Subgroups — C₁₃₇×#22666 #46872052

Order 35072.

978 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2352×C₁₃₇	C₂	1	2	yes	maximal
4	#1949×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1854×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1853×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1809×C₁₃₇	C₂	1	2	yes	maximal, complemented
9	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
14	#436×C₁₃₇	C₄	1	8	yes	complemented
15	#428×C₁₃₇	—	2	8	no	—
16	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#388×C₁₃₇	—	2	2	no	—
27	#387×C₁₃₇	—	2	2	no	—
28	#386×C₁₃₇	—	2	2	no	—
29	#382×C₁₃₇	C₂²	1	2	yes	—
30	#380×C₁₃₇	C₂²	1	2	yes	—
31	#380×C₁₃₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
33	C₄×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
34	#340×C₁₃₇	C₂²	1	2	yes	—
35	#340×C₁₃₇	C₂²	1	2	yes	—
36	#340×C₁₃₇	C₂²	1	2	yes	—
37	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₃₇	—	2	2	no	—
39	C₂³×C₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂×M₁₆×C₁₃₇	—	2	4	no	—
41	C₂×M₁₆×C₁₃₇	—	2	4	no	—
42	C₂×M₁₆×C₁₃₇	C₂×C₄	1	8	yes	complemented
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₄×D₄×C₁₃₇	Q₈	1	4	yes	—
48	#117×C₁₃₇	—	2	4	no	—
49	#117×C₁₃₇	—	4	2	no	—
50	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₇	—	4	2	no	—
53	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	2	yes	—
62	C₂×C₄²×C₁₃₇	—	2	2	no	—
63	C₂×C₄²×C₁₃₇	—	2	2	no	—
64	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
65	#107×C₁₃₇	—	2	4	no	—
66	#107×C₁₃₇	—	2	4	no	—
67	#106×C₁₃₇	—	2	2	no	—
68	#106×C₁₃₇	—	2	4	no	—
69	#106×C₁₃₇	—	4	1	no	—
70	#106×C₁₃₇	—	4	1	no	—
71	#106×C₁₃₇	D₄	1	4	yes	complemented
72	C₄⋊C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
73	C₂²⋊C₈×C₁₃₇	C₂×C₄	1	8	yes	complemented
74	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
75	#95×C₁₃₇	—	2	2	no	—
76	#95×C₁₃₇	—	2	2	no	—
77	#95×C₁₃₇	—	2	2	no	—
78	#95×C₁₃₇	—	2	2	no	—
79	#95×C₁₃₇	—	2	2	no	—
80	C₂⁴×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₁₃₇	—	4	2	no	—
82	C₂×D₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₁₃₇	—	2	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	—	2	2	no	—
87	C₂²×C₄×C₁₃₇	—	2	2	no	—
88	C₂²×C₄×C₁₃₇	—	2	2	no	—
89	C₂²×C₄×C₁₃₇	—	2	2	no	—
90	C₂²×C₄×C₁₃₇	—	2	2	no	—
91	C₂²×C₄×C₁₃₇	—	4	4	no	—
92	C₂²×C₄×C₁₃₇	—	4	4	no	—
93	C₂²×C₄×C₁₃₇	—	8	2	no	—
94	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
99	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
100	M₁₆×C₁₃₇	—	2	16	no	—
101	C₂×C₈×C₁₃₇	—	2	2	no	—
102	C₂×C₈×C₁₃₇	—	2	2	no	—
103	C₂×C₈×C₁₃₇	—	2	1	no	—
104	C₂×C₈×C₁₃₇	—	2	1	no	—
105	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
107	C₂×C₈×C₁₃₇	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃₇	#41	1	2	yes	—
110	C₄⋊C₄×C₁₃₇	—	2	2	no	—
111	C₄⋊C₄×C₁₃₇	—	2	2	no	—
112	C₄⋊C₄×C₁₃₇	—	2	2	no	—
113	C₄⋊C₄×C₁₃₇	—	2	2	no	—
114	C₄⋊C₄×C₁₃₇	—	2	2	no	—
115	C₄⋊C₄×C₁₃₇	—	2	2	no	—
116	C₄⋊C₄×C₁₃₇	—	2	2	no	—
117	C₄⋊C₄×C₁₃₇	—	4	4	no	—
118	C₄⋊C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
119	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
