Subgroups — C₁₃₇×#26501 #46875887

Order 35072.

724 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3164×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3081×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#2948×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2063×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#2027×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1978×C₁₃₇	C₂	1	1	yes	maximal, characteristic
7	#1970×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1952×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1945×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1856×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1772×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
13	#544×C₁₃₇	C₂²	1	2	yes	—
14	#518×C₁₃₇	C₂²	1	2	yes	—
15	#501×C₁₃₇	—	2	2	no	—
16	#500×C₁₃₇	—	2	1	no	—
17	#499×C₁₃₇	—	2	1	no	—
18	#478×C₁₃₇	—	2	2	no	—
19	#478×C₁₃₇	—	2	2	no	—
20	#476×C₁₃₇	—	2	2	no	—
21	#474×C₁₃₇	—	2	2	no	—
22	#442×C₁₃₇	—	2	2	no	—
23	C₄×D₈×C₁₃₇	—	2	1	no	—
24	C₄×D₈×C₁₃₇	—	2	1	no	—
25	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
27	#428×C₁₃₇	C₂²	1	2	yes	—
28	#428×C₁₃₇	C₂²	1	2	yes	—
29	#427×C₁₃₇	C₂²	1	1	yes	characteristic
30	#427×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
32	#424×C₁₃₇	C₂²	1	2	yes	—
33	#418×C₁₃₇	C₂²	1	2	yes	—
34	#418×C₁₃₇	C₂²	1	2	yes	—
35	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
36	C₂×#100×C₁₃₇	C₂²	1	2	yes	—
37	#405×C₁₃₇	C₂²	1	1	yes	characteristic
38	#405×C₁₃₇	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
40	#390×C₁₃₇	C₂²	1	2	yes	—
41	#343×C₁₃₇	C₂²	1	2	yes	—
42	#339×C₁₃₇	C₂²	1	1	yes	characteristic
43	#339×C₁₃₇	C₂²	1	2	yes	—
44	#339×C₁₃₇	C₂²	1	1	yes	characteristic
45	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
46	C₂×D₈×C₁₃₇	—	2	1	no	—
47	C₂×D₈×C₁₃₇	—	2	1	no	—
48	C₂×D₈×C₁₃₇	—	2	2	no	—
49	C₂×D₈×C₁₃₇	Q₈	1	4	yes	complemented
50	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
52	C₂×M₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
57	#125×C₁₃₇	—	2	2	no	—
58	#125×C₁₃₇	—	2	2	no	—
59	C₂²⋊Q₈×C₁₃₇	—	4	2	no	—
60	C₄×D₄×C₁₃₇	—	2	2	no	—
61	C₄×D₄×C₁₃₇	—	2	2	no	—
62	C₄×D₄×C₁₃₇	—	2	2	no	—
63	C₄×D₄×C₁₃₇	—	2	2	no	—
64	#117×C₁₃₇	C₂³	1	1	yes	characteristic
65	#117×C₁₃₇	C₂³	1	1	yes	characteristic
66	#117×C₁₃₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₁₃₇	—	4	2	no	—
69	C₂×C₂²⋊C₄×C₁₃₇	—	4	2	no	—
70	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
71	#108×C₁₃₇	—	2	2	no	—
72	#108×C₁₃₇	—	2	2	no	—
73	#107×C₁₃₇	—	2	1	no	—
74	#107×C₁₃₇	—	2	1	no	—
75	#107×C₁₃₇	—	2	2	no	—
76	#107×C₁₃₇	—	2	2	no	—
77	#107×C₁₃₇	D₄	1	1	yes	characteristic
78	#107×C₁₃₇	D₄	1	1	yes	characteristic
79	#107×C₁₃₇	D₄	1	1	yes	characteristic
80	#107×C₁₃₇	D₄	1	1	yes	characteristic
81	#106×C₁₃₇	—	2	2	no	—
82	#106×C₁₃₇	—	2	2	no	—
83	#106×C₁₃₇	D₄	1	2	yes	—
84	#106×C₁₃₇	D₄	1	1	yes	characteristic
85	#106×C₁₃₇	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₁₃₇	—	2	2	no	—
87	C₄⋊C₈×C₁₃₇	—	2	2	no	—
88	#102×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	#102×C₁₃₇	—	2	2	no	—
94	#100×C₁₃₇	—	2	4	no	—
95	#97×C₁₃₇	—	2	1	no	—
96	#97×C₁₃₇	—	2	1	no	—
97	C₄×C₈×C₁₃₇	—	2	1	no	—
98	C₄×C₈×C₁₃₇	—	2	1	no	—
99	C₄×C₈×C₁₃₇	—	2	2	no	—
100	C₂⁴×C₁₃₇	—	2	2	no	—
101	C₂×Q₈×C₁₃₇	—	4	2	no	—
102	C₂×D₄×C₁₃₇	—	2	2	no	—
