Subgroups — C₁₃₇×#20675 #46870061

Order 35072.

1308 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2911×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#421×C₁₃₇	C₂	1	2	yes	maximal
5	#1814×C₁₃₇	C₂	1	8	yes	maximal, complemented
6	C₂×#344×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₁₃₇	C₂²	1	1	yes	characteristic
8	#568×C₁₃₇	—	2	4	no	—
9	#568×C₁₃₇	C₄	1	8	yes	complemented
10	C₂×#131×C₁₃₇	—	2	2	no	—
11	C₂×#131×C₁₃₇	C₂²	1	4	yes	—
12	C₂²×M₁₆×C₁₃₇	—	2	2	no	—
13	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₁₃₇	C₂²	1	2	yes	—
17	#430×C₁₃₇	—	2	2	no	—
18	#430×C₁₃₇	—	2	2	no	—
19	C₂×#108×C₁₃₇	C₂²	1	4	yes	complemented
20	#428×C₁₃₇	—	2	4	no	—
21	#427×C₁₃₇	C₂²	1	4	yes	complemented
22	#421×C₁₃₇	—	2	4	no	—
23	#421×C₁₃₇	C₂²	1	8	yes	—
24	C₂×#104×C₁₃₇	—	2	4	no	—
25	#344×C₁₃₇	C₂²	1	8	yes	complemented
26	C₂×#41×C₁₃₇	—	2	1	no	—
27	C₂×#41×C₁₃₇	—	2	2	no	—
28	C₂×#41×C₁₃₇	—	2	2	no	—
29	C₂×#41×C₁₃₇	—	2	1	no	—
30	C₂×#41×C₁₃₇	C₂×C₄	1	4	yes	—
31	C₂×#41×C₁₃₇	D₄	1	4	yes	—
32	C₂×#41×C₁₃₇	Q₈	1	2	yes	—
33	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
34	C₂²×Q₈×C₁₃₇	Q₈	1	1	yes	characteristic
35	C₂²×D₄×C₁₃₇	—	2	1	no	—
36	C₂²×D₄×C₁₃₇	Q₈	1	1	yes	characteristic
37	C₂³×C₄×C₁₃₇	—	2	1	no	—
38	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
39	#131×C₁₃₇	—	2	16	no	—
40	#131×C₁₃₇	—	2	8	no	—
41	#131×C₁₃₇	C₂×C₄	1	16	yes	complemented
42	C₂×M₁₆×C₁₃₇	—	2	1	no	—
43	C₂×M₁₆×C₁₃₇	—	2	2	no	—
44	C₂×M₁₆×C₁₃₇	—	2	16	no	—
45	C₂×M₁₆×C₁₃₇	—	2	2	no	—
46	C₂×M₁₆×C₁₃₇	—	2	4	no	—
47	C₂×M₁₆×C₁₃₇	—	2	2	no	—
48	C₂×M₁₆×C₁₃₇	—	2	2	no	—
49	C₂×M₁₆×C₁₃₇	—	2	2	no	—
50	C₂×M₁₆×C₁₃₇	—	2	2	no	—
51	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₁₃₇	D₄	1	4	yes	—
54	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
55	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
56	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
57	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₁₃₇	—	2	1	no	—
59	C₂²×C₈×C₁₃₇	—	2	4	no	—
60	C₂²×C₈×C₁₃₇	—	2	2	no	—
61	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
62	#117×C₁₃₇	—	2	2	no	—
63	#117×C₁₃₇	—	2	2	no	—
64	#117×C₁₃₇	—	2	2	no	—
65	#117×C₁₃₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₄²×C₁₃₇	—	2	2	no	—
69	#108×C₁₃₇	—	2	8	no	—
70	#107×C₁₃₇	—	2	4	no	—
71	#106×C₁₃₇	—	2	4	no	—
72	#104×C₁₃₇	—	2	16	no	—
73	C₂⁴×C₁₃₇	—	2	1	no	—
74	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
75	#41×C₁₃₇	—	2	4	no	—
76	#41×C₁₃₇	—	2	4	no	—
77	#41×C₁₃₇	—	2	4	no	—
78	#41×C₁₃₇	—	2	4	no	—
79	#41×C₁₃₇	—	2	4	no	—
80	#41×C₁₃₇	—	2	2	no	—
81	#41×C₁₃₇	—	2	2	no	—
82	#41×C₁₃₇	C₄⋊C₄	1	8	yes	—
83	#41×C₁₃₇	C₂²×C₄	1	4	yes	—
84	#41×C₁₃₇	C₂×D₄	1	4	yes	—
85	C₂×Q₈×C₁₃₇	—	2	1	no	—
86	C₂×Q₈×C₁₃₇	—	2	2	no	—
87	C₂×Q₈×C₁₃₇	C₄⋊C₄	1	4	yes	—
88	C₂×Q₈×C₁₃₇	C₂×Q₈	1	2	yes	—
89	C₂×D₄×C₁₃₇	—	2	4	no	—
90	C₂×D₄×C₁₃₇	—	2	4	no	—
91	C₂×D₄×C₁₃₇	—	2	2	no	—
92	C₂×D₄×C₁₃₇	—	2	2	no	—
93	C₂×D₄×C₁₃₇	—	2	2	no	—
94	C₂×D₄×C₁₃₇	—	2	1	no	—
95	C₂×D₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
96	C₂×D₄×C₁₃₇	C₂×Q₈	1	2	yes	—
97	C₂²×C₄×C₁₃₇	—	2	1	no	—
98	C₂²×C₄×C₁₃₇	—	2	4	no	—
99	C₂²×C₄×C₁₃₇	—	2	4	no	—
100	C₂²×C₄×C₁₃₇	—	2	2	no	—
101	C₂²×C₄×C₁₃₇	—	2	2	no	—
102	C₂²×C₄×C₁₃₇	—	4	2	no	—
103	C₂²×C₄×C₁₃₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
109	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
112	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
