Subgroups — C₁₃₇×#24742 #46874128

Order 35072.

1104 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#460×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×#460×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#440×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2541×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2351×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#2067×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#2041×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1961×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1940×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1880×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1804×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₃₇	C₂²	1	2	yes	—
16	C₂×#121×C₁₃₇	—	2	2	no	—
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
20	#460×C₁₃₇	—	2	8	no	—
21	#460×C₁₃₇	—	2	4	no	—
22	#440×C₁₃₇	—	2	4	no	—
23	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#102×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×#102×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
33	#392×C₁₃₇	—	2	1	no	—
34	#392×C₁₃₇	—	2	1	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×D₄×C₁₃₇	—	2	1	no	—
46	C₂²×D₄×C₁₃₇	—	2	2	no	—
47	C₂²×D₄×C₁₃₇	—	2	2	no	—
48	C₂²×D₄×C₁₃₇	—	2	1	no	—
49	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
50	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₁₃₇	—	2	1	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
53	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
54	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
55	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
56	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
57	C₂×SD₁₆×C₁₃₇	D₄	1	4	yes	complemented
58	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
60	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
61	#127×C₁₃₇	—	2	4	no	—
62	#127×C₁₃₇	—	2	2	no	—
63	#127×C₁₃₇	—	2	2	no	—
64	#127×C₁₃₇	—	4	2	no	—
65	#121×C₁₃₇	—	4	8	no	—
66	C₄×Q₈×C₁₃₇	—	2	2	no	—
67	C₄×Q₈×C₁₃₇	—	2	4	no	—
68	C₄×Q₈×C₁₃₇	—	4	2	no	—
69	C₄×Q₈×C₁₃₇	D₄	1	4	yes	complemented
70	C₄×D₄×C₁₃₇	—	2	2	no	—
71	C₄×D₄×C₁₃₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
82	C₂×C₄²×C₁₃₇	—	2	1	no	—
83	C₂×C₄²×C₁₃₇	—	2	2	no	—
84	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
86	#107×C₁₃₇	—	2	2	no	—
87	C₄⋊C₈×C₁₃₇	—	2	4	no	—
88	C₄⋊C₈×C₁₃₇	—	2	2	no	—
89	#103×C₁₃₇	—	2	2	no	—
90	#102×C₁₃₇	—	2	4	no	—
91	#102×C₁₃₇	—	2	4	no	—
92	#102×C₁₃₇	—	2	2	no	—
93	#102×C₁₃₇	—	2	4	no	—
94	#102×C₁₃₇	—	4	2	no	—
95	#97×C₁₃₇	—	2	2	no	—
96	#95×C₁₃₇	—	2	1	no	—
97	#95×C₁₃₇	—	2	2	no	—
98	#95×C₁₃₇	—	2	1	no	—
99	#95×C₁₃₇	D₄	1	1	yes	characteristic
100	#95×C₁₃₇	D₄	1	1	yes	characteristic
101	C₂⁴×C₁₃₇	—	2	1	no	—
102	C₂⁴×C₁₃₇	—	2	2	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	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂×Q₈×C₁₃₇	#41	1	2	yes	—
