Subgroups — C₁₃₇×#24752 #46874138

Order 35072.

992 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#462×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2857×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂²⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#2068×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2056×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1981×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1947×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1892×C₁₃₇	C₂	1	2	yes	maximal
11	#1803×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
19	#467×C₁₃₇	—	2	8	no	—
20	#462×C₁₃₇	—	2	4	no	—
21	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₁₃₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
30	#391×C₁₃₇	C₂²	1	1	yes	characteristic
31	#391×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	—	2	1	no	—
33	#389×C₁₃₇	—	2	1	no	—
34	#389×C₁₃₇	C₂²	1	1	yes	characteristic
35	#387×C₁₃₇	—	2	1	no	—
36	#385×C₁₃₇	—	2	1	no	—
37	#384×C₁₃₇	C₂²	1	1	yes	characteristic
38	#380×C₁₃₇	—	2	1	no	—
39	#380×C₁₃₇	—	2	1	no	—
40	#380×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
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×Q₈×C₁₃₇	C₂³	1	2	yes	—
47	C₂²×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
52	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
53	C₂×Q₁₆×C₁₃₇	—	4	2	no	—
54	C₂²×C₈×C₁₃₇	—	2	4	no	—
55	C₂²×C₈×C₁₃₇	—	2	4	no	—
56	C₂²×C₈×C₁₃₇	—	4	2	no	—
57	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
61	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
62	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
63	C₂²⋊Q₈×C₁₃₇	—	2	4	no	—
64	C₂²⋊Q₈×C₁₃₇	D₄	1	4	yes	—
65	C₄×Q₈×C₁₃₇	—	2	2	no	—
66	C₄×Q₈×C₁₃₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
82	C₂×C₄²×C₁₃₇	—	2	2	no	—
83	C₂×C₄²×C₁₃₇	—	2	2	no	—
84	C₂×C₄²×C₁₃₇	—	2	1	no	—
85	C₂×C₄²×C₁₃₇	—	2	1	no	—
86	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
90	#107×C₁₃₇	—	2	4	no	—
91	C₄⋊C₈×C₁₃₇	—	2	4	no	—
92	C₄⋊C₈×C₁₃₇	—	2	2	no	—
93	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	—
94	#103×C₁₃₇	—	2	4	no	—
95	#103×C₁₃₇	—	2	4	no	—
96	#103×C₁₃₇	—	2	4	no	—
97	#103×C₁₃₇	—	4	2	no	—
98	#95×C₁₃₇	—	2	1	no	—
99	#95×C₁₃₇	—	2	1	no	—
100	#95×C₁₃₇	—	2	1	no	—
101	#95×C₁₃₇	—	2	1	no	—
102	#95×C₁₃₇	—	2	1	no	—
103	#95×C₁₃₇	—	2	2	no	—
104	#95×C₁₃₇	—	2	2	no	—
105	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂×Q₈×C₁₃₇	—	2	1	no	—
107	C₂×Q₈×C₁₃₇	—	2	1	no	—
108	C₂×Q₈×C₁₃₇	—	2	2	no	—
109	C₂×Q₈×C₁₃₇	—	2	2	no	—
110	C₂×Q₈×C₁₃₇	—	2	1	no	—
111	C₂×Q₈×C₁₃₇	—	2	2	no	—
112	C₂×Q₈×C₁₃₇	—	2	2	no	—
113	C₂×Q₈×C₁₃₇	—	2	1	no	—
114	C₂×Q₈×C₁₃₇	—	2	2	no	—
115	C₂×Q₈×C₁₃₇	—	2	1	no	—
116	C₂×Q₈×C₁₃₇	—	2	2	no	—
117	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₁₃₇	—	2	2	no	—
119	C₂²×C₄×C₁₃₇	—	2	2	no	—
120	C₂²×C₄×C₁₃₇	—	2	1	no	—
121	C₂²×C₄×C₁₃₇	—	2	1	no	—
122	C₂²×C₄×C₁₃₇	—	2	1	no	—
123	C₂²×C₄×C₁₃₇	—	2	1	no	—
124	C₂²×C₄×C₁₃₇	—	2	2	no	—
125	C₂²×C₄×C₁₃₇	—	2	1	no	—
126	C₂²×C₄×C₁₃₇	—	2	1	no	—
127	C₂²×C₄×C₁₃₇	—	2	2	no	—
