Subgroups — C₁₃₇×#24753 #46874139

Order 35072.

1132 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#465×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2856×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂²⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2065×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2055×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1982×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1948×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1893×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1803×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₁₃₇	—	2	1	no	—
17	C₂×#121×C₁₃₇	—	2	1	no	—
18	C₂×#121×C₁₃₇	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
21	#465×C₁₃₇	—	2	8	no	—
22	#460×C₁₃₇	—	2	4	no	—
23	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₁₃₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
32	#391×C₁₃₇	C₂²	1	1	yes	characteristic
33	#390×C₁₃₇	—	2	1	no	—
34	#390×C₁₃₇	—	2	1	no	—
35	#390×C₁₃₇	C₂²	1	2	yes	—
36	#389×C₁₃₇	C₂²	1	1	yes	characteristic
37	#387×C₁₃₇	—	2	1	no	—
38	#385×C₁₃₇	—	2	1	no	—
39	#384×C₁₃₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
47	C₂²×D₄×C₁₃₇	—	2	2	no	—
48	C₂²×D₄×C₁₃₇	—	2	1	no	—
49	C₂²×D₄×C₁₃₇	—	2	1	no	—
50	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₁₃₇	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
55	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
56	C₂×SD₁₆×C₁₃₇	—	4	2	no	—
57	C₂²×C₈×C₁₃₇	—	2	4	no	—
58	C₂²×C₈×C₁₃₇	—	2	4	no	—
59	C₂²×C₈×C₁₃₇	—	4	2	no	—
60	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
62	#127×C₁₃₇	—	2	2	no	—
63	#127×C₁₃₇	—	2	2	no	—
64	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
65	C₂²⋊Q₈×C₁₃₇	D₄	1	4	yes	complemented
66	#121×C₁₃₇	—	2	4	no	—
67	#121×C₁₃₇	—	2	4	no	—
68	#121×C₁₃₇	—	4	4	no	—
69	#121×C₁₃₇	—	4	4	no	—
70	C₄×Q₈×C₁₃₇	—	2	2	no	—
71	C₄×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	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	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
86	C₂×C₄²×C₁₃₇	—	2	2	no	—
87	C₂×C₄²×C₁₃₇	—	2	2	no	—
88	C₂×C₄²×C₁₃₇	—	2	1	no	—
89	C₂×C₄²×C₁₃₇	—	2	1	no	—
90	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
94	#106×C₁₃₇	—	2	4	no	—
95	C₄⋊C₈×C₁₃₇	—	2	4	no	—
96	C₄⋊C₈×C₁₃₇	—	2	2	no	—
97	C₄⋊C₈×C₁₃₇	D₄	1	4	yes	complemented
98	#103×C₁₃₇	—	2	4	no	—
99	#102×C₁₃₇	—	2	4	no	—
100	#102×C₁₃₇	—	2	4	no	—
101	#102×C₁₃₇	—	4	2	no	—
102	#95×C₁₃₇	—	2	1	no	—
103	#95×C₁₃₇	—	2	1	no	—
104	#95×C₁₃₇	—	2	1	no	—
105	#95×C₁₃₇	—	2	1	no	—
106	#95×C₁₃₇	—	2	1	no	—
107	C₂⁴×C₁₃₇	—	2	2	no	—
108	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×Q₈×C₁₃₇	—	2	1	no	—
110	C₂×Q₈×C₁₃₇	—	2	1	no	—
