Subgroups — C₁₃₇×#23520 #46872906

Order 35072.

1350 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2896×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	C₂×#413×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2047×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2046×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#2045×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂×#341×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₃₇	C₂²	1	2	yes	—
10	C₂×#135×C₁₃₇	—	2	8	no	—
11	C₂×#135×C₁₃₇	—	2	2	no	—
12	C₂×#135×C₁₃₇	—	2	2	no	—
13	C₂²×Q₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
15	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₁₃₇	C₂²	1	2	yes	—
18	C₂³×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
20	#424×C₁₃₇	—	2	2	no	—
21	#424×C₁₃₇	C₂²	1	2	yes	—
22	#424×C₁₃₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
24	#419×C₁₃₇	C₂²	1	4	yes	—
25	#418×C₁₃₇	C₂²	1	4	yes	—
26	#417×C₁₃₇	—	2	4	no	—
27	#416×C₁₃₇	—	2	4	no	—
28	#413×C₁₃₇	—	2	8	no	—
29	C₂×#101×C₁₃₇	C₂²	1	4	yes	—
30	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
31	#341×C₁₃₇	C₂²	1	2	yes	—
32	#341×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#41×C₁₃₇	—	2	4	no	—
34	C₂×#41×C₁₃₇	—	2	2	no	—
35	C₂×#41×C₁₃₇	—	2	2	no	—
36	C₂×#41×C₁₃₇	—	2	2	no	—
37	C₂×#41×C₁₃₇	—	2	2	no	—
38	C₂×#41×C₁₃₇	—	2	4	no	—
39	C₂×#41×C₁₃₇	—	2	4	no	—
40	C₂×#41×C₁₃₇	D₄	1	8	yes	—
41	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×D₄×C₁₃₇	—	2	2	no	—
43	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
44	C₂³×C₄×C₁₃₇	—	2	2	no	—
45	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	#135×C₁₃₇	—	2	16	no	—
47	#135×C₁₃₇	—	2	16	no	—
48	C₂×Q₁₆×C₁₃₇	—	2	4	no	—
49	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
50	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
53	C₂×D₈×C₁₃₇	—	2	4	no	—
54	C₂×D₈×C₁₃₇	—	2	2	no	—
55	C₂×M₁₆×C₁₃₇	—	2	2	no	—
56	C₂×M₁₆×C₁₃₇	—	2	1	no	—
57	C₂×M₁₆×C₁₃₇	—	2	4	no	—
58	C₂×M₁₆×C₁₃₇	—	2	4	no	—
59	C₂×M₁₆×C₁₃₇	—	2	4	no	—
60	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
62	C₂²×C₈×C₁₃₇	—	2	4	no	—
63	C₂²×C₈×C₁₃₇	—	2	2	no	—
64	C₂²×C₈×C₁₃₇	—	2	1	no	—
65	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
66	C₂²×C₈×C₁₃₇	D₄	1	2	yes	—
67	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
69	#117×C₁₃₇	—	2	1	no	—
70	#117×C₁₃₇	—	2	1	no	—
71	#117×C₁₃₇	D₄	1	2	yes	—
72	#117×C₁₃₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
74	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
75	C₂×C₄²×C₁₃₇	—	2	1	no	—
76	C₄⋊C₈×C₁₃₇	—	2	4	no	—
77	C₄⋊C₈×C₁₃₇	—	2	4	no	—
