	no	—
139	C₄⋊C₄×C₁₃₇	—	2	1	no	—
140	C₄⋊C₄×C₁₃₇	—	2	1	no	—
141	C₄⋊C₄×C₁₃₇	—	2	2	no	—
142	C₄⋊C₄×C₁₃₇	—	2	1	no	—
143	C₄⋊C₄×C₁₃₇	—	2	1	no	—
144	C₄⋊C₄×C₁₃₇	—	2	2	no	—
145	C₄⋊C₄×C₁₃₇	—	2	1	no	—
146	C₄⋊C₄×C₁₃₇	—	4	2	no	—
147	C₄⋊C₄×C₁₃₇	—	4	2	no	—
148	C₄⋊C₄×C₁₃₇	—	8	2	no	—
149	C₄⋊C₄×C₁₃₇	—	8	2	no	—
150	C₄⋊C₄×C₁₃₇	SD₁₆	1	4	yes	complemented
151	C₄⋊C₄×C₁₃₇	#41	1	2	yes	—
152	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
153	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
154	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
155	C₄²×C₁₃₇	—	2	2	no	—
156	C₄²×C₁₃₇	—	2	2	no	—
157	C₄²×C₁₃₇	—	4	2	no	—
158	C₄²×C₁₃₇	—	4	2	no	—
159	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
160	C₄²×C₁₃₇	#41	1	2	yes	—
161	C₂³×C₁₃₇	—	2	4	no	—
162	C₂³×C₁₃₇	—	4	2	no	—
163	C₂³×C₁₃₇	—	4	2	no	—
164	C₂³×C₁₃₇	—	4	2	no	—
165	C₂³×C₁₃₇	—	4	2	no	—
166	C₂³×C₁₃₇	—	4	2	no	—
167	C₂³×C₁₃₇	—	4	2	no	—
168	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₁₃₇	—	2	2	no	—
170	Q₈×C₁₃₇	—	2	2	no	—
171	Q₈×C₁₃₇	—	4	1	no	—
172	Q₈×C₁₃₇	—	4	1	no	—
173	Q₈×C₁₃₇	—	8	4	no	—
174	D₄×C₁₃₇	—	2	4	no	—
175	D₄×C₁₃₇	—	2	4	no	—
176	D₄×C₁₃₇	—	4	2	no	—
177	D₄×C₁₃₇	—	4	2	no	—
178	D₄×C₁₃₇	—	4	8	no	—
179	D₄×C₁₃₇	—	8	4	no	—
180	C₂×C₄×C₁₃₇	—	2	2	no	—
181	C₂×C₄×C₁₃₇	—	2	2	no	—
182	C₂×C₄×C₁₃₇	—	2	1	no	—
183	C₂×C₄×C₁₃₇	—	2	1	no	—
184	C₂×C₄×C₁₃₇	—	2	2	no	—
185	C₂×C₄×C₁₃₇	—	2	2	no	—
186	C₂×C₄×C₁₃₇	—	2	2	no	—
187	C₂×C₄×C₁₃₇	—	2	2	no	—
188	C₂×C₄×C₁₃₇	—	2	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	—	4	1	no	—
194	C₂×C₄×C₁₃₇	—	4	2	no	—
195	C₂×C₄×C₁₃₇	—	4	1	no	—
196	C₂×C₄×C₁₃₇	—	4	1	no	—
197	C₂×C₄×C₁₃₇	—	4	1	no	—
198	C₂×C₄×C₁₃₇	—	4	1	no	—
199	C₂×C₄×C₁₃₇	—	4	1	no	—
200	C₂×C₄×C₁₃₇	—	4	2	no	—
201	C₂×C₄×C₁₃₇	—	4	2	no	—
202	C₂×C₄×C₁₃₇	—	8	8	no	—
203	C₂×C₄×C₁₃₇	#121	1	2	yes	—
204	C₂×C₄×C₁₃₇	#121	1	2	yes	—
205	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
206	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
207	C₂×C₄×C₁₃₇	#137	1	2	yes	—
208	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
214	C₈×C₁₃₇	—	4	1	no	—
215	C₈×C₁₃₇	—	4	2	no	—
216	C₈×C₁₃₇	—	4	1	no	—
217	C₂²×C₁₃₇	—	4	4	no	—
218	C₂²×C₁₃₇	—	4	4	no	—
219	C₂²×C₁₃₇	—	4	4	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	C₂×#121	1	1	yes	characteristic
225	C₂²×C₁₃₇	#546	1	1	yes	characteristic
226	C₂²×C₁₃₇	#548	1	1	yes	characteristic
227	C₂²×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
228	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
229	C₂²×C₁₃₇	#576	1	1	yes	characteristic
230	C₂²×C₁₃₇	#576	1	1	yes	characteristic
231	C₄×C₁₃₇	—	2	4	no	—
232	C₄×C₁₃₇	—	2	2	no	—
233	C₄×C₁₃₇	—	4	2	no	—
234	C₄×C₁₃₇	—	4	2	no	—
235	C₄×C₁₃₇	—	4	2	no	—
236	C₄×C₁₃₇	—	8	1	no	—
237	C₄×C₁₃₇	—	8	1	no	—
238	C₄×C₁₃₇	—	8	2	no	—
239	C₄×C₁₃₇	#461	1	4	yes	—
240	C₄×C₁₃₇	#529	1	1	yes	characteristic
241	C₄×C₁₃₇	#542	1	1	yes	characteristic
242	C₄×C₁₃₇	#546	1	1	yes	characteristic
243	C₄×C₁₃₇	#547	1	1	yes	characteristic
244	C₂×C₁₃₇	—	8	4	no	—
245	C₂×C₁₃₇	#2427	1	1	yes	characteristic
246	C₂×C₁₃₇	C₂×#461	1	1	yes	characteristic
247	C₂×C₁₃₇	#3041	1	1	yes	characteristic
248	C₂×C₁₃₇	#3295	1	1	yes	characteristic
249	C₂×C₁₃₇	#3333	1	1	yes	characteristic
250	C₂×C₁₃₇	#3362	1	1	yes	characteristic
251	C₂×C₁₃₇	#3375	1	1	yes	characteristic