103	C₂×D₄×C₁₃₇	—	2	2	no	—
104	C₂×D₄×C₁₃₇	—	2	2	no	—
105	C₂×D₄×C₁₃₇	—	2	2	no	—
106	C₂×D₄×C₁₃₇	—	2	2	no	—
107	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
108	C₂²×C₄×C₁₃₇	—	2	1	no	—
109	C₂²×C₄×C₁₃₇	—	2	2	no	—
110	C₂²×C₄×C₁₃₇	—	4	2	no	—
111	C₂²×C₄×C₁₃₇	—	4	2	no	—
112	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
113	D₈×C₁₃₇	—	2	4	no	—
114	D₈×C₁₃₇	—	2	4	no	—
115	M₁₆×C₁₃₇	—	2	1	no	—
116	M₁₆×C₁₃₇	—	2	2	no	—
117	M₁₆×C₁₃₇	—	2	2	no	—
118	M₁₆×C₁₃₇	—	2	1	no	—
119	M₁₆×C₁₃₇	—	2	1	no	—
120	M₁₆×C₁₃₇	#41	1	2	yes	—
121	C₂×C₈×C₁₃₇	—	2	1	no	—
122	C₂×C₈×C₁₃₇	—	2	1	no	—
123	C₂×C₈×C₁₃₇	—	2	1	no	—
124	C₂×C₈×C₁₃₇	—	2	1	no	—
125	C₂×C₈×C₁₃₇	—	2	2	no	—
126	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
131	C₂×C₈×C₁₃₇	#41	1	2	yes	—
132	C₄⋊C₄×C₁₃₇	—	2	2	no	—
133	C₄⋊C₄×C₁₃₇	—	2	2	no	—
134	C₄⋊C₄×C₁₃₇	—	4	2	no	—
135	C₄⋊C₄×C₁₃₇	—	4	2	no	—
136	C₄⋊C₄×C₁₃₇	—	4	2	no	—
137	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
144	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
145	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
146	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
147	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
148	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
149	C₄²×C₁₃₇	—	2	1	no	—
150	C₄²×C₁₃₇	—	2	1	no	—
151	C₄²×C₁₃₇	—	2	2	no	—
152	C₄²×C₁₃₇	—	2	1	no	—
153	C₄²×C₁₃₇	—	2	1	no	—
154	C₂³×C₁₃₇	—	2	4	no	—
155	C₂³×C₁₃₇	—	4	2	no	—
156	C₂³×C₁₃₇	—	4	2	no	—
157	C₂³×C₁₃₇	—	8	2	no	—
158	C₂³×C₁₃₇	—	8	2	no	—
159	C₂³×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
160	Q₈×C₁₃₇	—	16	2	no	—
161	D₄×C₁₃₇	—	2	4	no	—
162	D₄×C₁₃₇	—	2	4	no	—
163	D₄×C₁₃₇	—	4	2	no	—
164	D₄×C₁₃₇	—	4	2	no	—
165	C₂×C₄×C₁₃₇	—	2	2	no	—
166	C₂×C₄×C₁₃₇	—	2	1	no	—
167	C₂×C₄×C₁₃₇	—	2	1	no	—
168	C₂×C₄×C₁₃₇	—	2	2	no	—
169	C₂×C₄×C₁₃₇	—	2	2	no	—
170	C₂×C₄×C₁₃₇	—	2	1	no	—
171	C₂×C₄×C₁₃₇	—	2	1	no	—
172	C₂×C₄×C₁₃₇	—	4	1	no	—
173	C₂×C₄×C₁₃₇	—	4	1	no	—
174	C₂×C₄×C₁₃₇	—	4	2	no	—
175	C₂×C₄×C₁₃₇	—	8	2	no	—
176	C₂×C₄×C₁₃₇	—	16	2	no	—
177	C₂×C₄×C₁₃₇	—	16	2	no	—
178	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈×C₁₃₇	—	2	1	no	—
185	C₈×C₁₃₇	—	2	1	no	—
186	C₈×C₁₃₇	—	2	2	no	—
187	C₈×C₁₃₇	—	4	2	no	—
188	C₈×C₁₃₇	—	4	1	no	—
189	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
190	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
191	C₂²×C₁₃₇	—	2	1	no	—
192	C₂²×C₁₃₇	—	2	1	no	—
193	C₂²×C₁₃₇	—	4	4	no	—
194	C₂²×C₁₃₇	—	8	4	no	—
195	C₂²×C₁₃₇	—	8	2	no	—
196	C₂²×C₁₃₇	—	8	2	no	—
197	C₂²×C₁₃₇	—	16	2	no	—
198	C₂²×C₁₃₇	#542	1	1	yes	characteristic
199	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
200	C₂²×C₁₃₇	#554	1	1	yes	characteristic
201	C₄×C₁₃₇	—	4	2	no	—
202	C₄×C₁₃₇	—	4	1	no	—
203	C₄×C₁₃₇	—	4	2	no	—
204	C₄×C₁₃₇	—	4	1	no	—
205	C₄×C₁₃₇	—	4	1	no	—
206	C₄×C₁₃₇	—	4	2	no	—
207	C₄×C₁₃₇	—	4	1	no	—
208	C₄×C₁₃₇	—	8	2	no	—
209	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
210	C₄×C₁₃₇	#532	1	1	yes	characteristic
211	C₄×C₁₃₇	#548	1	1	yes	characteristic
212	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
213	C₂×C₁₃₇	—	2	1	no	—
214	C₂×C₁₃₇	—	2	1	no	—
215	C₂×C₁₃₇	—	8	4	no	—
216	C₂×C₁₃₇	#2452	1	1	yes	characteristic
217	C₂×C₁₃₇	#3290	1	1	yes	characteristic
218	C₂×C₁₃₇	#3292	1	1	yes	characteristic
219	#26501	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₁₃₇	#26501	1	1	yes	characteristic, direct product factor, Sylow