128	C₂²×C₄×C₁₃₇	—	2	1	no	—
129	C₂²×C₄×C₁₃₇	—	2	1	no	—
130	C₂²×C₄×C₁₃₇	—	2	2	no	—
131	C₂²×C₄×C₁₃₇	—	2	2	no	—
132	C₂²×C₄×C₁₃₇	—	4	1	no	—
133	C₂²×C₄×C₁₃₇	Q₁₆	1	4	yes	complemented
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
141	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
142	Q₁₆×C₁₃₇	—	4	8	no	—
143	C₂×C₈×C₁₃₇	—	2	2	no	—
144	C₂×C₈×C₁₃₇	—	2	2	no	—
145	C₂×C₈×C₁₃₇	—	2	2	no	—
146	C₂×C₈×C₁₃₇	—	2	2	no	—
147	C₂×C₈×C₁₃₇	—	2	2	no	—
148	C₂×C₈×C₁₃₇	—	4	8	no	—
149	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
150	C₄⋊C₄×C₁₃₇	—	2	4	no	—
151	C₄⋊C₄×C₁₃₇	—	2	1	no	—
152	C₄⋊C₄×C₁₃₇	—	2	1	no	—
153	C₄⋊C₄×C₁₃₇	—	2	2	no	—
154	C₄⋊C₄×C₁₃₇	—	2	1	no	—
155	C₄⋊C₄×C₁₃₇	—	2	2	no	—
156	C₄⋊C₄×C₁₃₇	—	2	1	no	—
157	C₄⋊C₄×C₁₃₇	—	2	1	no	—
158	C₄⋊C₄×C₁₃₇	—	4	2	no	—
159	C₄⋊C₄×C₁₃₇	—	4	2	no	—
160	C₄⋊C₄×C₁₃₇	—	4	2	no	—
161	C₄⋊C₄×C₁₃₇	—	4	4	no	—
162	C₄⋊C₄×C₁₃₇	—	4	2	no	—
163	C₄⋊C₄×C₁₃₇	—	4	2	no	—
164	C₄⋊C₄×C₁₃₇	—	4	2	no	—
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
167	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
168	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
169	C₄²×C₁₃₇	—	2	2	no	—
170	C₄²×C₁₃₇	—	2	2	no	—
171	C₄²×C₁₃₇	—	2	1	no	—
172	C₄²×C₁₃₇	—	2	1	no	—
173	C₄²×C₁₃₇	—	4	2	no	—
174	C₄²×C₁₃₇	—	4	2	no	—
175	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
176	C₄²×C₁₃₇	#41	1	2	yes	—
177	C₂³×C₁₃₇	—	2	1	no	—
178	C₂³×C₁₃₇	—	2	1	no	—
179	C₂³×C₁₃₇	—	2	1	no	—
180	C₂³×C₁₃₇	—	2	1	no	—
181	C₂³×C₁₃₇	#121	1	2	yes	—
182	C₂³×C₁₃₇	C₂×Q₁₆	1	2	yes	—
183	C₂³×C₁₃₇	#137	1	2	yes	—
184	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₁₃₇	—	2	2	no	—
186	Q₈×C₁₃₇	—	2	2	no	—
187	Q₈×C₁₃₇	—	4	1	no	—
188	Q₈×C₁₃₇	—	4	1	no	—
189	Q₈×C₁₃₇	—	4	2	no	—
190	Q₈×C₁₃₇	—	4	2	no	—
191	Q₈×C₁₃₇	—	4	2	no	—
192	Q₈×C₁₃₇	—	4	2	no	—
193	C₂×C₄×C₁₃₇	—	2	1	no	—
194	C₂×C₄×C₁₃₇	—	2	1	no	—
195	C₂×C₄×C₁₃₇	—	2	1	no	—
196	C₂×C₄×C₁₃₇	—	2	1	no	—
197	C₂×C₄×C₁₃₇	—	2	1	no	—
198	C₂×C₄×C₁₃₇	—	2	1	no	—
199	C₂×C₄×C₁₃₇	—	2	2	no	—
200	C₂×C₄×C₁₃₇	—	2	2	no	—
201	C₂×C₄×C₁₃₇	—	2	2	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	2	8	no	—
204	C₂×C₄×C₁₃₇	—	2	1	no	—
205	C₂×C₄×C₁₃₇	—	2	2	no	—
206	C₂×C₄×C₁₃₇	—	2	2	no	—
207	C₂×C₄×C₁₃₇	—	2	1	no	—
208	C₂×C₄×C₁₃₇	—	2	1	no	—
209	C₂×C₄×C₁₃₇	—	4	4	no	—
210	C₂×C₄×C₁₃₇	—	4	1	no	—
211	C₂×C₄×C₁₃₇	—	4	1	no	—
212	C₂×C₄×C₁₃₇	—	4	1	no	—
213	C₂×C₄×C₁₃₇	—	4	1	no	—
214	C₂×C₄×C₁₃₇	—	4	1	no	—
215	C₂×C₄×C₁₃₇	—	4	2	no	—
216	C₂×C₄×C₁₃₇	—	4	2	no	—
217	C₂×C₄×C₁₃₇	—	4	2	no	—
218	C₂×C₄×C₁₃₇	—	4	1	no	—
219	C₂×C₄×C₁₃₇	—	4	2	no	—
220	C₂×C₄×C₁₃₇	—	4	1	no	—
221	C₂×C₄×C₁₃₇	—	4	2	no	—
222	C₂×C₄×C₁₃₇	—	4	1	no	—
223	C₂×C₄×C₁₃₇	—	4	1	no	—
224	C₂×C₄×C₁₃₇	—	4	2	no	—
225	C₂×C₄×C₁₃₇	—	4	1	no	—
226	C₂×C₄×C₁₃₇	#121	1	2	yes	—
227	C₂×C₄×C₁₃₇	#121	1	2	yes	—
228	C₂×C₄×C₁₃₇	#121	1	2	yes	—
229	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
230	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
231	C₂×C₄×C₁₃₇	#135	1	2	yes	—
232	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
236	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