78	#103×C₁₃₇	—	2	8	no	—
79	#102×C₁₃₇	—	2	8	no	—
80	#101×C₁₃₇	—	2	8	no	—
81	#100×C₁₃₇	—	2	8	no	—
82	C₂⁴×C₁₃₇	—	2	2	no	—
83	C₂⁴×C₁₃₇	#41	1	1	yes	characteristic
84	#41×C₁₃₇	—	2	8	no	—
85	#41×C₁₃₇	—	2	8	no	—
86	#41×C₁₃₇	—	2	8	no	—
87	#41×C₁₃₇	—	2	8	no	—
88	#41×C₁₃₇	—	4	4	no	—
89	#41×C₁₃₇	—	4	4	no	—
90	#41×C₁₃₇	—	4	4	no	—
91	#41×C₁₃₇	—	4	4	no	—
92	C₂×Q₈×C₁₃₇	—	2	2	no	—
93	C₂×Q₈×C₁₃₇	—	2	2	no	—
94	C₂×Q₈×C₁₃₇	—	2	2	no	—
95	C₂×Q₈×C₁₃₇	—	2	2	no	—
96	C₂×Q₈×C₁₃₇	—	2	2	no	—
97	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
98	C₂×D₄×C₁₃₇	—	2	8	no	—
99	C₂×D₄×C₁₃₇	—	2	2	no	—
100	C₂×D₄×C₁₃₇	—	2	2	no	—
101	C₂×D₄×C₁₃₇	—	2	2	no	—
102	C₂×D₄×C₁₃₇	—	2	2	no	—
103	C₂×D₄×C₁₃₇	—	2	2	no	—
104	C₂×D₄×C₁₃₇	—	4	2	no	—
105	C₂×D₄×C₁₃₇	—	4	2	no	—
106	C₂×D₄×C₁₃₇	—	4	4	no	—
107	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
108	C₂²×C₄×C₁₃₇	—	2	2	no	—
109	C₂²×C₄×C₁₃₇	—	2	2	no	—
110	C₂²×C₄×C₁₃₇	—	2	8	no	—
111	C₂²×C₄×C₁₃₇	—	4	2	no	—
112	C₂²×C₄×C₁₃₇	—	4	2	no	—
113	C₂²×C₄×C₁₃₇	—	4	4	no	—
114	C₂²×C₄×C₁₃₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
120	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
121	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
122	Q₁₆×C₁₃₇	—	2	8	no	—
123	SD₁₆×C₁₃₇	—	2	16	no	—
124	D₈×C₁₃₇	—	2	8	no	—
125	M₁₆×C₁₃₇	—	2	4	no	—
126	M₁₆×C₁₃₇	—	2	4	no	—
127	M₁₆×C₁₃₇	—	2	4	no	—
128	M₁₆×C₁₃₇	—	4	2	no	—
129	M₁₆×C₁₃₇	—	4	4	no	—
130	C₂×C₈×C₁₃₇	—	2	4	no	—
131	C₂×C₈×C₁₃₇	—	2	4	no	—
132	C₂×C₈×C₁₃₇	—	2	2	no	—
133	C₂×C₈×C₁₃₇	—	2	2	no	—
134	C₂×C₈×C₁₃₇	—	2	4	no	—
135	C₂×C₈×C₁₃₇	—	2	2	no	—
136	C₂×C₈×C₁₃₇	—	4	2	no	—
137	C₂×C₈×C₁₃₇	—	4	4	no	—
138	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₁₃₇	—	2	2	no	—
141	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
142	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
143	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
144	C₄²×C₁₃₇	—	2	2	no	—
145	C₄²×C₁₃₇	—	2	2	no	—
146	C₂³×C₁₃₇	—	2	4	no	—
147	C₂³×C₁₃₇	—	4	2	no	—
148	C₂³×C₁₃₇	—	4	2	no	—
149	C₂³×C₁₃₇	—	4	2	no	—
150	C₂³×C₁₃₇	—	8	4	no	—
151	C₂³×C₁₃₇	#121	1	1	yes	characteristic
152	C₂³×C₁₃₇	#121	1	1	yes	characteristic
153	C₂³×C₁₃₇	#123	1	4	yes	—
154	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
155	Q₈×C₁₃₇	—	2	4	no	—
156	Q₈×C₁₃₇	—	2	4	no	—
157	Q₈×C₁₃₇	—	4	2	no	—
158	Q₈×C₁₃₇	—	4	2	no	—
159	D₄×C₁₃₇	—	2	4	no	—
160	D₄×C₁₃₇	—	2	4	no	—
161	D₄×C₁₃₇	—	4	8	no	—
162	D₄×C₁₃₇	—	4	8	no	—
163	D₄×C₁₃₇	—	4	2	no	—
164	D₄×C₁₃₇	—	4	2	no	—
165	C₂×C₄×C₁₃₇	—	2	4	no	—
166	C₂×C₄×C₁₃₇	—	2	4	no	—
167	C₂×C₄×C₁₃₇	—	4	4	no	—
168	C₂×C₄×C₁₃₇	—	4	2	no	—
169	C₂×C₄×C₁₃₇	—	4	2	no	—
170	C₂×C₄×C₁₃₇	—	4	8	no	—