111	C₂×Q₈×C₁₃₇	—	2	1	no	—
112	C₂×Q₈×C₁₃₇	—	2	1	no	—
113	C₂×Q₈×C₁₃₇	—	2	1	no	—
114	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
120	C₂×D₄×C₁₃₇	—	2	2	no	—
121	C₂×D₄×C₁₃₇	—	4	2	no	—
122	C₂×D₄×C₁₃₇	—	4	2	no	—
123	C₂×D₄×C₁₃₇	—	4	4	no	—
124	C₂×D₄×C₁₃₇	—	4	4	no	—
125	C₂×D₄×C₁₃₇	—	4	4	no	—
126	C₂×D₄×C₁₃₇	—	4	2	no	—
127	C₂×D₄×C₁₃₇	—	4	2	no	—
128	C₂×D₄×C₁₃₇	—	4	2	no	—
129	C₂×D₄×C₁₃₇	—	4	2	no	—
130	C₂²×C₄×C₁₃₇	—	2	2	no	—
131	C₂²×C₄×C₁₃₇	—	2	2	no	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	1	no	—
134	C₂²×C₄×C₁₃₇	—	2	1	no	—
135	C₂²×C₄×C₁₃₇	—	2	1	no	—
136	C₂²×C₄×C₁₃₇	—	2	2	no	—
137	C₂²×C₄×C₁₃₇	—	2	1	no	—
138	C₂²×C₄×C₁₃₇	—	2	1	no	—
139	C₂²×C₄×C₁₃₇	—	2	2	no	—
140	C₂²×C₄×C₁₃₇	—	2	1	no	—
141	C₂²×C₄×C₁₃₇	—	2	1	no	—
142	C₂²×C₄×C₁₃₇	—	2	2	no	—
143	C₂²×C₄×C₁₃₇	—	4	1	no	—
144	C₂²×C₄×C₁₃₇	SD₁₆	1	4	yes	complemented
145	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
152	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
153	SD₁₆×C₁₃₇	—	4	8	no	—
154	C₂×C₈×C₁₃₇	—	2	2	no	—
155	C₂×C₈×C₁₃₇	—	2	2	no	—
156	C₂×C₈×C₁₃₇	—	2	2	no	—
157	C₂×C₈×C₁₃₇	—	2	2	no	—
158	C₂×C₈×C₁₃₇	—	2	2	no	—
159	C₂×C₈×C₁₃₇	—	4	8	no	—
160	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
161	C₄⋊C₄×C₁₃₇	—	2	4	no	—
162	C₄⋊C₄×C₁₃₇	—	2	1	no	—
163	C₄⋊C₄×C₁₃₇	—	2	1	no	—
164	C₄⋊C₄×C₁₃₇	—	2	2	no	—
165	C₄⋊C₄×C₁₃₇	—	2	1	no	—
166	C₄⋊C₄×C₁₃₇	—	2	2	no	—
167	C₄⋊C₄×C₁₃₇	—	2	1	no	—
168	C₄⋊C₄×C₁₃₇	—	2	1	no	—
169	C₄⋊C₄×C₁₃₇	—	4	2	no	—
170	C₄⋊C₄×C₁₃₇	—	4	2	no	—
171	C₄⋊C₄×C₁₃₇	—	4	2	no	—
172	C₄⋊C₄×C₁₃₇	—	4	2	no	—
173	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
174	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
175	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
176	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
177	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
178	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
179	C₄²×C₁₃₇	—	2	2	no	—
180	C₄²×C₁₃₇	—	2	2	no	—
181	C₄²×C₁₃₇	—	2	1	no	—
182	C₄²×C₁₃₇	—	2	1	no	—
183	C₄²×C₁₃₇	—	4	2	no	—
184	C₄²×C₁₃₇	—	4	2	no	—
185	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
186	C₄²×C₁₃₇	#41	1	2	yes	—
187	C₂³×C₁₃₇	—	2	1	no	—
188	C₂³×C₁₃₇	—	2	1	no	—
189	C₂³×C₁₃₇	—	2	1	no	—
190	C₂³×C₁₃₇	—	2	1	no	—
191	C₂³×C₁₃₇	—	4	2	no	—
192	C₂³×C₁₃₇	—	4	2	no	—
193	C₂³×C₁₃₇	—	4	2	no	—
194	C₂³×C₁₃₇	—	4	2	no	—
195	C₂³×C₁₃₇	—	4	2	no	—
196	C₂³×C₁₃₇	—	4	2	no	—
197	C₂³×C₁₃₇	—	4	2	no	—
198	C₂³×C₁₃₇	#121	1	2	yes	—
199	C₂³×C₁₃₇	C₂×SD₁₆	1	2	yes	—
200	C₂³×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
201	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₁₃₇	—	2	2	no	—
203	Q₈×C₁₃₇	—	2	2	no	—
204	Q₈×C₁₃₇	—	4	1	no	—
205	Q₈×C₁₃₇	—	4	1	no	—
206	D₄×C₁₃₇	—	4	2	no	—
207	D₄×C₁₃₇	—	4	2	no	—
208	D₄×C₁₃₇	—	4	2	no	—
