Subgroups — C₁₃₇×#19392 #46868778

Order 35072.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3226×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3036×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#461×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2972×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1681×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1673×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1571×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1546×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1540×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₁₃₇	—	2	1	no	—
11	C₂²×SD₁₆×C₁₃₇	—	2	1	no	—
12	#532×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
15	#500×C₁₃₇	C₂²	1	2	yes	complemented
16	#486×C₁₃₇	C₂²	1	2	yes	complemented
17	#480×C₁₃₇	—	2	2	no	—
18	#474×C₁₃₇	—	2	2	no	—
19	#463×C₁₃₇	—	2	2	no	—
20	#461×C₁₃₇	—	2	2	no	—
21	#461×C₁₃₇	C₂²	1	2	yes	—
22	#461×C₁₃₇	C₂²	1	4	yes	—
23	#459×C₁₃₇	C₂²	1	2	yes	—
24	C₈×D₄×C₁₃₇	—	2	2	no	—
25	C₈×D₄×C₁₃₇	C₂²	1	4	yes	—
26	#431×C₁₃₇	C₂²	1	1	yes	characteristic
27	#431×C₁₃₇	C₂²	1	1	yes	characteristic
28	#423×C₁₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#417×C₁₃₇	—	2	1	no	—
31	C₂×#103×C₁₃₇	—	2	1	no	—
32	C₂×C₂²⋊C₈×C₁₃₇	—	2	2	no	—
33	#335×C₁₃₇	C₂²	1	4	yes	complemented
34	#332×C₁₃₇	C₂²	1	2	yes	complemented
35	#332×C₁₃₇	C₂²	1	4	yes	complemented
36	#331×C₁₃₇	C₂²	1	2	yes	complemented
37	C₂×#41×C₁₃₇	—	2	1	no	—
38	C₂²×Q₈×C₁₃₇	—	2	1	no	—
39	C₂²×D₄×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₁₃₇	—	2	1	no	—
41	AGL(1,ℤ/8)×C₁₃₇	—	4	4	no	—
42	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
43	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
44	C₂×SD₁₆×C₁₃₇	—	4	4	no	—
45	C₂×D₈×C₁₃₇	—	2	2	no	—
46	C₂×M₁₆×C₁₃₇	—	2	1	no	—
47	C₂²×C₈×C₁₃₇	—	2	1	no	—
48	C₂²×C₈×C₁₃₇	—	2	4	no	—
49	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
50	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
51	C₄⋊Q₈×C₁₃₇	—	2	1	no	—
52	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	#127×C₁₃₇	C₂³	1	1	yes	characteristic
55	#124×C₁₃₇	—	2	1	no	—
56	#121×C₁₃₇	—	2	1	no	—
57	#121×C₁₃₇	—	2	1	no	—
58	C₄×D₄×C₁₃₇	—	2	1	no	—
59	C₄×D₄×C₁₃₇	—	2	1	no	—
60	C₄×D₄×C₁₃₇	—	2	1	no	—
61	C₄×D₄×C₁₃₇	—	2	1	no	—
62	C₄×D₄×C₁₃₇	D₄	1	2	yes	—
63	C₄×D₄×C₁₃₇	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
68	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
69	#107×C₁₃₇	—	2	4	no	—
70	C₄⋊C₈×C₁₃₇	—	2	1	no	—
71	C₄⋊C₈×C₁₃₇	—	2	1	no	—
72	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
73	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
74	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
75	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
76	#103×C₁₃₇	—	2	4	no	—
77	#103×C₁₃₇	—	2	2	no	—
78	#102×C₁₃₇	—	2	4	no	—
79	#102×C₁₃₇	—	2	2	no	—
80	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
81	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
82	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
83	C₄×C₈×C₁₃₇	—	2	1	no	—
84	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
85	C₂⁴×C₁₃₇	—	2	1	no	—