237	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
238	C₈×C₁₃₇	—	2	4	no	—
239	C₈×C₁₃₇	—	4	2	no	—
240	C₂²×C₁₃₇	—	2	2	no	—
241	C₂²×C₁₃₇	—	2	2	no	—
242	C₂²×C₁₃₇	—	2	2	no	—
243	C₂²×C₁₃₇	—	2	2	no	—
244	C₂²×C₁₃₇	—	2	2	no	—
245	C₂²×C₁₃₇	—	2	2	no	—
246	C₂²×C₁₃₇	#462	1	4	yes	complemented
247	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
248	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
249	C₂²×C₁₃₇	#537	1	1	yes	characteristic
250	C₂²×C₁₃₇	C₂²×Q₁₆	1	1	yes	characteristic
251	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
252	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
253	C₂²×C₁₃₇	#575	1	1	yes	characteristic
254	C₄×C₁₃₇	—	2	2	no	—
255	C₄×C₁₃₇	—	2	2	no	—
256	C₄×C₁₃₇	—	2	2	no	—
257	C₄×C₁₃₇	—	4	2	no	—
258	C₄×C₁₃₇	—	4	2	no	—
259	C₄×C₁₃₇	—	4	1	no	—
260	C₄×C₁₃₇	—	8	2	no	—
261	C₄×C₁₃₇	—	8	1	no	—
262	C₄×C₁₃₇	—	8	2	no	—
263	C₄×C₁₃₇	—	8	1	no	—
264	C₄×C₁₃₇	#467	1	4	yes	complemented
265	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
266	C₄×C₁₃₇	#547	1	1	yes	characteristic
267	C₄×C₁₃₇	#548	1	1	yes	characteristic
268	C₄×C₁₃₇	#553	1	1	yes	characteristic
269	C₂×C₁₃₇	—	2	4	no	—
270	C₂×C₁₃₇	#2416	1	1	yes	characteristic
271	C₂×C₁₃₇	C₂×#462	1	1	yes	characteristic
272	C₂×C₁₃₇	#3035	1	1	yes	characteristic
273	C₂×C₁₃₇	C₂×#467	1	1	yes	characteristic
274	C₂×C₁₃₇	#3056	1	1	yes	characteristic
275	C₂×C₁₃₇	#3248	1	1	yes	characteristic
276	C₂×C₁₃₇	#3249	1	1	yes	characteristic
277	#24752	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₁₃₇	#24752	1	1	yes	characteristic, direct product factor, Sylow
279	C₂×#467	C₂×C₁₃₇	1	2	yes	—
280	C₂×#462	C₂×C₁₃₇	1	1	yes	characteristic, complemented
281	C₂²×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic
282	#2857	C₂×C₁₃₇	1	1	yes	characteristic
283	C₄×C₂²⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic
284	#2068	C₂×C₁₃₇	1	2	yes	complemented
285	#2056	C₂×C₁₃₇	1	2	yes	complemented
286	#1981	C₂×C₁₃₇	1	1	yes	characteristic, complemented
287	#1947	C₂×C₁₃₇	1	1	yes	characteristic, complemented
288	#1892	C₂×C₁₃₇	1	2	yes	—
289	#1803	C₂×C₁₃₇	1	1	yes	characteristic, complemented
290	C₂²×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
291	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
293	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
295	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
296	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
297	#467	—	2	8	no	—
298	#462	—	2	4	no	—
299	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
300	C₂×C₄⋊C₈	—	2	4	no	—
301	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
302	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
304	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
305	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
306	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
307	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
308	#391	C₂²×C₁₃₇	1	1	yes	characteristic
309	#391	C₂²×C₁₃₇	1	2	yes	—
310	#389	—	2	1	no	—
311	#389	—	2	1	no	—