209	D₄×C₁₃₇	—	4	2	no	—
210	D₄×C₁₃₇	—	8	4	no	—
211	D₄×C₁₃₇	—	8	8	no	—
212	D₄×C₁₃₇	—	8	4	no	—
213	C₂×C₄×C₁₃₇	—	2	1	no	—
214	C₂×C₄×C₁₃₇	—	2	1	no	—
215	C₂×C₄×C₁₃₇	—	2	1	no	—
216	C₂×C₄×C₁₃₇	—	2	1	no	—
217	C₂×C₄×C₁₃₇	—	2	1	no	—
218	C₂×C₄×C₁₃₇	—	2	1	no	—
219	C₂×C₄×C₁₃₇	—	2	2	no	—
220	C₂×C₄×C₁₃₇	—	2	2	no	—
221	C₂×C₄×C₁₃₇	—	2	2	no	—
222	C₂×C₄×C₁₃₇	—	2	2	no	—
223	C₂×C₄×C₁₃₇	—	2	8	no	—
224	C₂×C₄×C₁₃₇	—	2	1	no	—
225	C₂×C₄×C₁₃₇	—	2	2	no	—
226	C₂×C₄×C₁₃₇	—	2	2	no	—
227	C₂×C₄×C₁₃₇	—	2	1	no	—
228	C₂×C₄×C₁₃₇	—	2	1	no	—
229	C₂×C₄×C₁₃₇	—	4	4	no	—
230	C₂×C₄×C₁₃₇	—	4	1	no	—
231	C₂×C₄×C₁₃₇	—	4	1	no	—
232	C₂×C₄×C₁₃₇	—	4	1	no	—
233	C₂×C₄×C₁₃₇	—	4	1	no	—
234	C₂×C₄×C₁₃₇	—	4	1	no	—
235	C₂×C₄×C₁₃₇	—	4	2	no	—
236	C₂×C₄×C₁₃₇	—	4	1	no	—
237	C₂×C₄×C₁₃₇	—	4	2	no	—
238	C₂×C₄×C₁₃₇	—	4	1	no	—
239	C₂×C₄×C₁₃₇	—	4	2	no	—
240	C₂×C₄×C₁₃₇	—	4	1	no	—
241	C₂×C₄×C₁₃₇	—	4	1	no	—
242	C₂×C₄×C₁₃₇	—	4	1	no	—
243	C₂×C₄×C₁₃₇	#121	1	2	yes	—
244	C₂×C₄×C₁₃₇	#121	1	2	yes	—
245	C₂×C₄×C₁₃₇	#121	1	2	yes	—
246	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
247	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
248	C₂×C₄×C₁₃₇	#135	1	2	yes	—
249	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
255	C₈×C₁₃₇	—	2	4	no	—
256	C₈×C₁₃₇	—	4	2	no	—
257	C₂²×C₁₃₇	—	2	2	no	—
258	C₂²×C₁₃₇	—	2	2	no	—
259	C₂²×C₁₃₇	—	2	2	no	—
260	C₂²×C₁₃₇	—	2	2	no	—
261	C₂²×C₁₃₇	—	2	2	no	—
262	C₂²×C₁₃₇	—	2	2	no	—
263	C₂²×C₁₃₇	—	8	2	no	—
264	C₂²×C₁₃₇	—	8	2	no	—
265	C₂²×C₁₃₇	—	8	2	no	—
266	C₂²×C₁₃₇	—	8	2	no	—
267	C₂²×C₁₃₇	—	8	2	no	—
268	C₂²×C₁₃₇	—	8	2	no	—
269	C₂²×C₁₃₇	—	8	2	no	—
270	C₂²×C₁₃₇	#460	1	4	yes	complemented
271	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
272	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
273	C₂²×C₁₃₇	#537	1	1	yes	characteristic
274	C₂²×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
275	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
276	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₂²×C₁₃₇	#575	1	1	yes	characteristic
278	C₄×C₁₃₇	—	2	2	no	—
279	C₄×C₁₃₇	—	2	2	no	—
280	C₄×C₁₃₇	—	2	2	no	—
281	C₄×C₁₃₇	—	4	2	no	—
282	C₄×C₁₃₇	—	4	2	no	—
283	C₄×C₁₃₇	—	4	1	no	—
284	C₄×C₁₃₇	—	8	1	no	—
285	C₄×C₁₃₇	—	8	2	no	—
286	C₄×C₁₃₇	—	8	1	no	—
287	C₄×C₁₃₇	#465	1	4	yes	complemented
288	C₄×C₁₃₇	D₄²	1	1	yes	characteristic
289	C₄×C₁₃₇	#547	1	1	yes	characteristic
290	C₄×C₁₃₇	#548	1	1	yes	characteristic
291	C₄×C₁₃₇	#553	1	1	yes	characteristic
292	C₂×C₁₃₇	—	2	4	no	—
293	C₂×C₁₃₇	—	16	2	no	—
294	C₂×C₁₃₇	#2416	1	1	yes	characteristic
295	C₂×C₁₃₇	C₂×#460	1	1	yes	characteristic
296	C₂×C₁₃₇	#3035	1	1	yes	characteristic
297	C₂×C₁₃₇	C₂×#465	1	1	yes	characteristic
298	C₂×C₁₃₇	#3055	1	1	yes	characteristic
299	C₂×C₁₃₇	#3246	1	1	yes	characteristic