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	C₄⋊C₄	1	8	yes	—
124	C₄²×C₁₃₇	—	4	4	no	—
125	C₄²×C₁₃₇	—	8	2	no	—
126	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
127	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
128	C₂³×C₁₃₇	—	2	2	no	—
129	C₂³×C₁₃₇	—	2	2	no	—
130	C₂³×C₁₃₇	—	2	2	no	—
131	C₂³×C₁₃₇	—	2	2	no	—
132	C₂³×C₁₃₇	—	2	2	no	—
133	C₂³×C₁₃₇	—	2	2	no	—
134	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
135	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
136	D₄×C₁₃₇	—	4	4	no	—
137	C₂×C₄×C₁₃₇	—	2	2	no	—
138	C₂×C₄×C₁₃₇	—	2	2	no	—
139	C₂×C₄×C₁₃₇	—	2	2	no	—
140	C₂×C₄×C₁₃₇	—	2	2	no	—
141	C₂×C₄×C₁₃₇	—	2	4	no	—
142	C₂×C₄×C₁₃₇	—	2	4	no	—
143	C₂×C₄×C₁₃₇	—	2	4	no	—
144	C₂×C₄×C₁₃₇	—	2	4	no	—
145	C₂×C₄×C₁₃₇	—	2	4	no	—
146	C₂×C₄×C₁₃₇	—	4	2	no	—
147	C₂×C₄×C₁₃₇	—	4	2	no	—
148	C₂×C₄×C₁₃₇	—	4	2	no	—
149	C₂×C₄×C₁₃₇	—	4	2	no	—
150	C₂×C₄×C₁₃₇	—	4	2	no	—
151	C₂×C₄×C₁₃₇	—	4	2	no	—
152	C₂×C₄×C₁₃₇	—	4	2	no	—
153	C₂×C₄×C₁₃₇	—	4	2	no	—
154	C₂×C₄×C₁₃₇	—	4	2	no	—
155	C₂×C₄×C₁₃₇	—	4	2	no	—
156	C₂×C₄×C₁₃₇	—	4	2	no	—
157	C₂×C₄×C₁₃₇	—	8	8	no	—
158	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
160	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
161	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₁₃₇	—	2	4	no	—
168	C₈×C₁₃₇	—	2	2	no	—
169	C₈×C₁₃₇	C₄×D₄	1	4	yes	—
170	C₂²×C₁₃₇	—	2	4	no	—
171	C₂²×C₁₃₇	—	2	4	no	—
172	C₂²×C₁₃₇	—	2	4	no	—
173	C₂²×C₁₃₇	—	4	2	no	—
174	C₂²×C₁₃₇	—	4	2	no	—
175	C₂²×C₁₃₇	—	4	2	no	—
176	C₂²×C₁₃₇	—	4	2	no	—
177	C₂²×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
179	C₂²×C₁₃₇	#547	1	1	yes	characteristic
180	C₂²×C₁₃₇	#575	1	2	yes	—
181	C₂²×C₁₃₇	#577	1	2	yes	—
182	C₄×C₁₃₇	—	2	4	no	—
183	C₄×C₁₃₇	—	2	2	no	—
184	C₄×C₁₃₇	—	2	2	no	—
185	C₄×C₁₃₇	—	4	4	no	—
186	C₄×C₁₃₇	—	8	2	no	—
187	C₄×C₁₃₇	—	8	2	no	—
188	C₄×C₁₃₇	—	8	2	no	—
189	C₄×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
190	C₄×C₁₃₇	#520	1	1	yes	characteristic
191	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
192	C₄×C₁₃₇	#554	1	1	yes	characteristic
193	C₂×C₁₃₇	—	4	4	no	—
194	C₂×C₁₃₇	D₄×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₁₃₇	#2911	1	1	yes	characteristic
196	C₂×C₁₃₇	#2912	1	1	yes	characteristic
197	C₂×C₁₃₇	#3275	1	2	yes	—
198	C₂×C₁₃₇	#3351	1	2	yes	—
199	#22666	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₁₃₇	#22666	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#436	C₂×C₁₃₇	1	1	yes	characteristic
202	C₂×#428	C₂×C₁₃₇	1	2	yes	complemented
203	#2352	C₂×C₁₃₇	1	2	yes	—
204	#1949	C₂×C₁₃₇	1	2	yes	complemented
205	#1854	C₂×C₁₃₇	1	2	yes	complemented
206	#1853	C₂×C₁₃₇	1	2	yes	complemented
207	#1846	C₂×C₁₃₇	1	1	yes	characteristic, complemented
208	#1809	C₂×C₁₃₇	1	2	yes	complemented
209	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic, complemented
210	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
211	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
212	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
213	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
214	#436	C₄×C₁₃₇	1	8	yes	complemented
215	#428	—	2	8	no	—