113	M₁₆×C₁₃₇	—	2	4	no	—
114	M₁₆×C₁₃₇	—	2	2	no	—
115	M₁₆×C₁₃₇	—	2	16	no	—
116	M₁₆×C₁₃₇	—	2	4	no	—
117	M₁₆×C₁₃₇	—	2	4	no	—
118	M₁₆×C₁₃₇	—	4	2	no	—
119	M₁₆×C₁₃₇	—	4	2	no	—
120	M₁₆×C₁₃₇	C₂²⋊C₄	1	8	yes	—
121	M₁₆×C₁₃₇	C₂²⋊C₄	1	8	yes	—
122	M₁₆×C₁₃₇	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₁₃₇	—	2	4	no	—
124	C₂×C₈×C₁₃₇	—	2	2	no	—
125	C₂×C₈×C₁₃₇	—	2	16	no	—
126	C₂×C₈×C₁₃₇	—	2	4	no	—
127	C₂×C₈×C₁₃₇	—	4	2	no	—
128	C₂×C₈×C₁₃₇	—	4	2	no	—
129	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
130	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
132	C₄⋊C₄×C₁₃₇	—	2	4	no	—
133	C₄⋊C₄×C₁₃₇	—	4	2	no	—
134	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
135	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
136	C₄²×C₁₃₇	—	2	4	no	—
137	C₄²×C₁₃₇	—	4	2	no	—
138	C₂³×C₁₃₇	—	2	2	no	—
139	C₂³×C₁₃₇	—	2	4	no	—
140	C₂³×C₁₃₇	—	8	1	no	—
141	C₂³×C₁₃₇	—	8	2	no	—
142	C₂³×C₁₃₇	#117	1	2	yes	—
143	C₂³×C₁₃₇	#121	1	2	yes	—
144	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
145	C₂³×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
147	Q₈×C₁₃₇	—	2	2	no	—
148	Q₈×C₁₃₇	—	2	4	no	—
149	Q₈×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
150	D₄×C₁₃₇	—	2	4	no	—
151	D₄×C₁₃₇	—	2	4	no	—
152	D₄×C₁₃₇	—	2	8	no	—
153	D₄×C₁₃₇	—	2	2	no	—
154	D₄×C₁₃₇	—	2	4	no	—
155	D₄×C₁₃₇	C₂×C₄⋊C₄	1	4	yes	—
156	C₂×C₄×C₁₃₇	—	2	2	no	—
157	C₂×C₄×C₁₃₇	—	2	4	no	—
158	C₂×C₄×C₁₃₇	—	2	4	no	—
159	C₂×C₄×C₁₃₇	—	2	4	no	—
160	C₂×C₄×C₁₃₇	—	2	8	no	—
161	C₂×C₄×C₁₃₇	—	4	4	no	—
162	C₂×C₄×C₁₃₇	—	8	4	no	—
163	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
164	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
166	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₁₃₇	#117	1	2	yes	—
168	C₂×C₄×C₁₃₇	#121	1	2	yes	—
169	C₂×C₄×C₁₃₇	#121	1	2	yes	—
170	C₂×C₄×C₁₃₇	#121	1	2	yes	—
171	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃₇	C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
181	C₈×C₁₃₇	—	2	4	no	—
182	C₈×C₁₃₇	—	4	4	no	—
183	C₈×C₁₃₇	C₂×C₂²⋊C₄	1	8	yes	—
184	C₂²×C₁₃₇	—	2	4	no	—
185	C₂²×C₁₃₇	—	4	2	no	—
186	C₂²×C₁₃₇	—	4	1	no	—
187	C₂²×C₁₃₇	—	4	2	no	—
188	C₂²×C₁₃₇	—	4	1	no	—
189	C₂²×C₁₃₇	—	4	1	no	—
190	C₂²×C₁₃₇	—	4	4	no	—
191	C₂²×C₁₃₇	—	8	4	no	—
192	C₂²×C₁₃₇	#380	1	2	yes	—
193	C₂²×C₁₃₇	#380	1	2	yes	—
194	C₂²×C₁₃₇	C₂×#117	1	1	yes	characteristic
195	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
196	C₂²×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
197	C₄×C₁₃₇	—	2	4	no	—
198	C₄×C₁₃₇	—	8	4	no	—
199	C₄×C₁₃₇	#380	1	2	yes	—
200	C₄×C₁₃₇	#380	1	2	yes	—
201	C₄×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
202	C₄×C₁₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₁₃₇	C₂×#121	1	1	yes	characteristic
204	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₂×C₁₃₇	—	2	1	no	—
206	C₂×C₁₃₇	—	2	1	no	—
207	C₂×C₁₃₇	—	2	2	no	—
208	C₂×C₁₃₇	—	2	1	no	—
209	C₂×C₁₃₇	—	2	2	no	—
210	C₂×C₁₃₇	—	4	4	no	—
211	C₂×C₁₃₇	C₂×#380	1	1	yes	characteristic
212	#20675	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₁₃₇	#20675	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#568	C₂×C₁₃₇	1	1	yes	characteristic
215	C₂×#430	C₂×C₁₃₇	1	1	yes	characteristic, complemented
216	#2911	C₂×C₁₃₇	1	1	yes	characteristic, complemented
217	C₂×#421	C₂×C₁₃₇	1	2	yes	—
218	#1814	C₂×C₁₃₇	1	8	yes	complemented
219	C₂×#344	C₂×C₁₃₇	1	2	yes	complemented