312	#389	C₂²×C₁₃₇	1	1	yes	characteristic
313	#387	—	2	1	no	—
314	#385	—	2	1	no	—
315	#384	C₂²×C₁₃₇	1	1	yes	characteristic
316	#380	—	2	1	no	—
317	#380	—	2	1	no	—
318	#380	C₂²×C₁₃₇	1	1	yes	characteristic
319	C₄×C₄⋊C₄	—	2	1	no	—
320	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
321	C₄×C₂²⋊C₄	—	2	1	no	—
322	#340	C₂²×C₁₃₇	1	2	yes	—
323	#340	C₂²×C₁₃₇	1	2	yes	—
324	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
325	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
326	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
327	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
328	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
329	C₂×Q₁₆	—	2	4	no	—
330	C₂×Q₁₆	—	2	4	no	—
331	C₂×Q₁₆	—	4	2	no	—
332	C₂²×C₈	—	2	4	no	—
333	C₂²×C₈	—	2	4	no	—
334	C₂²×C₈	—	4	2	no	—
335	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
336	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
337	C₄⋊Q₈	—	2	2	no	—
338	C₄⋊Q₈	—	2	2	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	4	no	—
341	C₂²⋊Q₈	—	2	4	no	—
342	C₂²⋊Q₈	D₄×C₁₃₇	1	4	yes	—
343	C₄×Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₂²⋊C₄	—	2	2	no	—
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₄²	—	2	2	no	—
361	C₂×C₄²	—	2	2	no	—
362	C₂×C₄²	—	2	1	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
365	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
368	#107	—	2	4	no	—
369	C₄⋊C₈	—	2	4	no	—
370	C₄⋊C₈	—	2	2	no	—
371	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	—
372	#103	—	2	4	no	—
373	#103	—	2	4	no	—
374	#103	—	2	4	no	—
375	#103	—	4	2	no	—
376	#95	—	2	1	no	—
377	#95	—	2	1	no	—
378	#95	—	2	1	no	—
379	#95	—	2	1	no	—
380	#95	—	2	1	no	—
381	#95	—	2	2	no	—
382	#95	—	2	2	no	—
383	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
384	C₂×Q₈	—	2	1	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	2	2	no	—
393	C₂×Q₈	—	2	1	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	4	1	no	—
411	C₂²×C₄	Q₁₆×C₁₃₇	1	4	yes	complemented
412	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
414	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
415	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
416	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
417	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
418	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
419	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
420	Q₁₆	—	4	8	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	4	8	no	—
427	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	4	2	no	—
437	C₄⋊C₄	—	4	2	no	—
438	C₄⋊C₄	—	4	2	no	—
439	C₄⋊C₄	—	4	4	no	—
440	C₄⋊C₄	—	4	2	no	—
441	C₄⋊C₄	—	4	2	no	—
442	C₄⋊C₄	—	4	2	no	—
443	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	4	4	no	—
446	C₂²⋊C₄	—	4	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	4	2	no	—
452	C₄²	—	4	2	no	—
453	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
454	C₄²	#41×C₁₃₇	1	2	yes	—
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	1	no	—
459	C₂³	#121×C₁₃₇	1	2	yes	—