109	C₂×D₄×C₁₃₇	—	2	1	no	—
110	C₂×D₄×C₁₃₇	—	2	2	no	—
111	C₂×D₄×C₁₃₇	—	2	2	no	—
112	C₂×D₄×C₁₃₇	—	2	1	no	—
113	C₂×D₄×C₁₃₇	—	2	1	no	—
114	C₂×D₄×C₁₃₇	—	2	2	no	—
115	C₂×D₄×C₁₃₇	—	2	2	no	—
116	C₂×D₄×C₁₃₇	—	2	1	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	2	1	no	—
119	C₂×D₄×C₁₃₇	—	2	2	no	—
120	C₂×D₄×C₁₃₇	—	4	4	no	—
121	C₂×D₄×C₁₃₇	—	4	4	no	—
122	C₂×D₄×C₁₃₇	—	4	4	no	—
123	C₂×D₄×C₁₃₇	—	4	4	no	—
124	C₂×D₄×C₁₃₇	—	4	2	no	—
125	C₂×D₄×C₁₃₇	—	4	2	no	—
126	C₂×D₄×C₁₃₇	—	4	4	no	—
127	C₂×D₄×C₁₃₇	—	8	2	no	—
128	C₂×D₄×C₁₃₇	—	8	2	no	—
129	C₂×D₄×C₁₃₇	—	8	2	no	—
130	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	2	no	—
134	C₂²×C₄×C₁₃₇	—	2	1	no	—
135	C₂²×C₄×C₁₃₇	—	4	4	no	—
136	C₂²×C₄×C₁₃₇	—	4	2	no	—
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	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
142	SD₁₆×C₁₃₇	—	2	8	no	—
143	SD₁₆×C₁₃₇	—	4	8	no	—
144	C₂×C₈×C₁₃₇	—	2	1	no	—
145	C₂×C₈×C₁₃₇	—	2	1	no	—
146	C₂×C₈×C₁₃₇	—	2	1	no	—
147	C₂×C₈×C₁₃₇	—	2	2	no	—
148	C₂×C₈×C₁₃₇	—	2	2	no	—
149	C₂×C₈×C₁₃₇	—	2	2	no	—
150	C₂×C₈×C₁₃₇	—	2	1	no	—
151	C₂×C₈×C₁₃₇	—	2	1	no	—
152	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
153	C₄⋊C₄×C₁₃₇	—	2	4	no	—
154	C₄⋊C₄×C₁₃₇	—	2	4	no	—
155	C₄⋊C₄×C₁₃₇	—	2	1	no	—
156	C₄⋊C₄×C₁₃₇	—	2	2	no	—
157	C₄⋊C₄×C₁₃₇	—	2	1	no	—
158	C₄⋊C₄×C₁₃₇	—	2	1	no	—
159	C₄⋊C₄×C₁₃₇	—	2	2	no	—
160	C₄⋊C₄×C₁₃₇	—	4	4	no	—
161	C₄⋊C₄×C₁₃₇	—	4	4	no	—
162	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
163	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
164	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
165	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
166	C₄²×C₁₃₇	—	2	4	no	—
167	C₄²×C₁₃₇	—	2	2	no	—
168	C₄²×C₁₃₇	—	2	2	no	—
169	C₄²×C₁₃₇	—	4	4	no	—
170	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
171	C₄²×C₁₃₇	#41	1	2	yes	—
172	C₂³×C₁₃₇	—	2	2	no	—
173	C₂³×C₁₃₇	—	4	1	no	—
174	C₂³×C₁₃₇	—	4	1	no	—
175	C₂³×C₁₃₇	—	4	2	no	—
176	C₂³×C₁₃₇	—	4	2	no	—
177	C₂³×C₁₃₇	—	4	2	no	—
178	C₂³×C₁₃₇	—	4	2	no	—
179	C₂³×C₁₃₇	—	4	2	no	—
180	C₂³×C₁₃₇	—	4	2	no	—
181	C₂³×C₁₃₇	—	4	2	no	—
182	C₂³×C₁₃₇	—	4	2	no	—
183	C₂³×C₁₃₇	—	4	2	no	—
184	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₁₃₇	—	2	2	no	—
186	Q₈×C₁₃₇	—	2	2	no	—
187	Q₈×C₁₃₇	—	2	2	no	—
188	Q₈×C₁₃₇	#121	1	4	yes	complemented
189	D₄×C₁₃₇	—	2	2	no	—
190	D₄×C₁₃₇	—	2	2	no	—
191	D₄×C₁₃₇	—	4	2	no	—
192	D₄×C₁₃₇	—	4	1	no	—
193	D₄×C₁₃₇	—	4	2	no	—
194	D₄×C₁₃₇	—	4	2	no	—
195	D₄×C₁₃₇	—	4	1	no	—
196	D₄×C₁₃₇	—	4	2	no	—
197	D₄×C₁₃₇	—	8	8	no	—
198	D₄×C₁₃₇	—	8	4	no	—
199	D₄×C₁₃₇	—	8	4	no	—
200	D₄×C₁₃₇	—	8	8	no	—
201	C₂×C₄×C₁₃₇	—	2	2	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	2	2	no	—
204	C₂×C₄×C₁₃₇	—	2	2	no	—
205	C₂×C₄×C₁₃₇	—	2	2	no	—
206	C₂×C₄×C₁₃₇	—	2	2	no	—
207	C₂×C₄×C₁₃₇	—	2	2	no	—
208	C₂×C₄×C₁₃₇	—	2	2	no	—