86	#41×C₁₃₇	—	4	2	no	—
87	#41×C₁₃₇	—	4	2	no	—
88	C₂×Q₈×C₁₃₇	—	2	1	no	—
89	C₂×Q₈×C₁₃₇	—	2	2	no	—
90	C₂×Q₈×C₁₃₇	—	4	1	no	—
91	C₂×Q₈×C₁₃₇	—	4	2	no	—
92	C₂×Q₈×C₁₃₇	—	4	2	no	—
93	C₂×D₄×C₁₃₇	—	2	1	no	—
94	C₂×D₄×C₁₃₇	—	2	2	no	—
95	C₂×D₄×C₁₃₇	—	2	1	no	—
96	C₂×D₄×C₁₃₇	—	2	1	no	—
97	C₂×D₄×C₁₃₇	—	2	1	no	—
98	C₂×D₄×C₁₃₇	—	4	1	no	—
99	C₂×D₄×C₁₃₇	D₈	1	4	yes	—
100	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₃₇	—	2	1	no	—
102	C₂²×C₄×C₁₃₇	—	2	2	no	—
103	C₂²×C₄×C₁₃₇	—	2	2	no	—
104	C₂²×C₄×C₁₃₇	—	4	1	no	—
105	C₂²×C₄×C₁₃₇	—	4	1	no	—
106	C₂²×C₄×C₁₃₇	—	8	2	no	—
107	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
110	SD₁₆×C₁₃₇	—	4	8	no	—
111	SD₁₆×C₁₃₇	—	4	4	no	—
112	D₈×C₁₃₇	—	4	4	no	—
113	M₁₆×C₁₃₇	—	4	2	no	—
114	C₂×C₈×C₁₃₇	—	2	2	no	—
115	C₂×C₈×C₁₃₇	—	2	2	no	—
116	C₂×C₈×C₁₃₇	—	2	2	no	—
117	C₂×C₈×C₁₃₇	—	4	2	no	—
118	C₂×C₈×C₁₃₇	—	8	4	no	—
119	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
120	C₄⋊C₄×C₁₃₇	—	2	1	no	—
121	C₄⋊C₄×C₁₃₇	—	2	2	no	—
122	C₄⋊C₄×C₁₃₇	—	2	1	no	—
123	C₄⋊C₄×C₁₃₇	—	2	2	no	—
124	C₄⋊C₄×C₁₃₇	—	2	1	no	—
125	C₄⋊C₄×C₁₃₇	—	4	1	no	—
126	C₄⋊C₄×C₁₃₇	—	4	1	no	—
127	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
128	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
129	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
130	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
131	C₂²⋊C₄×C₁₃₇	—	4	1	no	—
132	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₁₃₇	#41	1	1	yes	characteristic
135	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
136	C₂³×C₁₃₇	—	2	2	no	—
137	C₂³×C₁₃₇	—	2	4	no	—
138	C₂³×C₁₃₇	—	4	1	no	—
139	C₂³×C₁₃₇	—	8	1	no	—
140	C₂³×C₁₃₇	—	8	1	no	—
141	C₂³×C₁₃₇	#121	1	1	yes	characteristic
142	Q₈×C₁₃₇	—	4	2	no	—
143	Q₈×C₁₃₇	—	4	1	no	—
144	Q₈×C₁₃₇	—	4	2	no	—
145	Q₈×C₁₃₇	—	4	1	no	—
146	Q₈×C₁₃₇	—	8	2	no	—
147	D₄×C₁₃₇	—	2	2	no	—
148	D₄×C₁₃₇	—	2	4	no	—
149	D₄×C₁₃₇	—	4	1	no	—
150	D₄×C₁₃₇	—	4	1	no	—
151	D₄×C₁₃₇	—	4	2	no	—
152	D₄×C₁₃₇	—	8	1	no	—
153	D₄×C₁₃₇	—	8	2	no	—
154	D₄×C₁₃₇	C₂×D₈	1	4	yes	—
155	C₂×C₄×C₁₃₇	—	2	2	no	—
156	C₂×C₄×C₁₃₇	—	2	2	no	—
157	C₂×C₄×C₁₃₇	—	4	1	no	—
158	C₂×C₄×C₁₃₇	—	4	4	no	—
159	C₂×C₄×C₁₃₇	—	4	1	no	—
160	C₂×C₄×C₁₃₇	—	4	2	no	—
161	C₂×C₄×C₁₃₇	—	8	2	no	—
162	C₂×C₄×C₁₃₇	—	8	1	no	—
163	C₂×C₄×C₁₃₇	—	8	2	no	—
164	C₂×C₄×C₁₃₇	—	8	2	no	—
165	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
166	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
167	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
168	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
169	C₂×C₄×C₁₃₇	#135	1	2	yes	—
170	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
173	C₈×C₁₃₇	—	2	4	no	—
174	C₈×C₁₃₇	—	4	2	no	—
175	C₈×C₁₃₇	—	4	2	no	—
176	C₈×C₁₃₇	—	4	2	no	—
177	C₂²×C₁₃₇	—	2	1	no	—
178	C₂²×C₁₃₇	—	2	1	no	—
179	C₂²×C₁₃₇	—	2	4	no	—
180	C₂²×C₁₃₇	—	4	2	no	—
181	C₂²×C₁₃₇	—	4	2	no	—
182	C₂²×C₁₃₇	—	8	4	no	—
183	C₂²×C₁₃₇	—	8	1	no	—
184	C₂²×C₁₃₇	—	8	1	no	—
185	C₂²×C₁₃₇	—	8	1	no	—
186	C₂²×C₁₃₇	#466	1	2	yes	—
187	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
188	C₄×C₁₃₇	—	2	1	no	—
189	C₄×C₁₃₇	—	2	1	no	—
190	C₄×C₁₃₇	—	2	1	no	—
191	C₄×C₁₃₇	—	4	2	no	—
192	C₄×C₁₃₇	—	8	1	no	—
193	C₄×C₁₃₇	—	8	2	no	—