300	C₂×C₁₃₇	#3247	1	1	yes	characteristic
301	#24753	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₁₃₇	#24753	1	1	yes	characteristic, direct product factor, Sylow
303	C₂×#465	C₂×C₁₃₇	1	2	yes	complemented
304	C₂×#460	C₂×C₁₃₇	1	1	yes	characteristic, complemented
305	C₂²×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
306	#2856	C₂×C₁₃₇	1	1	yes	characteristic
307	C₄×C₂²⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
308	#2065	C₂×C₁₃₇	1	2	yes	complemented
309	#2055	C₂×C₁₃₇	1	2	yes	complemented
310	#1982	C₂×C₁₃₇	1	1	yes	characteristic, complemented
311	#1948	C₂×C₁₃₇	1	1	yes	characteristic, complemented
312	#1893	C₂×C₁₃₇	1	2	yes	complemented
313	#1803	C₂×C₁₃₇	1	1	yes	characteristic, complemented
314	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
315	C₂³×C₈	C₂²×C₁₃₇	1	2	yes	—
316	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
317	C₂×C₂²⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
318	C₂×#121	—	2	1	no	—
319	C₂×#121	—	2	1	no	—
320	C₂×#121	C₂²×C₁₃₇	1	2	yes	—
321	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
322	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
323	#465	—	2	8	no	—
324	#460	—	2	4	no	—
325	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
326	C₂×C₄⋊C₈	—	2	4	no	—
327	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
328	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
329	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
330	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
331	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
332	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
333	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
334	#391	C₂²×C₁₃₇	1	1	yes	characteristic
335	#390	—	2	1	no	—
336	#390	—	2	1	no	—
337	#390	C₂²×C₁₃₇	1	2	yes	—
338	#389	C₂²×C₁₃₇	1	1	yes	characteristic
339	#387	—	2	1	no	—
340	#385	—	2	1	no	—
341	#384	C₂²×C₁₃₇	1	1	yes	characteristic
342	#380	C₂²×C₁₃₇	1	1	yes	characteristic
343	C₄×C₄⋊C₄	—	2	1	no	—
344	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	—	2	1	no	—
346	#340	C₂²×C₁₃₇	1	2	yes	—
347	#340	C₂²×C₁₃₇	1	2	yes	—
348	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	—	2	1	no	—
352	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
353	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
354	C₂³×C₄	D₄×C₁₃₇	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
356	C₂×SD₁₆	—	2	4	no	—
357	C₂×SD₁₆	—	2	4	no	—
358	C₂×SD₁₆	—	4	2	no	—
359	C₂²×C₈	—	2	4	no	—
360	C₂²×C₈	—	2	4	no	—
361	C₂²×C₈	—	4	2	no	—
362	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
363	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
364	#127	—	2	2	no	—
