Subgroups — C₁₃₇×#22671 #46872057

Order 35072.

1018 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#440×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×C₄×SD₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1944×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1863×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1862×C₁₃₇	C₂	1	2	yes	maximal
8	#1845×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1803×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1802×C₁₃₇	C₂	1	1	yes	maximal, characteristic
11	#1770×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
18	#440×C₁₃₇	—	2	8	no	—
19	C₄×SD₁₆×C₁₃₇	C₄	1	8	yes	complemented
20	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
21	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
22	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#389×C₁₃₇	—	2	2	no	—
31	#389×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	C₂²	1	2	yes	—
33	#385×C₁₃₇	—	2	2	no	—
34	#384×C₁₃₇	—	2	1	no	—
35	#384×C₁₃₇	C₂²	1	1	yes	characteristic
36	#384×C₁₃₇	C₂²	1	1	yes	characteristic
37	#380×C₁₃₇	—	2	1	no	—
38	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
39	C₄×C₄⋊C₄×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	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₃₇	—	2	1	no	—
46	C₂³×C₄×C₁₃₇	—	2	2	no	—
47	C₂×SD₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₂×SD₁₆×C₁₃₇	D₄	1	4	yes	—
49	C₂×SD₁₆×C₁₃₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₇	—	2	2	no	—
54	C₄×Q₈×C₁₃₇	—	2	4	no	—
55	C₄×Q₈×C₁₃₇	—	4	2	no	—
56	C₄×Q₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
57	C₄×D₄×C₁₃₇	—	2	4	no	—
58	C₄×D₄×C₁₃₇	—	2	2	no	—
59	C₄×D₄×C₁₃₇	—	4	2	no	—
60	C₄×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₄⋊C₄×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	—	4	4	no	—
70	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₃₇	C₂³	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	1	no	—
77	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
78	C₂×C₄²×C₁₃₇	—	2	1	no	—
79	C₂×C₄²×C₁₃₇	—	2	2	no	—
80	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
82	#107×C₁₃₇	—	2	4	no	—
83	#107×C₁₃₇	—	4	2	no	—
84	#106×C₁₃₇	C₂×C₄	1	4	yes	complemented
85	C₄⋊C₈×C₁₃₇	—	2	2	no	—
86	#103×C₁₃₇	—	2	4	no	—
87	#103×C₁₃₇	C₂×C₄	1	4	yes	complemented
88	#102×C₁₃₇	—	2	4	no	—
89	#102×C₁₃₇	C₂×C₄	1	4	yes	—
90	#97×C₁₃₇	—	2	4	no	—
91	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
92	#95×C₁₃₇	—	2	2	no	—
93	#95×C₁₃₇	—	2	2	no	—
94	#95×C₁₃₇	D₄	1	2	yes	—
95	C₂⁴×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	C₂²×C₄	1	2	yes	—
100	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₂×Q₈×C₁₃₇	C₂×Q₈	1	2	yes	—
102	C₂×D₄×C₁₃₇	—	2	1	no	—
103	C₂×D₄×C₁₃₇	—	2	1	no	—
104	C₂×D₄×C₁₃₇	—	2	1	no	—
105	C₂×D₄×C₁₃₇	C₂²×C₄	1	2	yes	—
106	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
107	C₂×D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₁₃₇	—	2	1	no	—
109	C₂²×C₄×C₁₃₇	—	2	1	no	—
110	C₂²×C₄×C₁₃₇	—	2	1	no	—
111	C₂²×C₄×C₁₃₇	—	2	4	no	—
112	C₂²×C₄×C₁₃₇	—	2	2	no	—
113	C₂²×C₄×C₁₃₇	—	2	2	no	—
114	C₂²×C₄×C₁₃₇	—	4	4	no	—
115	C₂²×C₄×C₁₃₇	—	4	4	no	—
116	C₂²×C₄×C₁₃₇	—	4	2	no	—
117	C₂²×C₄×C₁₃₇	—	4	2	no	—
118	C₂²×C₄×C₁₃₇	—	8	2	no	—
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
124	SD₁₆×C₁₃₇	C₄⋊C₄	1	16	yes	complemented
125	C₂×C₈×C₁₃₇	—	2	1	no	—
126	C₂×C₈×C₁₃₇	—	2	1	no	—
127	C₂×C₈×C₁₃₇	—	2	2	no	—
128	C₂×C₈×C₁₃₇	—	2	2	no	—
129	C₂×C₈×C₁₃₇	—	2	2	no	—
130	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
134	C₄⋊C₄×C₁₃₇	—	2	4	no	—
135	C₄⋊C₄×C₁₃₇	—	2	2	no	—
136	C₄⋊C₄×C₁₃₇	—	2	2	no	—
137	C₄⋊C₄×C₁₃₇	—	2	1	no	—
138	C₄⋊C₄×C₁₃₇	—	2	1	no	—
139	C₄⋊C₄×C₁₃₇	—	2	2	no	—
140	C₄⋊C₄×C₁₃₇	—	2	2	no	—