221	#3164	C₂×C₁₃₇	1	1	yes	characteristic, complemented
222	#3081	C₂×C₁₃₇	1	2	yes	complemented
223	#2948	C₂×C₁₃₇	1	1	yes	characteristic
224	#2063	C₂×C₁₃₇	1	2	yes	complemented
225	#2027	C₂×C₁₃₇	1	2	yes	complemented
226	#1978	C₂×C₁₃₇	1	1	yes	characteristic
227	#1970	C₂×C₁₃₇	1	1	yes	characteristic
228	#1952	C₂×C₁₃₇	1	1	yes	characteristic, complemented
229	#1945	C₂×C₁₃₇	1	1	yes	characteristic, complemented
230	#1856	C₂×C₁₃₇	1	2	yes	complemented
231	#1772	C₂×C₁₃₇	1	1	yes	characteristic, complemented
232	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
233	#544	C₂²×C₁₃₇	1	2	yes	—
234	#518	C₂²×C₁₃₇	1	2	yes	—
235	#501	—	2	2	no	—
236	#500	—	2	1	no	—
237	#499	—	2	1	no	—
238	#478	—	2	2	no	—
239	#478	—	2	2	no	—
240	#476	—	2	2	no	—
241	#474	—	2	2	no	—
242	#442	—	2	2	no	—
243	C₄×D₈	—	2	1	no	—
244	C₄×D₈	—	2	1	no	—
245	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
246	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
247	#428	C₂²×C₁₃₇	1	2	yes	—
248	#428	C₂²×C₁₃₇	1	2	yes	—
249	#427	C₂²×C₁₃₇	1	1	yes	characteristic
250	#427	C₂²×C₁₃₇	1	1	yes	characteristic
251	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
252	#424	C₂²×C₁₃₇	1	2	yes	—
253	#418	C₂²×C₁₃₇	1	2	yes	—
254	#418	C₂²×C₁₃₇	1	2	yes	—
255	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
256	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
257	#405	C₂²×C₁₃₇	1	1	yes	characteristic
258	#405	C₂²×C₁₃₇	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
260	#390	C₂²×C₁₃₇	1	2	yes	—
261	#343	C₂²×C₁₃₇	1	2	yes	—
262	#339	C₂²×C₁₃₇	1	1	yes	characteristic
263	#339	C₂²×C₁₃₇	1	2	yes	—
264	#339	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
266	C₂×D₈	—	2	1	no	—
267	C₂×D₈	—	2	1	no	—
268	C₂×D₈	—	2	2	no	—
269	C₂×D₈	Q₈×C₁₃₇	1	4	yes	complemented
270	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
271	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
272	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
275	C₄⋊Q₈	—	2	2	no	—
276	C₄⋊Q₈	—	2	2	no	—
277	#125	—	2	2	no	—
278	#125	—	2	2	no	—
279	C₂²⋊Q₈	—	4	2	no	—
280	C₄×D₄	—	2	2	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	#117	C₂³×C₁₃₇	1	1	yes	characteristic
285	#117	C₂³×C₁₃₇	1	2	yes	—
286	#117	C₂³×C₁₃₇	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
288	C₂×C₂²⋊C₄	—	4	2	no	—
289	C₂×C₂²⋊C₄	—	4	2	no	—
290	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
291	#108	—	2	2	no	—
292	#108	—	2	2	no	—
293	#107	—	2	1	no	—
294	#107	—	2	1	no	—
295	#107	—	2	2	no	—
296	#107	—	2	2	no	—
297	#107	D₄×C₁₃₇	1	1	yes	characteristic
298	#107	D₄×C₁₃₇	1	1	yes	characteristic
299	#107	D₄×C₁₃₇	1	1	yes	characteristic
300	#107	D₄×C₁₃₇	1	1	yes	characteristic
301	#106	—	2	2	no	—
302	#106	—	2	2	no	—
303	#106	D₄×C₁₃₇	1	2	yes	—
304	#106	D₄×C₁₃₇	1	1	yes	characteristic
305	#106	D₄×C₁₃₇	1	1	yes	characteristic
306	C₄⋊C₈	—	2	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#100	—	2	4	no	—
315	#97	—	2	1	no	—
316	#97	—	2	1	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	—	2	1	no	—
319	C₄×C₈	—	2	2	no	—
320	C₂⁴	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
333	D₈	—	2	4	no	—
334	D₈	—	2	4	no	—
335	M₁₆	—	2	1	no	—
336	M₁₆	—	2	2	no	—
337	M₁₆	—	2	2	no	—
338	M₁₆	—	2	1	no	—
339	M₁₆	—	2	1	no	—
340	M₁₆	#41×C₁₃₇	1	2	yes	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