209	C₂×C₄×C₁₃₇	—	2	2	no	—
210	C₂×C₄×C₁₃₇	—	2	2	no	—
211	C₂×C₄×C₁₃₇	—	2	2	no	—
212	C₂×C₄×C₁₃₇	—	4	2	no	—
213	C₂×C₄×C₁₃₇	—	4	1	no	—
214	C₂×C₄×C₁₃₇	—	4	2	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	—	8	4	no	—
219	C₂×C₄×C₁₃₇	#121	1	2	yes	—
220	C₂×C₄×C₁₃₇	#121	1	2	yes	—
221	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
222	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
228	C₈×C₁₃₇	—	2	2	no	—
229	C₈×C₁₃₇	—	4	2	no	—
230	C₈×C₁₃₇	—	4	1	no	—
231	C₂²×C₁₃₇	—	4	2	no	—
232	C₂²×C₁₃₇	—	4	4	no	—
233	C₂²×C₁₃₇	—	8	2	no	—
234	C₂²×C₁₃₇	—	8	1	no	—
235	C₂²×C₁₃₇	—	8	1	no	—
236	C₂²×C₁₃₇	—	8	2	no	—
237	C₂²×C₁₃₇	—	8	2	no	—
238	C₂²×C₁₃₇	—	8	2	no	—
239	C₂²×C₁₃₇	—	8	2	no	—
240	C₂²×C₁₃₇	—	8	2	no	—
241	C₂²×C₁₃₇	—	8	2	no	—
242	C₂²×C₁₃₇	—	8	2	no	—
243	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
244	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
245	C₂²×C₁₃₇	#548	1	1	yes	characteristic
246	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
247	C₂²×C₁₃₇	#577	1	2	yes	—
248	C₄×C₁₃₇	—	2	4	no	—
249	C₄×C₁₃₇	—	2	4	no	—
250	C₄×C₁₃₇	—	2	2	no	—
251	C₄×C₁₃₇	—	2	2	no	—
252	C₄×C₁₃₇	—	4	2	no	—
253	C₄×C₁₃₇	—	4	2	no	—
254	C₄×C₁₃₇	—	8	2	no	—
255	C₄×C₁₃₇	—	8	1	no	—
256	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
257	C₄×C₁₃₇	#537	1	1	yes	characteristic
258	C₄×C₁₃₇	#547	1	1	yes	characteristic
259	C₄×C₁₃₇	#553	1	1	yes	characteristic
260	C₂×C₁₃₇	—	8	2	no	—
261	C₂×C₁₃₇	—	16	2	no	—
262	C₂×C₁₃₇	#2416	1	1	yes	characteristic
263	C₂×C₁₃₇	#3038	1	1	yes	characteristic
264	C₂×C₁₃₇	#3043	1	1	yes	characteristic
265	C₂×C₁₃₇	#3265	1	2	yes	—
266	C₂×C₁₃₇	#3368	1	2	yes	—
267	#24742	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₁₃₇	#24742	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×#460	C₂×C₁₃₇	1	2	yes	complemented
270	C₂×#460	C₂×C₁₃₇	1	1	yes	characteristic, complemented
271	C₂×#440	C₂×C₁₃₇	1	1	yes	characteristic, complemented
272	#2541	C₂×C₁₃₇	1	1	yes	characteristic
273	#2351	C₂×C₁₃₇	1	1	yes	characteristic, complemented
274	#2067	C₂×C₁₃₇	1	2	yes	complemented
275	#2041	C₂×C₁₃₇	1	2	yes	complemented
276	#1961	C₂×C₁₃₇	1	1	yes	characteristic, complemented
277	#1940	C₂×C₁₃₇	1	1	yes	characteristic, complemented
278	#1880	C₂×C₁₃₇	1	2	yes	complemented
279	#1804	C₂×C₁₃₇	1	1	yes	characteristic, complemented
280	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
281	C₂²×SD₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
282	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
283	C₂×#127	C₂²×C₁₃₇	1	2	yes	—
284	C₂×#121	—	2	2	no	—
285	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
286	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
288	#460	—	2	8	no	—
289	#460	—	2	4	no	—
290	#440	—	2	4	no	—
291	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
292	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