252	#25322	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₁₃₇	#25322	1	1	yes	characteristic, direct product factor, Sylow
254	C₂×#486	C₂×C₁₃₇	1	1	yes	characteristic
255	C₂×#460	C₂×C₁₃₇	1	1	yes	characteristic
256	#2567	C₂×C₁₃₇	1	1	yes	characteristic, complemented
257	#2366	C₂×C₁₃₇	1	1	yes	characteristic
258	#2071	C₂×C₁₃₇	1	2	yes	complemented
259	#2055	C₂×C₁₃₇	1	2	yes	complemented
260	#1987	C₂×C₁₃₇	1	1	yes	characteristic, complemented
261	#1939	C₂×C₁₃₇	1	1	yes	characteristic, complemented
262	#1877	C₂×C₁₃₇	1	2	yes	complemented
263	#1802	C₂×C₁₃₇	1	2	yes	complemented
264	C₄×#103	C₂×C₁₃₇	1	1	yes	characteristic, complemented
265	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
269	C₂×C₄×D₄	—	2	1	no	—
270	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
271	#486	—	2	4	no	—
272	#460	—	2	4	no	—
273	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
275	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
276	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
277	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
278	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
279	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
280	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
281	#395	—	2	2	no	—
282	#391	C₂²×C₁₃₇	1	2	yes	—
283	#390	—	2	1	no	—
284	#390	C₂²×C₁₃₇	1	2	yes	—
285	#389	—	2	1	no	—
286	#389	—	2	1	no	—
287	#386	—	2	2	no	—
288	#384	C₂²×C₁₃₇	1	2	yes	—
289	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
290	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
291	#340	C₂²×C₁₃₇	1	2	yes	—
292	#340	C₂²×C₁₃₇	1	1	yes	characteristic
293	#340	C₂²×C₁₃₇	1	2	yes	—
294	#340	C₂²×C₁₃₇	1	1	yes	characteristic
295	#336	C₂²×C₁₃₇	1	2	yes	—
296	C₂²×Q₈	—	2	1	no	—
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	C₂³×C₄	—	2	2	no	—
302	C₂×SD₁₆	—	2	4	no	—
303	C₂×SD₁₆	—	2	4	no	—
304	C₂×SD₁₆	—	4	2	no	—
305	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
306	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	2	no	—
310	#127	—	2	2	no	—
311	#127	D₄×C₁₃₇	1	4	yes	—
312	C₄×Q₈	—	2	2	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×D₄	—	2	4	no	—
315	C₄×D₄	—	4	2	no	—
316	C₄×D₄	—	4	4	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
326	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
333	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
334	C₄⋊C₈	—	2	2	no	—
335	C₄⋊C₈	—	4	2	no	—
336	#103	—	2	2	no	—
337	#103	—	2	2	no	—
338	#102	—	2	4	no	—
339	#102	—	2	4	no	—
340	#102	—	2	4	no	—
341	C₄×C₈	—	2	2	no	—
342	#95	—	2	1	no	—
343	#95	—	2	2	no	—
344	#95	—	2	2	no	—
345	#95	D₄×C₁₃₇	1	2	yes	—
346	C₂⁴	—	2	2	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	4	2	no	—
353	C₂×Q₈	—	8	1	no	—
354	C₂×Q₈	—	8	1	no	—
355	C₂×Q₈	#41×C₁₃₇	1	2	yes	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	2	4	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	4	2	no	—
364	C₂×D₄	—	4	2	no	—
365	C₂×D₄	—	8	1	no	—
366	C₂×D₄	—	8	1	no	—
367	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	4	4	no	—
374	C₂²×C₄	—	4	4	no	—
375	C₂²×C₄	—	8	2	no	—
376	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
381	SD₁₆	—	4	8	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	4	2	no	—
400	C₄⋊C₄	—	4	2	no	—
401	C₄⋊C₄	—	8	2	no	—