141	C₄⋊C₄×C₁₃₇	—	2	2	no	—
142	C₄⋊C₄×C₁₃₇	—	2	2	no	—
143	C₄⋊C₄×C₁₃₇	—	4	4	no	—
144	C₄⋊C₄×C₁₃₇	—	4	4	no	—
145	C₄⋊C₄×C₁₃₇	—	4	2	no	—
146	C₄⋊C₄×C₁₃₇	—	4	2	no	—
147	C₄⋊C₄×C₁₃₇	—	8	1	no	—
148	C₄⋊C₄×C₁₃₇	—	8	1	no	—
149	C₄⋊C₄×C₁₃₇	—	8	2	no	—
150	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
151	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
152	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
153	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
154	C₄²×C₁₃₇	—	2	4	no	—
155	C₄²×C₁₃₇	—	2	2	no	—
156	C₄²×C₁₃₇	—	2	2	no	—
157	C₄²×C₁₃₇	—	4	4	no	—
158	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₁₃₇	#41	1	2	yes	—
160	C₂³×C₁₃₇	—	2	2	no	—
161	C₂³×C₁₃₇	—	2	2	no	—
162	C₂³×C₁₃₇	—	2	2	no	—
163	C₂³×C₁₃₇	—	4	1	no	—
164	C₂³×C₁₃₇	—	4	1	no	—
165	C₂³×C₁₃₇	—	4	1	no	—
166	C₂³×C₁₃₇	—	4	1	no	—
167	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₁₃₇	—	2	2	no	—
169	Q₈×C₁₃₇	—	2	2	no	—
170	Q₈×C₁₃₇	—	2	2	no	—
171	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
172	D₄×C₁₃₇	—	2	2	no	—
173	D₄×C₁₃₇	—	2	2	no	—
174	D₄×C₁₃₇	—	2	2	no	—
175	D₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
176	C₂×C₄×C₁₃₇	—	2	2	no	—
177	C₂×C₄×C₁₃₇	—	2	2	no	—
178	C₂×C₄×C₁₃₇	—	2	2	no	—
179	C₂×C₄×C₁₃₇	—	2	1	no	—
180	C₂×C₄×C₁₃₇	—	2	1	no	—
181	C₂×C₄×C₁₃₇	—	2	2	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	2	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	8	no	—
194	C₂×C₄×C₁₃₇	—	4	2	no	—
195	C₂×C₄×C₁₃₇	—	4	2	no	—
196	C₂×C₄×C₁₃₇	—	4	2	no	—
197	C₂×C₄×C₁₃₇	—	4	2	no	—
198	C₂×C₄×C₁₃₇	—	4	2	no	—
199	C₂×C₄×C₁₃₇	—	4	2	no	—
200	C₂×C₄×C₁₃₇	—	8	8	no	—
201	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
202	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
204	C₂×C₄×C₁₃₇	#137	1	2	yes	—
205	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
206	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
208	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₁₃₇	—	2	2	no	—
212	C₈×C₁₃₇	—	4	2	no	—
213	C₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	complemented
214	C₂²×C₁₃₇	—	2	4	no	—
215	C₂²×C₁₃₇	—	4	2	no	—
216	C₂²×C₁₃₇	—	4	2	no	—
217	C₂²×C₁₃₇	—	4	2	no	—
218	C₂²×C₁₃₇	—	4	2	no	—
219	C₂²×C₁₃₇	—	4	2	no	—
220	C₂²×C₁₃₇	—	4	2	no	—
221	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
223	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
224	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
226	C₂²×C₁₃₇	#576	1	1	yes	characteristic
227	C₂²×C₁₃₇	#578	1	1	yes	characteristic
228	C₄×C₁₃₇	—	2	4	no	—
229	C₄×C₁₃₇	—	2	2	no	—
230	C₄×C₁₃₇	—	4	2	no	—
231	C₄×C₁₃₇	—	4	2	no	—
232	C₄×C₁₃₇	—	4	2	no	—
233	C₄×C₁₃₇	—	4	2	no	—
234	C₄×C₁₃₇	—	8	2	no	—
235	C₄×C₁₃₇	—	8	2	no	—
236	C₄×C₁₃₇	#440	1	4	yes	complemented
237	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
238	C₄×C₁₃₇	#520	1	1	yes	characteristic
239	C₄×C₁₃₇	#547	1	1	yes	characteristic
240	C₄×C₁₃₇	#554	1	1	yes	characteristic
241	C₂×C₁₃₇	—	4	4	no	—
242	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₁₃₇	C₂×#440	1	1	yes	characteristic
244	C₂×C₁₃₇	#2951	1	1	yes	characteristic
245	C₂×C₁₃₇	#3274	1	1	yes	characteristic
246	C₂×C₁₃₇	#3276	1	1	yes	characteristic
247	C₂×C₁₃₇	#3385	1	1	yes	characteristic
248	C₂×C₁₃₇	#3386	1	1	yes	characteristic
249	#22671	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₁₃₇	#22671	1	1	yes	characteristic, direct product factor, Sylow
251	C₂×#440	C₂×C₁₃₇	1	2	yes	—
252	C₂×C₄×SD₁₆	C₂×C₁₃₇	1	1	yes	characteristic
253	Q₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
254	D₄×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
255	#1944	C₂×C₁₃₇	1	2	yes	complemented
256	#1863	C₂×C₁₃₇	1	2	yes	complemented
257	#1862	C₂×C₁₃₇	1	2	yes	—