460	C₂³	C₂×Q₁₆×C₁₃₇	1	2	yes	—
461	C₂³	#137×C₁₃₇	1	2	yes	—
462	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	4	1	no	—
466	Q₈	—	4	1	no	—
467	Q₈	—	4	2	no	—
468	Q₈	—	4	2	no	—
469	Q₈	—	4	2	no	—
470	Q₈	—	4	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	#121×C₁₃₇	1	2	yes	—
505	C₂×C₄	#121×C₁₃₇	1	2	yes	—
506	C₂×C₄	#121×C₁₃₇	1	2	yes	—
507	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
508	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
509	C₂×C₄	#135×C₁₃₇	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
512	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
513	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
514	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
515	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
516	C₈	—	2	4	no	—
517	C₈	—	4	2	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	#462×C₁₃₇	1	4	yes	complemented
525	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
526	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
527	C₂²	#537×C₁₃₇	1	1	yes	characteristic
528	C₂²	C₂²×Q₁₆×C₁₃₇	1	1	yes	characteristic
529	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
530	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
531	C₂²	#575×C₁₃₇	1	1	yes	characteristic
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	2	2	no	—
535	C₄	—	4	1	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	8	1	no	—
539	C₄	—	8	2	no	—
540	C₄	—	8	1	no	—
541	C₄	—	8	2	no	—
542	C₄	#467×C₁₃₇	1	4	yes	complemented
543	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
544	C₄	#547×C₁₃₇	1	1	yes	characteristic
545	C₄	#548×C₁₃₇	1	1	yes	characteristic
546	C₄	#553×C₁₃₇	1	1	yes	characteristic
547	C₂	—	2	4	no	—
548	C₂	#2416×C₁₃₇	1	1	yes	characteristic
549	C₂	C₂×#462×C₁₃₇	1	1	yes	characteristic
550	C₂	#3035×C₁₃₇	1	1	yes	characteristic
551	C₂	C₂×#467×C₁₃₇	1	1	yes	characteristic
552	C₂	#3056×C₁₃₇	1	1	yes	characteristic
553	C₂	#3248×C₁₃₇	1	1	yes	characteristic
554	C₂	#3249×C₁₃₇	1	1	yes	characteristic

	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 ◁ C₂²×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#462×C₁₃₇	C₂	semidirect
6. #2068×C₁₃₇	C₂	semidirect
7. #2056×C₁₃₇	C₂	semidirect
8. #1981×C₁₃₇	C₂	semidirect
9. #1947×C₁₃₇	C₂	semidirect
11. #1803×C₁₃₇	C₂	semidirect
133. C₂²×C₄×C₁₃₇	Q₁₆	semidirect
246. C₂²×C₁₃₇	#462	semidirect
264. C₄×C₁₃₇	#467	semidirect
277. #24752	C₁₃₇	direct (N × H)
278. C₁₃₇	#24752	direct (N × H)
280. C₂×#462	C₂×C₁₃₇	semidirect
284. #2068	C₂×C₁₃₇	semidirect
285. #2056	C₂×C₁₃₇	semidirect
286. #1981	C₂×C₁₃₇	semidirect
287. #1947	C₂×C₁₃₇	semidirect
289. #1803	C₂×C₁₃₇	semidirect
411. C₂²×C₄	Q₁₆×C₁₃₇	semidirect
524. C₂²	#462×C₁₃₇	semidirect
542. C₄	#467×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#467×C₁₃₇	2
C₂×#462×C₁₃₇	2
C₂²×C₄⋊C₈×C₁₃₇	2
#2857×C₁₃₇	2
C₄×C₂²⋊Q₈×C₁₃₇	2
#2068×C₁₃₇	2
#2056×C₁₃₇	2
#1981×C₁₃₇	2
#1947×C₁₃₇	2
#1892×C₁₃₇	2
#1803×C₁₃₇	2
#24752	137

Subgroups — C137×#24752 #46874138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24752 #46874138