220	C₂²×#41	C₂²×C₁₃₇	1	1	yes	characteristic
221	#568	—	2	4	no	—
222	#568	C₄×C₁₃₇	1	8	yes	complemented
223	C₂×#131	—	2	2	no	—
224	C₂×#131	C₂²×C₁₃₇	1	4	yes	—
225	C₂²×M₁₆	—	2	2	no	—
226	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
227	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
228	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
229	C₂×#117	C₂²×C₁₃₇	1	2	yes	—
230	#430	—	2	2	no	—
231	#430	—	2	2	no	—
232	C₂×#108	C₂²×C₁₃₇	1	4	yes	complemented
233	#428	—	2	4	no	—
234	#427	C₂²×C₁₃₇	1	4	yes	complemented
235	#421	—	2	4	no	—
236	#421	C₂²×C₁₃₇	1	8	yes	—
237	C₂×#104	—	2	4	no	—
238	#344	C₂²×C₁₃₇	1	8	yes	complemented
239	C₂×#41	—	2	1	no	—
240	C₂×#41	—	2	2	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	2	1	no	—
243	C₂×#41	C₂×C₄×C₁₃₇	1	4	yes	—
244	C₂×#41	D₄×C₁₃₇	1	4	yes	—
245	C₂×#41	Q₈×C₁₃₇	1	2	yes	—
246	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
247	C₂²×Q₈	Q₈×C₁₃₇	1	1	yes	characteristic
248	C₂²×D₄	—	2	1	no	—
249	C₂²×D₄	Q₈×C₁₃₇	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
252	#131	—	2	16	no	—
253	#131	—	2	8	no	—
254	#131	C₂×C₄×C₁₃₇	1	16	yes	complemented
255	C₂×M₁₆	—	2	1	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	16	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	4	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
265	C₂×M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
266	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
267	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
268	C₂×M₁₆	D₄×C₁₃₇	1	4	yes	—
269	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
270	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	4	no	—
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
275	#117	—	2	2	no	—
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	#117	C₂³×C₁₃₇	1	4	yes	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	2	no	—
282	#108	—	2	8	no	—
283	#107	—	2	4	no	—
284	#106	—	2	4	no	—
285	#104	—	2	16	no	—
286	C₂⁴	—	2	1	no	—
287	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
288	#41	—	2	4	no	—
289	#41	—	2	4	no	—
290	#41	—	2	4	no	—
291	#41	—	2	4	no	—
292	#41	—	2	4	no	—
293	#41	—	2	2	no	—
294	#41	—	2	2	no	—
295	#41	C₄⋊C₄×C₁₃₇	1	8	yes	—
296	#41	C₂²×C₄×C₁₃₇	1	4	yes	—
297	#41	C₂×D₄×C₁₃₇	1	4	yes	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	C₄⋊C₄×C₁₃₇	1	4	yes	—
301	C₂×Q₈	C₂×Q₈×C₁₃₇	1	2	yes	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
309	C₂×D₄	C₂×Q₈×C₁₃₇	1	2	yes	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	4	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	4	2	no	—
316	C₂²×C₄	C₂²×C₄×C₁₃₇	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
321	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
322	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
323	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
324	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
326	M₁₆	—	2	4	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	16	no	—
329	M₁₆	—	2	4	no	—
330	M₁₆	—	2	4	no	—
331	M₁₆	—	4	2	no	—
332	M₁₆	—	4	2	no	—
333	M₁₆	C₂²⋊C₄×C₁₃₇	1	8	yes	—
334	M₁₆	C₂²⋊C₄×C₁₃₇	1	8	yes	—
335	M₁₆	C₂²×C₄×C₁₃₇	1	4	yes	—
336	C₂×C₈	—	2	4	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	16	no	—
339	C₂×C₈	—	2	4	no	—
340	C₂×C₈	—	4	2	no	—
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
343	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