258	#1845	C₂×C₁₃₇	1	1	yes	characteristic, complemented
259	#1803	C₂×C₁₃₇	1	1	yes	characteristic, complemented
260	#1802	C₂×C₁₃₇	1	1	yes	characteristic
261	#1770	C₂×C₁₃₇	1	1	yes	characteristic, complemented
262	C₂²×SD₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
264	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
266	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	—	2	1	no	—
268	#440	—	2	8	no	—
269	C₄×SD₁₆	C₄×C₁₃₇	1	8	yes	complemented
270	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
271	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
272	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
274	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
275	C₂×#103	C₂²×C₁₃₇	1	1	yes	characteristic
276	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
277	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
278	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
279	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
280	#389	—	2	2	no	—
281	#389	C₂²×C₁₃₇	1	2	yes	—
282	#389	C₂²×C₁₃₇	1	2	yes	—
283	#385	—	2	2	no	—
284	#384	—	2	1	no	—
285	#384	C₂²×C₁₃₇	1	1	yes	characteristic
286	#384	C₂²×C₁₃₇	1	1	yes	characteristic
287	#380	—	2	1	no	—
288	C₄×C₄⋊C₄	—	2	1	no	—
289	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
290	#340	C₂²×C₁₃₇	1	2	yes	—
291	#340	C₂²×C₁₃₇	1	2	yes	—
292	#336	C₂²×C₁₃₇	1	2	yes	—
293	C₂²×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
294	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
295	C₂³×C₄	—	2	1	no	—
296	C₂³×C₄	—	2	2	no	—
297	C₂×SD₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
298	C₂×SD₁₆	D₄×C₁₃₇	1	4	yes	—
299	C₂×SD₁₆	Q₈×C₁₃₇	1	4	yes	—
300	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
301	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
303	C₄×Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×Q₈	—	4	2	no	—
306	C₄×Q₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	4	2	no	—
310	C₄×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	4	4	no	—
320	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
321	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
323	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
331	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
332	#107	—	2	4	no	—
333	#107	—	4	2	no	—
334	#106	C₂×C₄×C₁₃₇	1	4	yes	complemented
335	C₄⋊C₈	—	2	2	no	—
336	#103	—	2	4	no	—
337	#103	C₂×C₄×C₁₃₇	1	4	yes	complemented
338	#102	—	2	4	no	—
339	#102	C₂×C₄×C₁₃₇	1	4	yes	—
340	#97	—	2	4	no	—
341	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	D₄×C₁₃₇	1	2	yes	—
345	C₂⁴	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	C₂²×C₄×C₁₃₇	1	2	yes	—
350	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
351	C₂×Q₈	C₂×Q₈×C₁₃₇	1	2	yes	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	C₂²×C₄×C₁₃₇	1	2	yes	—
356	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
357	C₂×D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	—	4	4	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	8	2	no	—
369	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
373	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
374	SD₁₆	C₄⋊C₄×C₁₃₇	1	16	yes	complemented
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
381	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
382	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
383	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	4	4	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	8	1	no	—
398	C₄⋊C₄	—	8	1	no	—
399	C₄⋊C₄	—	8	2	no	—
400	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
401	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₄²	—	2	4	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	4	4	no	—
