Subgroups — C₁₃₇×#19125 #46868511

Order 35072.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3035×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	C₈×#41×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1682×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1681×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1674×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1673×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1668×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1667×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#335×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#1540×C₁₃₇	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₁₃₇	—	2	2	no	—
13	#532×C₁₃₇	C₂²	1	2	yes	—
14	C₄×#41×C₁₃₇	C₂²	1	1	yes	characteristic
15	#500×C₁₃₇	C₂²	1	1	yes	characteristic
16	#493×C₁₃₇	C₂²	1	1	yes	characteristic
17	#486×C₁₃₇	C₂²	1	2	yes	—
18	#466×C₁₃₇	—	2	1	no	—
19	#466×C₁₃₇	—	2	1	no	—
20	#464×C₁₃₇	—	2	2	no	—
21	#463×C₁₃₇	—	2	2	no	—
22	#462×C₁₃₇	C₂²	1	2	yes	—
23	#461×C₁₃₇	C₂²	1	2	yes	—
24	#460×C₁₃₇	C₂²	1	2	yes	—
25	#459×C₁₃₇	C₂²	1	2	yes	—
26	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₁₃₇	—	2	1	no	—
29	C₄×D₈×C₁₃₇	—	2	1	no	—
30	C₈×D₄×C₁₃₇	—	2	1	no	—
31	C₈×D₄×C₁₃₇	—	2	1	no	—
32	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
34	#431×C₁₃₇	—	2	1	no	—
35	#431×C₁₃₇	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
37	#416×C₁₃₇	—	2	2	no	—
38	C₂×C₄×C₈×C₁₃₇	—	2	1	no	—
39	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
40	#335×C₁₃₇	C₂²	1	1	yes	characteristic
41	#335×C₁₃₇	C₂²	1	1	yes	characteristic
42	#335×C₁₃₇	C₂²	1	2	yes	—
43	#332×C₁₃₇	C₂²	1	2	yes	—
44	#332×C₁₃₇	C₂²	1	2	yes	—
45	#331×C₁₃₇	C₂²	1	2	yes	—
46	#331×C₁₃₇	C₂²	1	2	yes	—
47	C₂×#41×C₁₃₇	—	2	2	no	—
48	C₂×#41×C₁₃₇	D₄	1	1	yes	characteristic
49	#135×C₁₃₇	—	4	8	no	—
50	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
51	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
52	C₂×SD₁₆×C₁₃₇	—	2	2	no	—
53	C₂×D₈×C₁₃₇	—	2	2	no	—
54	C₂×D₈×C₁₃₇	—	2	1	no	—
55	C₂×D₈×C₁₃₇	—	2	1	no	—
56	C₂²×C₈×C₁₃₇	—	2	2	no	—
57	C₂²×C₈×C₁₃₇	—	2	1	no	—
58	C₂²×C₈×C₁₃₇	—	2	1	no	—
59	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
61	C₄⋊Q₈×C₁₃₇	C₂³	1	2	yes	—
62	#127×C₁₃₇	C₂³	1	2	yes	—
63	#124×C₁₃₇	—	2	2	no	—
64	#121×C₁₃₇	—	2	2	no	—
65	#121×C₁₃₇	—	2	2	no	—
66	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
67	C₄×Q₈×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₁₃₇	—	2	1	no	—
69	C₄×D₄×C₁₃₇	—	2	1	no	—
70	C₄×D₄×C₁₃₇	—	2	2	no	—
71	C₄×D₄×C₁₃₇	—	2	2	no	—
72	C₄×D₄×C₁₃₇	D₄	1	1	yes	characteristic
73	C₄×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
74	#117×C₁₃₇	—	2	1	no	—
75	#117×C₁₃₇	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₁₃₇	—	2	1	no	—
77	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
78	#107×C₁₃₇	—	2	1	no	—
79	#107×C₁₃₇	—	2	1	no	—
80	C₄⋊C₈×C₁₃₇	—	2	1	no	—