365	#127	—	2	2	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₂²⋊Q₈	D₄×C₁₃₇	1	4	yes	complemented
368	#121	—	2	4	no	—
369	#121	—	2	4	no	—
370	#121	—	4	4	no	—
371	#121	—	4	4	no	—
372	C₄×Q₈	—	2	2	no	—
373	C₄×Q₈	—	2	2	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
383	C₂×C₂²⋊C₄	—	2	1	no	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₄²	—	2	2	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
393	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
396	#106	—	2	4	no	—
397	C₄⋊C₈	—	2	4	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	D₄×C₁₃₇	1	4	yes	complemented
400	#103	—	2	4	no	—
401	#102	—	2	4	no	—
402	#102	—	2	4	no	—
403	#102	—	4	2	no	—
404	#95	—	2	1	no	—
405	#95	—	2	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	C₂⁴	—	2	2	no	—
410	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	1	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	4	2	no	—
424	C₂×D₄	—	4	2	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	4	no	—
427	C₂×D₄	—	4	4	no	—
428	C₂×D₄	—	4	2	no	—
429	C₂×D₄	—	4	2	no	—
430	C₂×D₄	—	4	2	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	SD₁₆×C₁₃₇	1	4	yes	complemented
447	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
453	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
455	SD₁₆	—	4	8	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	4	8	no	—
462	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	4	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	—	4	2	no	—
475	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	4	4	no	—
478	C₂²⋊C₄	—	4	2	no	—
479	C₂²⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	4	4	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	4	2	no	—
486	C₄²	—	4	2	no	—
487	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
488	C₄²	#41×C₁₃₇	1	2	yes	—
489	C₂³	—	2	1	no	—
490	C₂³	—	2	1	no	—
491	C₂³	—	2	1	no	—
492	C₂³	—	2	1	no	—
493	C₂³	—	4	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	2	no	—
499	C₂³	—	4	2	no	—
500	C₂³	#121×C₁₃₇	1	2	yes	—
501	C₂³	C₂×SD₁₆×C₁₃₇	1	2	yes	—
502	C₂³	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
503	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
504	Q₈	—	2	2	no	—
505	Q₈	—	2	2	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	2	no	—
512	D₄	—	8	4	no	—
513	D₄	—	8	8	no	—
514	D₄	—	8	4	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	#121×C₁₃₇	1	2	yes	—
546	C₂×C₄	#121×C₁₃₇	1	2	yes	—
547	C₂×C₄	#121×C₁₃₇	1	2	yes	—
548	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
549	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
550	C₂×C₄	#135×C₁₃₇	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
554	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
555	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