408	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
409	C₄²	#41×C₁₃₇	1	2	yes	—
410	C₂³	—	2	2	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	2	no	—
413	C₂³	—	4	1	no	—
414	C₂³	—	4	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	—	4	1	no	—
417	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	2	2	no	—
421	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	complemented
422	D₄	—	2	2	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	2	no	—
425	D₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	8	8	no	—
451	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
452	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
453	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
454	C₂×C₄	#137×C₁₃₇	1	2	yes	—
455	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
459	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
460	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
461	C₈	—	2	2	no	—
462	C₈	—	4	2	no	—
463	C₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	complemented
464	C₂²	—	2	4	no	—
465	C₂²	—	4	2	no	—
466	C₂²	—	4	2	no	—
467	C₂²	—	4	2	no	—
468	C₂²	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
472	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
473	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
474	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
475	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
476	C₂²	#576×C₁₃₇	1	1	yes	characteristic
477	C₂²	#578×C₁₃₇	1	1	yes	characteristic
478	C₄	—	2	4	no	—
479	C₄	—	2	2	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	2	no	—
484	C₄	—	8	2	no	—
485	C₄	—	8	2	no	—
486	C₄	#440×C₁₃₇	1	4	yes	complemented
487	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
488	C₄	#520×C₁₃₇	1	1	yes	characteristic
489	C₄	#547×C₁₃₇	1	1	yes	characteristic
490	C₄	#554×C₁₃₇	1	1	yes	characteristic
491	C₂	—	4	4	no	—
492	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
493	C₂	C₂×#440×C₁₃₇	1	1	yes	characteristic
494	C₂	#2951×C₁₃₇	1	1	yes	characteristic
495	C₂	#3274×C₁₃₇	1	1	yes	characteristic
496	C₂	#3276×C₁₃₇	1	1	yes	characteristic
497	C₂	#3385×C₁₃₇	1	1	yes	characteristic
498	C₂	#3386×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
LCS	G ▷ C₂×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. Q₈×C₄⋊C₄×C₁₃₇	C₂	semidirect
5. #1944×C₁₃₇	C₂	semidirect
6. #1863×C₁₃₇	C₂	semidirect
8. #1845×C₁₃₇	C₂	semidirect
9. #1803×C₁₃₇	C₂	semidirect
11. #1770×C₁₃₇	C₂	semidirect
19. C₄×SD₁₆×C₁₃₇	C₄	semidirect
56. C₄×Q₈×C₁₃₇	C₂×C₄	semidirect
84. #106×C₁₃₇	C₂×C₄	semidirect
87. #103×C₁₃₇	C₂×C₄	semidirect
91. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
124. SD₁₆×C₁₃₇	C₄⋊C₄	semidirect
171. Q₈×C₁₃₇	C₂×C₄⋊C₄	semidirect
213. C₈×C₁₃₇	C₂×C₄⋊C₄	semidirect
236. C₄×C₁₃₇	#440	semidirect
249. #22671	C₁₃₇	direct (N × H)
250. C₁₃₇	#22671	direct (N × H)
253. Q₈×C₄⋊C₄	C₂×C₁₃₇	semidirect
255. #1944	C₂×C₁₃₇	semidirect
256. #1863	C₂×C₁₃₇	semidirect
258. #1845	C₂×C₁₃₇	semidirect
259. #1803	C₂×C₁₃₇	semidirect
261. #1770	C₂×C₁₃₇	semidirect
269. C₄×SD₁₆	C₄×C₁₃₇	semidirect
306. C₄×Q₈	C₂×C₄×C₁₃₇	semidirect
334. #106	C₂×C₄×C₁₃₇	semidirect
337. #103	C₂×C₄×C₁₃₇	semidirect
341. C₄×C₈	C₂×C₄×C₁₃₇	semidirect
374. SD₁₆	C₄⋊C₄×C₁₃₇	semidirect
421. Q₈	C₂×C₄⋊C₄×C₁₃₇	semidirect
463. C₈	C₂×C₄⋊C₄×C₁₃₇	semidirect
486. C₄	#440×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#440×C₁₃₇	2
C₂×C₄×SD₁₆×C₁₃₇	2
Q₈×C₄⋊C₄×C₁₃₇	2
D₄×C₄⋊C₄×C₁₃₇	2
#1944×C₁₃₇	2
#1863×C₁₃₇	2
#1862×C₁₃₇	2
#1845×C₁₃₇	2
#1803×C₁₃₇	2
#1802×C₁₃₇	2
#1770×C₁₃₇	2
#22671	137

Subgroups — C137×#22671 #46872057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22671 #46872057