81	C₄⋊C₈×C₁₃₇	—	2	1	no	—
82	C₄⋊C₈×C₁₃₇	—	2	2	no	—
83	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
84	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
85	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₁₃₇	C₂³	1	1	yes	characteristic
87	#103×C₁₃₇	—	2	2	no	—
88	#103×C₁₃₇	—	2	2	no	—
89	#102×C₁₃₇	—	2	2	no	—
90	#102×C₁₃₇	—	2	2	no	—
91	#102×C₁₃₇	—	2	2	no	—
92	#102×C₁₃₇	—	2	2	no	—
93	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
94	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
95	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
96	C₄×C₈×C₁₃₇	—	2	1	no	—
97	C₄×C₈×C₁₃₇	—	2	1	no	—
98	C₄×C₈×C₁₃₇	—	2	1	no	—
99	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
100	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
101	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
103	#41×C₁₃₇	—	2	2	no	—
104	#41×C₁₃₇	—	4	4	no	—
105	#41×C₁₃₇	—	4	4	no	—
106	#41×C₁₃₇	D₈	1	4	yes	complemented
107	C₂×Q₈×C₁₃₇	—	2	2	no	—
108	C₂×Q₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₁₃₇	—	2	1	no	—
110	C₂×D₄×C₁₃₇	—	2	2	no	—
111	C₂×D₄×C₁₃₇	—	2	2	no	—
112	C₂×D₄×C₁₃₇	—	2	2	no	—
113	C₂×D₄×C₁₃₇	—	4	2	no	—
114	C₂×D₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₁₃₇	—	2	1	no	—
116	C₂²×C₄×C₁₃₇	—	2	1	no	—
117	C₂²×C₄×C₁₃₇	—	2	1	no	—
118	C₂²×C₄×C₁₃₇	—	4	2	no	—
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
122	Q₁₆×C₁₃₇	—	4	4	no	—
123	SD₁₆×C₁₃₇	—	4	4	no	—
124	SD₁₆×C₁₃₇	—	4	4	no	—
125	D₈×C₁₃₇	—	4	2	no	—
126	D₈×C₁₃₇	—	4	4	no	—
127	D₈×C₁₃₇	—	4	2	no	—
128	C₂×C₈×C₁₃₇	—	2	2	no	—
129	C₂×C₈×C₁₃₇	—	2	1	no	—
130	C₂×C₈×C₁₃₇	—	2	1	no	—
131	C₂×C₈×C₁₃₇	—	2	2	no	—
132	C₂×C₈×C₁₃₇	—	2	2	no	—
133	C₂×C₈×C₁₃₇	—	4	1	no	—
134	C₂×C₈×C₁₃₇	—	4	2	no	—
135	C₂×C₈×C₁₃₇	—	4	4	no	—
136	C₂×C₈×C₁₃₇	—	4	2	no	—
137	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₁₃₇	—	2	1	no	—
142	C₄⋊C₄×C₁₃₇	—	2	1	no	—
143	C₄⋊C₄×C₁₃₇	—	2	2	no	—
144	C₄⋊C₄×C₁₃₇	—	2	2	no	—
145	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
146	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
147	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
148	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
149	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
150	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
151	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
152	C₄²×C₁₃₇	—	2	1	no	—
153	C₄²×C₁₃₇	—	2	1	no	—
154	C₄²×C₁₃₇	—	2	1	no	—
155	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₁₃₇	#41	1	1	yes	characteristic
158	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
159	C₂³×C₁₃₇	—	2	1	no	—
160	C₂³×C₁₃₇	—	4	2	no	—
161	C₂³×C₁₃₇	#121	1	1	yes	characteristic
162	Q₈×C₁₃₇	—	2	1	no	—
163	Q₈×C₁₃₇	—	4	2	no	—
164	Q₈×C₁₃₇	—	4	2	no	—
165	Q₈×C₁₃₇	C₂×D₈	1	2	yes	—
166	D₄×C₁₃₇	—	2	4	no	—
167	D₄×C₁₃₇	—	2	1	no	—
168	D₄×C₁₃₇	—	4	2	no	—
169	D₄×C₁₃₇	—	4	2	no	—
170	D₄×C₁₃₇	—	4	2	no	—