347	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
349	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
350	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
351	C₂×C₈	#41×C₁₃₇	1	2	yes	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
358	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
359	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
360	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
361	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
362	C₄⋊C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	2	2	no	—
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₂²⋊C₄	—	4	2	no	—
368	C₂²⋊C₄	—	8	4	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	1	no	—
373	C₄²	—	2	1	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	4	2	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	8	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
380	Q₈	—	16	2	no	—
381	D₄	—	2	4	no	—
382	D₄	—	2	4	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	2	no	—
396	C₂×C₄	—	16	2	no	—
397	C₂×C₄	—	16	2	no	—
398	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
400	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
402	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
403	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
404	C₈	—	2	1	no	—
405	C₈	—	2	1	no	—
406	C₈	—	2	2	no	—
407	C₈	—	4	2	no	—
408	C₈	—	4	1	no	—
409	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
410	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	1	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	8	4	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	8	2	no	—
417	C₂²	—	16	2	no	—
418	C₂²	#542×C₁₃₇	1	1	yes	characteristic
419	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
420	C₂²	#554×C₁₃₇	1	1	yes	characteristic
421	C₄	—	4	2	no	—
422	C₄	—	4	1	no	—
423	C₄	—	4	2	no	—
424	C₄	—	4	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	2	no	—
427	C₄	—	4	1	no	—
428	C₄	—	8	2	no	—
429	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
430	C₄	#532×C₁₃₇	1	1	yes	characteristic
431	C₄	#548×C₁₃₇	1	1	yes	characteristic
432	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
433	C₂	—	2	1	no	—
434	C₂	—	2	1	no	—
435	C₂	—	8	4	no	—
436	C₂	#2452×C₁₃₇	1	1	yes	characteristic
437	C₂	#3290×C₁₃₇	1	1	yes	characteristic
438	C₂	#3292×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3164×C₁₃₇	C₂	semidirect
2. #3081×C₁₃₇	C₂	semidirect
4. #2063×C₁₃₇	C₂	semidirect
5. #2027×C₁₃₇	C₂	semidirect
8. #1952×C₁₃₇	C₂	semidirect
9. #1945×C₁₃₇	C₂	semidirect
10. #1856×C₁₃₇	C₂	semidirect
11. #1772×C₁₃₇	C₂	semidirect
49. C₂×D₈×C₁₃₇	Q₈	semidirect
219. #26501	C₁₃₇	direct (N × H)
220. C₁₃₇	#26501	direct (N × H)
221. #3164	C₂×C₁₃₇	semidirect
222. #3081	C₂×C₁₃₇	semidirect
224. #2063	C₂×C₁₃₇	semidirect
225. #2027	C₂×C₁₃₇	semidirect
228. #1952	C₂×C₁₃₇	semidirect
229. #1945	C₂×C₁₃₇	semidirect
230. #1856	C₂×C₁₃₇	semidirect
231. #1772	C₂×C₁₃₇	semidirect
269. C₂×D₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3164×C₁₃₇	2
#3081×C₁₃₇	2
#2948×C₁₃₇	2
#2063×C₁₃₇	2
#2027×C₁₃₇	2
#1978×C₁₃₇	2
#1970×C₁₃₇	2
#1952×C₁₃₇	2
#1945×C₁₃₇	2
#1856×C₁₃₇	2
#1772×C₁₃₇	2
#26501	137

Subgroups — C137×#26501 #46875887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#26501 #46875887