194	C₄×C₁₃₇	#466	1	2	yes	—
195	C₄×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
197	C₄×C₁₃₇	#577	1	2	yes	—
198	C₂×C₁₃₇	—	2	2	no	—
199	C₂×C₁₃₇	—	4	4	no	—
200	C₂×C₁₃₇	—	16	1	no	—
201	C₂×C₁₃₇	#3042	1	1	yes	characteristic
202	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
203	C₂×C₁₃₇	#3069	1	1	yes	characteristic
204	#19392	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₁₃₇	#19392	1	1	yes	characteristic, direct product factor, Sylow
206	#3226	C₂×C₁₃₇	1	1	yes	characteristic, complemented
207	#3036	C₂×C₁₃₇	1	1	yes	characteristic
208	C₂×#461	C₂×C₁₃₇	1	1	yes	characteristic, complemented
209	#2972	C₂×C₁₃₇	1	1	yes	characteristic, complemented
210	#1681	C₂×C₁₃₇	1	4	yes	complemented
211	#1673	C₂×C₁₃₇	1	4	yes	complemented
212	#1571	C₂×C₁₃₇	1	1	yes	characteristic, complemented
213	#1546	C₂×C₁₃₇	1	1	yes	characteristic, complemented
214	#1540	C₂×C₁₃₇	1	1	yes	characteristic, complemented
215	C₂×AGL(1,ℤ/8)	—	2	1	no	—
216	C₂²×SD₁₆	—	2	1	no	—
217	#532	C₂²×C₁₃₇	1	1	yes	characteristic
218	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
219	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
220	#500	C₂²×C₁₃₇	1	2	yes	complemented
221	#486	C₂²×C₁₃₇	1	2	yes	complemented
222	#480	—	2	2	no	—
223	#474	—	2	2	no	—
224	#463	—	2	2	no	—
225	#461	—	2	2	no	—
226	#461	C₂²×C₁₃₇	1	2	yes	—
227	#461	C₂²×C₁₃₇	1	4	yes	—
228	#459	C₂²×C₁₃₇	1	2	yes	—
229	C₈×D₄	—	2	2	no	—
230	C₈×D₄	C₂²×C₁₃₇	1	4	yes	—
231	#431	C₂²×C₁₃₇	1	1	yes	characteristic
232	#431	C₂²×C₁₃₇	1	1	yes	characteristic
233	#423	C₂²×C₁₃₇	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
235	#417	—	2	1	no	—
236	C₂×#103	—	2	1	no	—
237	C₂×C₂²⋊C₈	—	2	2	no	—
238	#335	C₂²×C₁₃₇	1	4	yes	complemented
239	#332	C₂²×C₁₃₇	1	2	yes	complemented
240	#332	C₂²×C₁₃₇	1	4	yes	complemented
241	#331	C₂²×C₁₃₇	1	2	yes	complemented
242	C₂×#41	—	2	1	no	—
243	C₂²×Q₈	—	2	1	no	—
244	C₂²×D₄	D₄×C₁₃₇	1	1	yes	characteristic
245	C₂³×C₄	—	2	1	no	—
246	AGL(1,ℤ/8)	—	4	4	no	—
247	C₂×SD₁₆	—	2	4	no	—
248	C₂×SD₁₆	—	2	2	no	—
249	C₂×SD₁₆	—	4	4	no	—
250	C₂×D₈	—	2	2	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	4	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
258	C₄⋊Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
259	#127	C₂³×C₁₃₇	1	1	yes	characteristic
260	#124	—	2	1	no	—
261	#121	—	2	1	no	—
262	#121	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	2	1	no	—
267	C₄×D₄	D₄×C₁₃₇	1	2	yes	—
268	C₄×D₄	C₂³×C₁₃₇	1	2	yes	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
274	#107	—	2	4	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
278	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
279	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
280	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
281	#103	—	2	4	no	—
282	#103	—	2	2	no	—
283	#102	—	2	4	no	—
284	#102	—	2	2	no	—
285	C₂²⋊C₈	—	2	1	no	—
286	C₂²⋊C₈	—	2	4	no	—
287	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
290	C₂⁴	—	2	1	no	—
291	#41	—	4	2	no	—
292	#41	—	4	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	D₈×C₁₃₇	1	4	yes	—
305	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	8	2	no	—