294	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
295	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
296	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
297	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
298	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
299	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
300	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
301	#392	—	2	1	no	—
302	#392	—	2	1	no	—
303	#390	C₂²×C₁₃₇	1	2	yes	—
304	#389	—	2	2	no	—
305	#389	C₂²×C₁₃₇	1	2	yes	—
306	#382	—	2	1	no	—
307	#382	C₂²×C₁₃₇	1	1	yes	characteristic
308	C₄×C₄⋊C₄	—	2	1	no	—
309	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
310	#340	C₂²×C₁₃₇	1	2	yes	—
311	#336	C₂²×C₁₃₇	1	2	yes	—
312	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
318	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	—	2	4	no	—
323	C₂×SD₁₆	—	2	4	no	—
324	C₂×SD₁₆	—	2	4	no	—
325	C₂×SD₁₆	D₄×C₁₃₇	1	4	yes	complemented
326	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
327	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
328	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
329	#127	—	2	4	no	—
330	#127	—	2	2	no	—
331	#127	—	2	2	no	—
332	#127	—	4	2	no	—
333	#121	—	4	8	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	4	no	—
336	C₄×Q₈	—	4	2	no	—
337	C₄×Q₈	D₄×C₁₃₇	1	4	yes	complemented
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	2	no	—
342	C₂×C₄⋊C₄	—	2	2	no	—
343	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	—	2	2	no	—
352	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
353	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
354	#107	—	2	2	no	—
355	C₄⋊C₈	—	2	4	no	—
356	C₄⋊C₈	—	2	2	no	—
357	#103	—	2	2	no	—
358	#102	—	2	4	no	—
359	#102	—	2	4	no	—
360	#102	—	2	2	no	—
361	#102	—	2	4	no	—
362	#102	—	4	2	no	—
363	#97	—	2	2	no	—
364	#95	—	2	1	no	—
365	#95	—	2	2	no	—
366	#95	—	2	1	no	—
367	#95	D₄×C₁₃₇	1	1	yes	characteristic
368	#95	D₄×C₁₃₇	1	1	yes	characteristic
369	C₂⁴	—	2	1	no	—
370	C₂⁴	—	2	2	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
375	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
376	C₂×Q₈	#41×C₁₃₇	1	2	yes	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	4	no	—
390	C₂×D₄	—	4	4	no	—
391	C₂×D₄	—	4	4	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	4	no	—
395	C₂×D₄	—	8	2	no	—
396	C₂×D₄	—	8	2	no	—
397	C₂×D₄	—	8	2	no	—
398	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
410	SD₁₆	—	2	8	no	—
411	SD₁₆	—	4	8	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	4	4	no	—
429	C₄⋊C₄	—	4	4	no	—
430	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	4	2	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₄²	—	2	4	no	—
435	C₄²	—	2	2	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	4	4	no	—
438	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
439	C₄²	#41×C₁₃₇	1	2	yes	—