171	C₂×C₄×C₁₃₇	—	4	8	no	—
172	C₂×C₄×C₁₃₇	#120	1	2	yes	—
173	C₂×C₄×C₁₃₇	#120	1	2	yes	—
174	C₂×C₄×C₁₃₇	#121	1	4	yes	—
175	C₂×C₄×C₁₃₇	#121	1	4	yes	—
176	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
177	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
178	C₂×C₄×C₁₃₇	#123	1	4	yes	—
179	C₂×C₄×C₁₃₇	#124	1	2	yes	—
180	C₂×C₄×C₁₃₇	#124	1	2	yes	—
181	C₂×C₄×C₁₃₇	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
185	C₈×C₁₃₇	—	2	2	no	—
186	C₈×C₁₃₇	—	2	2	no	—
187	C₈×C₁₃₇	—	4	2	no	—
188	C₈×C₁₃₇	—	4	4	no	—
189	C₂²×C₁₃₇	—	2	2	no	—
190	C₂²×C₁₃₇	—	2	2	no	—
191	C₂²×C₁₃₇	—	2	2	no	—
192	C₂²×C₁₃₇	—	4	4	no	—
193	C₂²×C₁₃₇	—	4	4	no	—
194	C₂²×C₁₃₇	—	8	4	no	—
195	C₂²×C₁₃₇	—	8	4	no	—
196	C₂²×C₁₃₇	—	8	4	no	—
197	C₂²×C₁₃₇	#394	1	2	yes	—
198	C₂²×C₁₃₇	#394	1	2	yes	—
199	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
200	C₂²×C₁₃₇	C₂×#123	1	2	yes	—
201	C₄×C₁₃₇	—	4	4	no	—
202	C₄×C₁₃₇	—	4	4	no	—
203	C₄×C₁₃₇	#394	1	2	yes	—
204	C₄×C₁₃₇	#394	1	2	yes	—
205	C₄×C₁₃₇	C₂×#120	1	1	yes	characteristic
206	C₄×C₁₃₇	C₂×#121	1	2	yes	—
207	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
208	C₂×C₁₃₇	—	2	2	no	—
209	C₂×C₁₃₇	—	2	2	no	—
210	C₂×C₁₃₇	—	2	2	no	—
211	C₂×C₁₃₇	—	8	4	no	—
212	C₂×C₁₃₇	C₂×#394	1	1	yes	characteristic
213	C₂×C₁₃₇	#3067	1	2	yes	—
214	#23520	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₁₃₇	#23520	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²×#135	C₂×C₁₃₇	1	1	yes	characteristic
217	C₂×#424	C₂×C₁₃₇	1	1	yes	characteristic, complemented
218	#2896	C₂×C₁₃₇	1	2	yes	complemented
219	C₂×#413	C₂×C₁₃₇	1	2	yes	complemented
220	#2047	C₂×C₁₃₇	1	2	yes	complemented
221	#2046	C₂×C₁₃₇	1	4	yes	complemented
222	#2045	C₂×C₁₃₇	1	2	yes	complemented
223	C₂×#341	C₂×C₁₃₇	1	1	yes	characteristic, complemented
224	C₂²×#41	C₂²×C₁₃₇	1	2	yes	—
225	C₂×#135	—	2	8	no	—
226	C₂×#135	—	2	2	no	—
227	C₂×#135	—	2	2	no	—
228	C₂²×Q₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
229	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
230	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
231	C₂²×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
232	C₂²×M₁₆	C₂²×C₁₃₇	1	2	yes	—
233	C₂³×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
234	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
235	#424	—	2	2	no	—
236	#424	C₂²×C₁₃₇	1	2	yes	—
237	#424	C₂²×C₁₃₇	1	2	yes	—
238	C₂×C₄⋊C₈	—	2	2	no	—
239	#419	C₂²×C₁₃₇	1	4	yes	—
240	#418	C₂²×C₁₃₇	1	4	yes	—
241	#417	—	2	4	no	—
242	#416	—	2	4	no	—
243	#413	—	2	8	no	—
244	C₂×#101	C₂²×C₁₃₇	1	4	yes	—
245	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