312	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
314	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
315	SD₁₆	—	4	8	no	—
316	SD₁₆	—	4	4	no	—
317	D₈	—	4	4	no	—
318	M₁₆	—	4	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	4	2	no	—
323	C₂×C₈	—	8	4	no	—
324	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₂²⋊C₄	—	4	1	no	—
337	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
339	C₄²	#41×C₁₃₇	1	1	yes	characteristic
340	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	—	2	4	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	8	1	no	—
345	C₂³	—	8	1	no	—
346	C₂³	#121×C₁₃₇	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	4	1	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	8	2	no	—
352	D₄	—	2	4	no	—
353	D₄	—	2	2	no	—
354	D₄	—	4	1	no	—
355	D₄	—	4	1	no	—
356	D₄	—	4	2	no	—
357	D₄	—	8	1	no	—
358	D₄	—	8	2	no	—
359	D₄	C₂×D₈×C₁₃₇	1	4	yes	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
371	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
372	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
373	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
374	C₂×C₄	#135×C₁₃₇	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
378	C₈	—	2	4	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	8	4	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	#466×C₁₃₇	1	2	yes	—
392	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	4	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	#466×C₁₃₇	1	2	yes	—
400	C₄	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
401	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
402	C₄	#577×C₁₃₇	1	2	yes	—
403	C₂	—	2	2	no	—
404	C₂	—	4	4	no	—
405	C₂	—	16	1	no	—
406	C₂	#3042×C₁₃₇	1	1	yes	characteristic
407	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
408	C₂	#3069×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#3036×C₁₃₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3226×C₁₃₇	C₂	semidirect
3. C₂×#461×C₁₃₇	C₂	semidirect
4. #2972×C₁₃₇	C₂	semidirect
5. #1681×C₁₃₇	C₂	semidirect
6. #1673×C₁₃₇	C₂	semidirect
7. #1571×C₁₃₇	C₂	semidirect
8. #1546×C₁₃₇	C₂	semidirect
9. #1540×C₁₃₇	C₂	semidirect
15. #500×C₁₃₇	C₂²	semidirect
16. #486×C₁₃₇	C₂²	semidirect
33. #335×C₁₃₇	C₂²	semidirect
34. #332×C₁₃₇	C₂²	semidirect
35. #332×C₁₃₇	C₂²	semidirect
36. #331×C₁₃₇	C₂²	semidirect
204. #19392	C₁₃₇	direct (N × H)
205. C₁₃₇	#19392	direct (N × H)
206. #3226	C₂×C₁₃₇	semidirect
208. C₂×#461	C₂×C₁₃₇	semidirect
209. #2972	C₂×C₁₃₇	semidirect
210. #1681	C₂×C₁₃₇	semidirect
211. #1673	C₂×C₁₃₇	semidirect
212. #1571	C₂×C₁₃₇	semidirect
213. #1546	C₂×C₁₃₇	semidirect
214. #1540	C₂×C₁₃₇	semidirect
220. #500	C₂²×C₁₃₇	semidirect
221. #486	C₂²×C₁₃₇	semidirect
238. #335	C₂²×C₁₃₇	semidirect
239. #332	C₂²×C₁₃₇	semidirect
240. #332	C₂²×C₁₃₇	semidirect
241. #331	C₂²×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3226×C₁₃₇	2
#3036×C₁₃₇	2
C₂×#461×C₁₃₇	2
#2972×C₁₃₇	2
#1681×C₁₃₇	2
#1673×C₁₃₇	2
#1571×C₁₃₇	2
#1546×C₁₃₇	2
#1540×C₁₃₇	2
#19392	137

Subgroups — C137×#19392 #46868778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19392 #46868778