440	C₂³	—	2	2	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	2	no	—
456	Q₈	#121×C₁₃₇	1	4	yes	complemented
457	D₄	—	2	2	no	—
458	D₄	—	2	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	8	8	no	—
466	D₄	—	8	4	no	—
467	D₄	—	8	4	no	—
468	D₄	—	8	8	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	8	4	no	—
487	C₂×C₄	#121×C₁₃₇	1	2	yes	—
488	C₂×C₄	#121×C₁₃₇	1	2	yes	—
489	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
490	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
492	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
494	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
495	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
496	C₈	—	2	2	no	—
497	C₈	—	4	2	no	—
498	C₈	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	8	1	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
512	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
513	C₂²	#548×C₁₃₇	1	1	yes	characteristic
514	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
515	C₂²	#577×C₁₃₇	1	2	yes	—
516	C₄	—	2	2	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	4	no	—
519	C₄	—	2	4	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	8	1	no	—
523	C₄	—	8	2	no	—
524	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
525	C₄	#537×C₁₃₇	1	1	yes	characteristic
526	C₄	#547×C₁₃₇	1	1	yes	characteristic
527	C₄	#553×C₁₃₇	1	1	yes	characteristic
528	C₂	—	8	2	no	—
529	C₂	—	16	2	no	—
530	C₂	#2416×C₁₃₇	1	1	yes	characteristic
531	C₂	#3038×C₁₃₇	1	1	yes	characteristic
532	C₂	#3043×C₁₃₇	1	1	yes	characteristic
533	C₂	#3265×C₁₃₇	1	2	yes	—
534	C₂	#3368×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#2541×C₁₃₇	C₂
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×#460×C₁₃₇	C₂	semidirect
2. C₂×#460×C₁₃₇	C₂	semidirect
3. C₂×#440×C₁₃₇	C₂	semidirect
5. #2351×C₁₃₇	C₂	semidirect
6. #2067×C₁₃₇	C₂	semidirect
7. #2041×C₁₃₇	C₂	semidirect
8. #1961×C₁₃₇	C₂	semidirect
9. #1940×C₁₃₇	C₂	semidirect
10. #1880×C₁₃₇	C₂	semidirect
11. #1804×C₁₃₇	C₂	semidirect
57. C₂×SD₁₆×C₁₃₇	D₄	semidirect
69. C₄×Q₈×C₁₃₇	D₄	semidirect
188. Q₈×C₁₃₇	#121	semidirect
267. #24742	C₁₃₇	direct (N × H)
268. C₁₃₇	#24742	direct (N × H)
269. C₂×#460	C₂×C₁₃₇	semidirect
270. C₂×#460	C₂×C₁₃₇	semidirect
271. C₂×#440	C₂×C₁₃₇	semidirect
273. #2351	C₂×C₁₃₇	semidirect
274. #2067	C₂×C₁₃₇	semidirect
275. #2041	C₂×C₁₃₇	semidirect
276. #1961	C₂×C₁₃₇	semidirect
277. #1940	C₂×C₁₃₇	semidirect
278. #1880	C₂×C₁₃₇	semidirect
279. #1804	C₂×C₁₃₇	semidirect
325. C₂×SD₁₆	D₄×C₁₃₇	semidirect
337. C₄×Q₈	D₄×C₁₃₇	semidirect
456. Q₈	#121×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#460×C₁₃₇	2
C₂×#440×C₁₃₇	2
#2541×C₁₃₇	2
#2351×C₁₃₇	2
#2067×C₁₃₇	2
#2041×C₁₃₇	2
#1961×C₁₃₇	2
#1940×C₁₃₇	2
#1880×C₁₃₇	2
#1804×C₁₃₇	2
#24742	137

Subgroups — C137×#24742 #46874128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#24742 #46874128