246	#341	C₂²×C₁₃₇	1	2	yes	—
247	#341	C₂²×C₁₃₇	1	2	yes	—
248	C₂×#41	—	2	4	no	—
249	C₂×#41	—	2	2	no	—
250	C₂×#41	—	2	2	no	—
251	C₂×#41	—	2	4	no	—
252	C₂×#41	—	2	2	no	—
253	C₂×#41	—	2	2	no	—
254	C₂×#41	—	2	4	no	—
255	C₂×#41	D₄×C₁₃₇	1	8	yes	—
256	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
257	C₂²×D₄	—	2	2	no	—
258	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
259	C₂³×C₄	—	2	2	no	—
260	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
261	#135	—	2	16	no	—
262	#135	—	2	16	no	—
263	C₂×Q₁₆	—	2	4	no	—
264	C₂×Q₁₆	—	2	2	no	—
265	C₂×SD₁₆	—	2	4	no	—
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×D₈	—	2	4	no	—
269	C₂×D₈	—	2	2	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂×M₁₆	—	2	1	no	—
272	C₂×M₁₆	—	2	4	no	—
273	C₂×M₁₆	—	2	4	no	—
274	C₂×M₁₆	—	2	4	no	—
275	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
276	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
277	C₂²×C₈	—	2	4	no	—
278	C₂²×C₈	—	2	2	no	—
279	C₂²×C₈	—	2	1	no	—
280	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
281	C₂²×C₈	D₄×C₁₃₇	1	2	yes	—
282	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
284	#117	—	2	1	no	—
285	#117	—	2	1	no	—
286	#117	D₄×C₁₃₇	1	2	yes	—
287	#117	C₂³×C₁₃₇	1	2	yes	—
288	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₄²	—	2	1	no	—
291	C₄⋊C₈	—	2	4	no	—
292	C₄⋊C₈	—	2	4	no	—
293	#103	—	2	8	no	—
294	#102	—	2	8	no	—
295	#101	—	2	8	no	—
296	#100	—	2	8	no	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	#41×C₁₃₇	1	1	yes	characteristic
299	#41	—	2	8	no	—
300	#41	—	2	8	no	—
301	#41	—	2	8	no	—
302	#41	—	2	8	no	—
303	#41	—	4	4	no	—
304	#41	—	4	4	no	—
305	#41	—	4	4	no	—
306	#41	—	4	4	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
313	C₂×D₄	—	2	8	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	4	2	no	—
321	C₂×D₄	—	4	4	no	—
322	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	4	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	4	4	no	—
329	C₂²×C₄	C₂×D₄×C₁₃₇	1	4	yes	—
330	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
334	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
335	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
336	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
337	Q₁₆	—	2	8	no	—
338	SD₁₆	—	2	16	no	—
339	D₈	—	2	8	no	—
340	M₁₆	—	2	4	no	—
341	M₁₆	—	2	4	no	—
342	M₁₆	—	2	4	no	—
343	M₁₆	—	4	2	no	—
344	M₁₆	—	4	4	no	—
345	C₂×C₈	—	2	4	no	—
346	C₂×C₈	—	2	4	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	4	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	4	2	no	—
352	C₂×C₈	—	4	4	no	—