216	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
217	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
218	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
219	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
220	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
221	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
222	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
224	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
225	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
226	#388	—	2	2	no	—
227	#387	—	2	2	no	—
228	#386	—	2	2	no	—
229	#382	C₂²×C₁₃₇	1	2	yes	—
230	#380	C₂²×C₁₃₇	1	2	yes	—
231	#380	C₂²×C₁₃₇	1	2	yes	—
232	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
233	C₄×C₂²⋊C₄	—	2	2	no	—
234	#340	C₂²×C₁₃₇	1	2	yes	—
235	#340	C₂²×C₁₃₇	1	2	yes	—
236	#340	C₂²×C₁₃₇	1	2	yes	—
237	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
238	C₂³×C₄	—	2	2	no	—
239	C₂³×C₄	C₂³×C₁₃₇	1	2	yes	—
240	C₂×M₁₆	—	2	4	no	—
241	C₂×M₁₆	—	2	4	no	—
242	C₂×M₁₆	C₂×C₄×C₁₃₇	1	8	yes	complemented
243	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
244	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
246	C₄×D₄	C₂×C₄×C₁₃₇	1	4	yes	—
247	C₄×D₄	Q₈×C₁₃₇	1	4	yes	—
248	#117	—	2	4	no	—
249	#117	—	4	2	no	—
250	C₂×C₄⋊C₄	—	2	4	no	—
251	C₂×C₄⋊C₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	4	2	no	—
253	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
255	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
258	C₂×C₂²⋊C₄	—	2	2	no	—
259	C₂×C₂²⋊C₄	—	2	2	no	—
260	C₂×C₂²⋊C₄	—	2	2	no	—
261	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	2	yes	—
262	C₂×C₄²	—	2	2	no	—
263	C₂×C₄²	—	2	2	no	—
264	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
265	#107	—	2	4	no	—
266	#107	—	2	4	no	—
267	#106	—	2	2	no	—
268	#106	—	2	4	no	—
269	#106	—	4	1	no	—
270	#106	—	4	1	no	—
271	#106	D₄×C₁₃₇	1	4	yes	complemented
272	C₄⋊C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
273	C₂²⋊C₈	C₂×C₄×C₁₃₇	1	8	yes	complemented
274	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
275	#95	—	2	2	no	—
276	#95	—	2	2	no	—
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	C₂⁴	C₂×D₄×C₁₃₇	1	2	yes	—
281	C₂×D₄	—	4	2	no	—
282	C₂×D₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	4	4	no	—
292	C₂²×C₄	—	4	4	no	—
293	C₂²×C₄	—	8	2	no	—
294	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
295	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
297	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
298	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
299	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
300	M₁₆	—	2	16	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
306	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
307	C₂×C₈	C₂²×C₄×C₁₃₇	1	2	yes	—
308	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
309	C₂×C₈	#41×C₁₃₇	1	2	yes	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
319	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	—	4	4	no	—
322	C₂²⋊C₄	—	8	2	no	—
323	C₂²⋊C₄	C₄⋊C₄×C₁₃₇	1	8	yes	—
324	C₄²	—	4	4	no	—