171	D₄×C₁₃₇	—	4	2	no	—
172	D₄×C₁₃₇	—	8	2	no	—
173	D₄×C₁₃₇	—	8	4	no	—
174	D₄×C₁₃₇	C₂×D₈	1	2	yes	—
175	C₂×C₄×C₁₃₇	—	2	1	no	—
176	C₂×C₄×C₁₃₇	—	2	1	no	—
177	C₂×C₄×C₁₃₇	—	2	1	no	—
178	C₂×C₄×C₁₃₇	—	2	4	no	—
179	C₂×C₄×C₁₃₇	—	2	1	no	—
180	C₂×C₄×C₁₃₇	—	2	1	no	—
181	C₂×C₄×C₁₃₇	—	4	1	no	—
182	C₂×C₄×C₁₃₇	—	4	2	no	—
183	C₂×C₄×C₁₃₇	—	4	1	no	—
184	C₂×C₄×C₁₃₇	—	4	1	no	—
185	C₂×C₄×C₁₃₇	—	4	2	no	—
186	C₂×C₄×C₁₃₇	—	8	4	no	—
187	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
188	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
189	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
190	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
191	C₂×C₄×C₁₃₇	#135	1	2	yes	—
192	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
195	C₈×C₁₃₇	—	2	1	no	—
196	C₈×C₁₃₇	—	2	1	no	—
197	C₈×C₁₃₇	—	2	1	no	—
198	C₈×C₁₃₇	—	4	2	no	—
199	C₈×C₁₃₇	—	4	2	no	—
200	C₈×C₁₃₇	—	4	1	no	—
201	C₈×C₁₃₇	—	4	1	no	—
202	C₈×C₁₃₇	#135	1	2	yes	complemented
203	C₂²×C₁₃₇	—	2	1	no	—
204	C₂²×C₁₃₇	—	2	1	no	—
205	C₂²×C₁₃₇	—	2	2	no	—
206	C₂²×C₁₃₇	—	4	1	no	—
207	C₂²×C₁₃₇	—	4	1	no	—
208	C₂²×C₁₃₇	—	8	2	no	—
209	C₂²×C₁₃₇	—	8	2	no	—
210	C₂²×C₁₃₇	—	8	2	no	—
211	C₂²×C₁₃₇	#466	1	2	yes	—
212	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
213	C₄×C₁₃₇	—	2	1	no	—
214	C₄×C₁₃₇	—	2	2	no	—
215	C₄×C₁₃₇	—	2	1	no	—
216	C₄×C₁₃₇	—	2	1	no	—
217	C₄×C₁₃₇	—	4	1	no	—
218	C₄×C₁₃₇	—	4	1	no	—
219	C₄×C₁₃₇	—	8	2	no	—
220	C₄×C₁₃₇	#466	1	2	yes	—
221	C₄×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
222	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
223	C₄×C₁₃₇	C₂×#135	1	1	yes	characteristic
224	C₄×C₁₃₇	#575	1	1	yes	characteristic
225	C₂×C₁₃₇	—	2	2	no	—
226	C₂×C₁₃₇	—	4	2	no	—
227	C₂×C₁₃₇	—	16	2	no	—
228	C₂×C₁₃₇	#3035	1	1	yes	characteristic
229	C₂×C₁₃₇	C₂×#466	1	1	yes	characteristic
230	C₂×C₁₃₇	#3067	1	1	yes	characteristic
231	#19125	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₁₃₇	#19125	1	1	yes	characteristic, direct product factor, Sylow
233	#3169	C₂×C₁₃₇	1	1	yes	characteristic, complemented
234	#3035	C₂×C₁₃₇	1	2	yes	complemented
235	C₈×#41	C₂×C₁₃₇	1	1	yes	characteristic, complemented
236	#1682	C₂×C₁₃₇	1	2	yes	complemented
237	#1681	C₂×C₁₃₇	1	2	yes	complemented
238	#1674	C₂×C₁₃₇	1	1	yes	characteristic, complemented
239	#1673	C₂×C₁₃₇	1	1	yes	characteristic, complemented
240	#1668	C₂×C₁₃₇	1	1	yes	characteristic, complemented
241	#1667	C₂×C₁₃₇	1	1	yes	characteristic, complemented
242	C₂×#335	C₂×C₁₃₇	1	1	yes	characteristic, complemented
243	#1540	C₂×C₁₃₇	1	2	yes	complemented
244	C₂×#135	—	2	2	no	—
245	#532	C₂²×C₁₃₇	1	2	yes	—
246	C₄×#41	C₂²×C₁₃₇	1	1	yes	characteristic
247	#500	C₂²×C₁₃₇	1	1	yes	characteristic
248	#493	C₂²×C₁₃₇	1	1	yes	characteristic
249	#486	C₂²×C₁₃₇	1	2	yes	—
250	#466	—	2	1	no	—
251	#466	—	2	1	no	—
252	#464	—	2	2	no	—
253	#463	—	2	2	no	—
254	#462	C₂²×C₁₃₇	1	2	yes	—
255	#461	C₂²×C₁₃₇	1	2	yes	—