353	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
357	C₂²⋊C₄	—	2	2	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	2	no	—
365	C₂³	—	8	4	no	—
366	C₂³	#121×C₁₃₇	1	1	yes	characteristic
367	C₂³	#121×C₁₃₇	1	1	yes	characteristic
368	C₂³	#123×C₁₃₇	1	4	yes	—
369	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	4	2	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	4	no	—
376	D₄	—	4	8	no	—
377	D₄	—	4	8	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	4	4	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	8	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	#120×C₁₃₇	1	2	yes	—
388	C₂×C₄	#120×C₁₃₇	1	2	yes	—
389	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
390	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
391	C₂×C₄	#121×C₁₃₇	1	4	yes	—
392	C₂×C₄	#121×C₁₃₇	1	4	yes	—
393	C₂×C₄	#123×C₁₃₇	1	4	yes	—
394	C₂×C₄	#124×C₁₃₇	1	2	yes	—
395	C₂×C₄	#124×C₁₃₇	1	2	yes	—
396	C₂×C₄	C₂²×D₄×C₁₃₇	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
400	C₈	—	2	2	no	—
401	C₈	—	2	2	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	4	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	4	no	—
409	C₂²	—	8	4	no	—
410	C₂²	—	8	4	no	—
411	C₂²	—	8	4	no	—
412	C₂²	#394×C₁₃₇	1	2	yes	—
413	C₂²	#394×C₁₃₇	1	2	yes	—
414	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
415	C₂²	C₂×#123×C₁₃₇	1	2	yes	—
416	C₄	—	4	4	no	—
417	C₄	—	4	4	no	—
418	C₄	#394×C₁₃₇	1	2	yes	—
419	C₄	#394×C₁₃₇	1	2	yes	—
420	C₄	C₂×#120×C₁₃₇	1	1	yes	characteristic
421	C₄	C₂×#121×C₁₃₇	1	2	yes	—
422	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
423	C₂	—	2	2	no	—
424	C₂	—	2	2	no	—
425	C₂	—	2	2	no	—
426	C₂	—	8	4	no	—
427	C₂	C₂×#394×C₁₃₇	1	1	yes	characteristic
428	C₂	#3067×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#121
prime radical	C₂²×#135×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#424×C₁₃₇	C₂	semidirect
3. #2896×C₁₃₇	C₂	semidirect
4. C₂×#413×C₁₃₇	C₂	semidirect
5. #2047×C₁₃₇	C₂	semidirect
6. #2046×C₁₃₇	C₂	semidirect
7. #2045×C₁₃₇	C₂	semidirect
8. C₂×#341×C₁₃₇	C₂	semidirect
214. #23520	C₁₃₇	direct (N × H)
215. C₁₃₇	#23520	direct (N × H)
217. C₂×#424	C₂×C₁₃₇	semidirect
218. #2896	C₂×C₁₃₇	semidirect
219. C₂×#413	C₂×C₁₃₇	semidirect
220. #2047	C₂×C₁₃₇	semidirect
221. #2046	C₂×C₁₃₇	semidirect
222. #2045	C₂×C₁₃₇	semidirect
223. C₂×#341	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₁₃₇	2
C₂×#424×C₁₃₇	2
#2896×C₁₃₇	2
C₂×#413×C₁₃₇	2
#2047×C₁₃₇	2
#2046×C₁₃₇	2
#2045×C₁₃₇	2
C₂×#341×C₁₃₇	2
#23520	137

Subgroups — C137×#23520 #46872906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#23520 #46872906