325	C₄²	—	8	2	no	—
326	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
327	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
328	C₂³	—	2	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	2	2	no	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
335	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
336	D₄	—	4	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	8	no	—
358	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
359	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
360	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
361	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
363	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
364	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
365	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
366	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
367	C₈	—	2	4	no	—
368	C₈	—	2	2	no	—
369	C₈	C₄×D₄×C₁₃₇	1	4	yes	—
370	C₂²	—	2	4	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
378	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
379	C₂²	#547×C₁₃₇	1	1	yes	characteristic
380	C₂²	#575×C₁₃₇	1	2	yes	—
381	C₂²	#577×C₁₃₇	1	2	yes	—
382	C₄	—	2	4	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	2	no	—
385	C₄	—	4	4	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	8	2	no	—
389	C₄	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
390	C₄	#520×C₁₃₇	1	1	yes	characteristic
391	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
392	C₄	#554×C₁₃₇	1	1	yes	characteristic
393	C₂	—	4	4	no	—
394	C₂	D₄×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
395	C₂	#2911×C₁₃₇	1	1	yes	characteristic
396	C₂	#2912×C₁₃₇	1	1	yes	characteristic
397	C₂	#3275×C₁₃₇	1	2	yes	—
398	C₂	#3351×C₁₃₇	1	2	yes	—

	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	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×D₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₁₃₇	C₂	semidirect
4. #1949×C₁₃₇	C₂	semidirect
5. #1854×C₁₃₇	C₂	semidirect
6. #1853×C₁₃₇	C₂	semidirect
7. #1846×C₁₃₇	C₂	semidirect
8. #1809×C₁₃₇	C₂	semidirect
9. C₄×#106×C₁₃₇	C₂	semidirect
14. #436×C₁₃₇	C₄	semidirect
42. C₂×M₁₆×C₁₃₇	C₂×C₄	semidirect
71. #106×C₁₃₇	D₄	semidirect
72. C₄⋊C₈×C₁₃₇	C₂×C₄	semidirect
73. C₂²⋊C₈×C₁₃₇	C₂×C₄	semidirect
74. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
199. #22666	C₁₃₇	direct (N × H)
200. C₁₃₇	#22666	direct (N × H)
202. C₂×#428	C₂×C₁₃₇	semidirect
204. #1949	C₂×C₁₃₇	semidirect
205. #1854	C₂×C₁₃₇	semidirect
206. #1853	C₂×C₁₃₇	semidirect
207. #1846	C₂×C₁₃₇	semidirect
208. #1809	C₂×C₁₃₇	semidirect
209. C₄×#106	C₂×C₁₃₇	semidirect
214. #436	C₄×C₁₃₇	semidirect
242. C₂×M₁₆	C₂×C₄×C₁₃₇	semidirect
271. #106	D₄×C₁₃₇	semidirect
272. C₄⋊C₈	C₂×C₄×C₁₃₇	semidirect
273. C₂²⋊C₈	C₂×C₄×C₁₃₇	semidirect
274. C₄×C₈	C₂×C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₇	2
C₂×#428×C₁₃₇	2
#2352×C₁₃₇	2
#1949×C₁₃₇	2
#1854×C₁₃₇	2
#1853×C₁₃₇	2
#1846×C₁₃₇	2
#1809×C₁₃₇	2
C₄×#106×C₁₃₇	2
#22666	137

Subgroups — C137×#22666 #46872052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#22666 #46872052