256	#460	C₂²×C₁₃₇	1	2	yes	—
257	#459	C₂²×C₁₃₇	1	2	yes	—
258	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
259	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
260	C₄×D₈	—	2	1	no	—
261	C₄×D₈	—	2	1	no	—
262	C₈×D₄	—	2	1	no	—
263	C₈×D₄	—	2	1	no	—
264	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₈×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
266	#431	—	2	1	no	—
267	#431	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
269	#416	—	2	2	no	—
270	C₂×C₄×C₈	—	2	1	no	—
271	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
272	#335	C₂²×C₁₃₇	1	1	yes	characteristic
273	#335	C₂²×C₁₃₇	1	1	yes	characteristic
274	#335	C₂²×C₁₃₇	1	2	yes	—
275	#332	C₂²×C₁₃₇	1	2	yes	—
276	#332	C₂²×C₁₃₇	1	2	yes	—
277	#331	C₂²×C₁₃₇	1	2	yes	—
278	#331	C₂²×C₁₃₇	1	2	yes	—
279	C₂×#41	—	2	2	no	—
280	C₂×#41	D₄×C₁₃₇	1	1	yes	characteristic
281	#135	—	4	8	no	—
282	C₂×Q₁₆	—	2	2	no	—
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×SD₁₆	—	2	2	no	—
285	C₂×D₈	—	2	2	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂×D₈	—	2	1	no	—
288	C₂²×C₈	—	2	2	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	2	1	no	—
291	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
292	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
293	C₄⋊Q₈	C₂³×C₁₃₇	1	2	yes	—
294	#127	C₂³×C₁₃₇	1	2	yes	—
295	#124	—	2	2	no	—
296	#121	—	2	2	no	—
297	#121	—	2	2	no	—
298	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
299	C₄×Q₈	C₂³×C₁₃₇	1	1	yes	characteristic
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	D₄×C₁₃₇	1	1	yes	characteristic
305	C₄×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
306	#117	—	2	1	no	—
307	#117	D₄×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
310	#107	—	2	1	no	—
311	#107	—	2	1	no	—
312	C₄⋊C₈	—	2	1	no	—
313	C₄⋊C₈	—	2	1	no	—
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₁₃₇	1	1	yes	characteristic
317	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
318	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
319	#103	—	2	2	no	—
320	#103	—	2	2	no	—
321	#102	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
332	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
333	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
334	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
335	#41	—	2	2	no	—
336	#41	—	4	4	no	—
337	#41	—	4	4	no	—
338	#41	D₈×C₁₃₇	1	4	yes	complemented
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
353	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	4	no	—
356	SD₁₆	—	4	4	no	—
357	D₈	—	4	2	no	—
358	D₈	—	4	4	no	—
359	D₈	—	4	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	4	1	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	—	4	4	no	—
368	C₂×C₈	—	4	2	no	—
369	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
371	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