556	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
557	C₈	—	2	4	no	—
558	C₈	—	4	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	8	2	no	—
566	C₂²	—	8	2	no	—
567	C₂²	—	8	2	no	—
568	C₂²	—	8	2	no	—
569	C₂²	—	8	2	no	—
570	C₂²	—	8	2	no	—
571	C₂²	—	8	2	no	—
572	C₂²	#460×C₁₃₇	1	4	yes	complemented
573	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
574	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
575	C₂²	#537×C₁₃₇	1	1	yes	characteristic
576	C₂²	C₂²×SD₁₆×C₁₃₇	1	1	yes	characteristic
577	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
578	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
579	C₂²	#575×C₁₃₇	1	1	yes	characteristic
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	2	no	—
589	C₄	#465×C₁₃₇	1	4	yes	complemented
590	C₄	D₄²×C₁₃₇	1	1	yes	characteristic
591	C₄	#547×C₁₃₇	1	1	yes	characteristic
592	C₄	#548×C₁₃₇	1	1	yes	characteristic
593	C₄	#553×C₁₃₇	1	1	yes	characteristic
594	C₂	—	2	4	no	—
595	C₂	—	16	2	no	—
596	C₂	#2416×C₁₃₇	1	1	yes	characteristic
597	C₂	C₂×#460×C₁₃₇	1	1	yes	characteristic
598	C₂	#3035×C₁₃₇	1	1	yes	characteristic
599	C₂	C₂×#465×C₁₃₇	1	1	yes	characteristic
600	C₂	#3055×C₁₃₇	1	1	yes	characteristic
601	C₂	#3246×C₁₃₇	1	1	yes	characteristic
602	C₂	#3247×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#2856×C₁₃₇	C₂
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
1. C₂×#465×C₁₃₇	C₂	semidirect
2. C₂×#460×C₁₃₇	C₂	semidirect
3. C₂²×C₄⋊C₈×C₁₃₇	C₂	semidirect
5. C₄×C₂²⋊Q₈×C₁₃₇	C₂	semidirect
6. #2065×C₁₃₇	C₂	semidirect
7. #2055×C₁₃₇	C₂	semidirect
8. #1982×C₁₃₇	C₂	semidirect
9. #1948×C₁₃₇	C₂	semidirect
10. #1893×C₁₃₇	C₂	semidirect
11. #1803×C₁₃₇	C₂	semidirect
65. C₂²⋊Q₈×C₁₃₇	D₄	semidirect
97. C₄⋊C₈×C₁₃₇	D₄	semidirect
144. C₂²×C₄×C₁₃₇	SD₁₆	semidirect
270. C₂²×C₁₃₇	#460	semidirect
287. C₄×C₁₃₇	#465	semidirect
301. #24753	C₁₃₇	direct (N × H)
302. C₁₃₇	#24753	direct (N × H)
303. C₂×#465	C₂×C₁₃₇	semidirect
304. C₂×#460	C₂×C₁₃₇	semidirect
305. C₂²×C₄⋊C₈	C₂×C₁₃₇	semidirect
307. C₄×C₂²⋊Q₈	C₂×C₁₃₇	semidirect
308. #2065	C₂×C₁₃₇	semidirect
309. #2055	C₂×C₁₃₇	semidirect
310. #1982	C₂×C₁₃₇	semidirect
311. #1948	C₂×C₁₃₇	semidirect
312. #1893	C₂×C₁₃₇	semidirect
313. #1803	C₂×C₁₃₇	semidirect
367. C₂²⋊Q₈	D₄×C₁₃₇	semidirect
399. C₄⋊C₈	D₄×C₁₃₇	semidirect
446. C₂²×C₄	SD₁₆×C₁₃₇	semidirect
572. C₂²	#460×C₁₃₇	semidirect
589. C₄	#465×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#465×C₁₃₇	2
C₂×#460×C₁₃₇	2
C₂²×C₄⋊C₈×C₁₃₇	2
#2856×C₁₃₇	2
C₄×C₂²⋊Q₈×C₁₃₇	2
#2065×C₁₃₇	2
#2055×C₁₃₇	2
#1982×C₁₃₇	2
#1948×C₁₃₇	2
#1893×C₁₃₇	2
#1803×C₁₃₇	2
#24753	137

Subgroups — C137×#24753 #46874139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24753 #46874139