344	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	2	4	no	—
348	C₂²⋊C₄	—	4	2	no	—
349	C₄²	—	2	4	no	—
350	C₄²	—	4	2	no	—
351	C₂³	—	2	2	no	—
352	C₂³	—	2	4	no	—
353	C₂³	—	8	1	no	—
354	C₂³	—	8	2	no	—
355	C₂³	#117×C₁₃₇	1	2	yes	—
356	C₂³	#121×C₁₃₇	1	2	yes	—
357	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
358	C₂³	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
359	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
360	Q₈	—	2	4	no	—
361	Q₈	—	2	2	no	—
362	Q₈	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
363	D₄	—	2	8	no	—
364	D₄	—	2	4	no	—
365	D₄	—	2	4	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	2	no	—
368	D₄	C₂×C₄⋊C₄×C₁₃₇	1	4	yes	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	4	4	no	—
375	C₂×C₄	—	8	4	no	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	2	yes	—
378	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
379	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
380	C₂×C₄	#117×C₁₃₇	1	2	yes	—
381	C₂×C₄	#121×C₁₃₇	1	2	yes	—
382	C₂×C₄	#121×C₁₃₇	1	2	yes	—
383	C₂×C₄	#121×C₁₃₇	1	2	yes	—
384	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
385	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
386	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
387	C₂×C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
388	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
389	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
390	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
392	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
394	C₈	—	2	4	no	—
395	C₈	—	4	4	no	—
396	C₈	C₂×C₂²⋊C₄×C₁₃₇	1	8	yes	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	4	1	no	—
403	C₂²	—	4	4	no	—
404	C₂²	—	8	4	no	—
405	C₂²	#380×C₁₃₇	1	2	yes	—
406	C₂²	#380×C₁₃₇	1	2	yes	—
407	C₂²	C₂×#117×C₁₃₇	1	1	yes	characteristic
408	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
409	C₂²	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
410	C₄	—	2	4	no	—
411	C₄	—	8	4	no	—
412	C₄	#380×C₁₃₇	1	2	yes	—
413	C₄	#380×C₁₃₇	1	2	yes	—
414	C₄	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
415	C₄	C₂²×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
416	C₄	C₂×#121×C₁₃₇	1	1	yes	characteristic
417	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	2	no	—
421	C₂	—	2	2	no	—
422	C₂	—	2	1	no	—
423	C₂	—	4	4	no	—
424	C₂	C₂×#380×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂×C₁₃₇	C₂×#380
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#430×C₁₃₇	C₂	semidirect
3. #2911×C₁₃₇	C₂	semidirect
5. #1814×C₁₃₇	C₂	semidirect
6. C₂×#344×C₁₃₇	C₂	semidirect
9. #568×C₁₃₇	C₄	semidirect
19. C₂×#108×C₁₃₇	C₂²	semidirect
21. #427×C₁₃₇	C₂²	semidirect
25. #344×C₁₃₇	C₂²	semidirect
41. #131×C₁₃₇	C₂×C₄	semidirect
212. #20675	C₁₃₇	direct (N × H)
213. C₁₃₇	#20675	direct (N × H)
215. C₂×#430	C₂×C₁₃₇	semidirect
216. #2911	C₂×C₁₃₇	semidirect
218. #1814	C₂×C₁₃₇	semidirect
219. C₂×#344	C₂×C₁₃₇	semidirect
222. #568	C₄×C₁₃₇	semidirect
232. C₂×#108	C₂²×C₁₃₇	semidirect
234. #427	C₂²×C₁₃₇	semidirect
238. #344	C₂²×C₁₃₇	semidirect
254. #131	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×#568×C₁₃₇	2
C₂×#430×C₁₃₇	2
#2911×C₁₃₇	2
C₂×#421×C₁₃₇	2
#1814×C₁₃₇	2
C₂×#344×C₁₃₇	2
#20675	137

Subgroups — C137×#20675 #46870061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#20675 #46870061