372	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
378	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
389	C₄²	#41×C₁₃₇	1	1	yes	characteristic
390	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
391	C₂³	—	2	1	no	—
392	C₂³	—	4	2	no	—
393	C₂³	#121×C₁₃₇	1	1	yes	characteristic
394	Q₈	—	2	1	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	4	2	no	—
397	Q₈	C₂×D₈×C₁₃₇	1	2	yes	—
398	D₄	—	2	4	no	—
399	D₄	—	2	1	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	4	no	—
406	D₄	C₂×D₈×C₁₃₇	1	2	yes	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	8	4	no	—
419	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
420	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
421	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
422	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
423	C₂×C₄	#135×C₁₃₇	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
425	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	2	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	1	no	—
434	C₈	#135×C₁₃₇	1	2	yes	complemented
435	C₂²	—	2	1	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	#466×C₁₃₇	1	2	yes	—
444	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
445	C₄	—	2	2	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	8	2	no	—
452	C₄	#466×C₁₃₇	1	2	yes	—
453	C₄	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
454	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
455	C₄	C₂×#135×C₁₃₇	1	1	yes	characteristic
456	C₄	#575×C₁₃₇	1	1	yes	characteristic
457	C₂	—	2	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	16	2	no	—
460	C₂	#3035×C₁₃₇	1	1	yes	characteristic
461	C₂	C₂×#466×C₁₃₇	1	1	yes	characteristic
462	C₂	#3067×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3169×C₁₃₇	C₂	semidirect
2. #3035×C₁₃₇	C₂	semidirect
3. C₈×#41×C₁₃₇	C₂	semidirect
4. #1682×C₁₃₇	C₂	semidirect
5. #1681×C₁₃₇	C₂	semidirect
6. #1674×C₁₃₇	C₂	semidirect
7. #1673×C₁₃₇	C₂	semidirect
8. #1668×C₁₃₇	C₂	semidirect
9. #1667×C₁₃₇	C₂	semidirect
10. C₂×#335×C₁₃₇	C₂	semidirect
11. #1540×C₁₃₇	C₂	semidirect
106. #41×C₁₃₇	D₈	semidirect
202. C₈×C₁₃₇	#135	semidirect
231. #19125	C₁₃₇	direct (N × H)
232. C₁₃₇	#19125	direct (N × H)
233. #3169	C₂×C₁₃₇	semidirect
234. #3035	C₂×C₁₃₇	semidirect
235. C₈×#41	C₂×C₁₃₇	semidirect
236. #1682	C₂×C₁₃₇	semidirect
237. #1681	C₂×C₁₃₇	semidirect
238. #1674	C₂×C₁₃₇	semidirect
239. #1673	C₂×C₁₃₇	semidirect
240. #1668	C₂×C₁₃₇	semidirect
241. #1667	C₂×C₁₃₇	semidirect
242. C₂×#335	C₂×C₁₃₇	semidirect
243. #1540	C₂×C₁₃₇	semidirect
338. #41	D₈×C₁₃₇	semidirect
434. C₈	#135×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₇	2
#3035×C₁₃₇	2
C₈×#41×C₁₃₇	2
#1682×C₁₃₇	2
#1681×C₁₃₇	2
#1674×C₁₃₇	2
#1673×C₁₃₇	2
#1668×C₁₃₇	2
#1667×C₁₃₇	2
C₂×#335×C₁₃₇	2
#1540×C₁₃₇	2
#19125	137

Subgroups — C137×#19125 #46868511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19125 #46868511