402	C₄⋊C₄	—	8	2	no	—
403	C₄⋊C₄	SD₁₆×C₁₃₇	1	4	yes	complemented
404	C₄⋊C₄	#41×C₁₃₇	1	2	yes	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₂²⋊C₄	—	4	4	no	—
407	C₂²⋊C₄	—	8	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	4	2	no	—
411	C₄²	—	4	2	no	—
412	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
413	C₄²	#41×C₁₃₇	1	2	yes	—
414	C₂³	—	2	4	no	—
415	C₂³	—	4	2	no	—
416	C₂³	—	4	2	no	—
417	C₂³	—	4	2	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
422	Q₈	—	2	2	no	—
423	Q₈	—	2	2	no	—
424	Q₈	—	4	1	no	—
425	Q₈	—	4	1	no	—
426	Q₈	—	8	4	no	—
427	D₄	—	2	4	no	—
428	D₄	—	2	4	no	—
429	D₄	—	4	8	no	—
430	D₄	—	4	2	no	—
431	D₄	—	4	2	no	—
432	D₄	—	8	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	8	8	no	—
456	C₂×C₄	#121×C₁₃₇	1	2	yes	—
457	C₂×C₄	#121×C₁₃₇	1	2	yes	—
458	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
459	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
460	C₂×C₄	#137×C₁₃₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
464	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
465	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
466	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
467	C₈	—	4	2	no	—
468	C₈	—	4	1	no	—
469	C₈	—	4	1	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	8	2	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
478	C₂²	#546×C₁₃₇	1	1	yes	characteristic
479	C₂²	#548×C₁₃₇	1	1	yes	characteristic
480	C₂²	C₂²×SD₁₆×C₁₃₇	1	1	yes	characteristic
481	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
482	C₂²	#576×C₁₃₇	1	1	yes	characteristic
483	C₂²	#576×C₁₃₇	1	1	yes	characteristic
484	C₄	—	2	4	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	2	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	8	2	no	—
490	C₄	—	8	1	no	—
491	C₄	—	8	1	no	—
492	C₄	#461×C₁₃₇	1	4	yes	—
493	C₄	#529×C₁₃₇	1	1	yes	characteristic
494	C₄	#542×C₁₃₇	1	1	yes	characteristic
495	C₄	#546×C₁₃₇	1	1	yes	characteristic
496	C₄	#547×C₁₃₇	1	1	yes	characteristic
497	C₂	—	8	4	no	—
498	C₂	#2427×C₁₃₇	1	1	yes	characteristic
499	C₂	C₂×#461×C₁₃₇	1	1	yes	characteristic
500	C₂	#3041×C₁₃₇	1	1	yes	characteristic
501	C₂	#3295×C₁₃₇	1	1	yes	characteristic
502	C₂	#3333×C₁₃₇	1	1	yes	characteristic
503	C₂	#3362×C₁₃₇	1	1	yes	characteristic
504	C₂	#3375×C₁₃₇	1	1	yes	characteristic

	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
3. #2567×C₁₃₇	C₂	semidirect
5. #2071×C₁₃₇	C₂	semidirect
6. #2055×C₁₃₇	C₂	semidirect
7. #1987×C₁₃₇	C₂	semidirect
8. #1939×C₁₃₇	C₂	semidirect
9. #1877×C₁₃₇	C₂	semidirect
10. #1802×C₁₃₇	C₂	semidirect
11. C₄×#103×C₁₃₇	C₂	semidirect
150. C₄⋊C₄×C₁₃₇	SD₁₆	semidirect
252. #25322	C₁₃₇	direct (N × H)
253. C₁₃₇	#25322	direct (N × H)
256. #2567	C₂×C₁₃₇	semidirect
258. #2071	C₂×C₁₃₇	semidirect
259. #2055	C₂×C₁₃₇	semidirect
260. #1987	C₂×C₁₃₇	semidirect
261. #1939	C₂×C₁₃₇	semidirect
262. #1877	C₂×C₁₃₇	semidirect
263. #1802	C₂×C₁₃₇	semidirect
264. C₄×#103	C₂×C₁₃₇	semidirect
403. C₄⋊C₄	SD₁₆×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₁₃₇	2
C₂×#460×C₁₃₇	2
#2567×C₁₃₇	2
#2366×C₁₃₇	2
#2071×C₁₃₇	2
#2055×C₁₃₇	2
#1987×C₁₃₇	2
#1939×C₁₃₇	2
#1877×C₁₃₇	2
#1802×C₁₃₇	2
C₄×#103×C₁₃₇	2
#25322	137

Subgroups — C137×#25322 #46874708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25322 #46874708