Subgroups — C₁₃₇×#26504 #46875890

Order 35072.

720 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3163×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	#1972×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1951×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1946×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#1856×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1773×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	#499×C₁₃₇	—	2	1	no	—
17	#498×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	1	no	—
23	#442×C₁₃₇	—	2	1	no	—
24	C₄×D₈×C₁₃₇	—	2	2	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×#106×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×#97×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	2	no	—
74	#107×C₁₃₇	—	2	2	no	—
75	#107×C₁₃₇	D₄	1	2	yes	—
76	#107×C₁₃₇	D₄	1	2	yes	—
77	#106×C₁₃₇	—	2	1	no	—
78	#106×C₁₃₇	—	2	1	no	—
79	#106×C₁₃₇	—	2	2	no	—
80	#106×C₁₃₇	—	2	2	no	—
81	#106×C₁₃₇	D₄	1	1	yes	characteristic
82	#106×C₁₃₇	D₄	1	1	yes	characteristic
83	#106×C₁₃₇	D₄	1	2	yes	—
84	C₄⋊C₈×C₁₃₇	—	2	2	no	—
85	C₄⋊C₈×C₁₃₇	—	2	2	no	—
86	#102×C₁₃₇	—	2	2	no	—
87	#102×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	#100×C₁₃₇	—	2	4	no	—
93	#97×C₁₃₇	—	2	1	no	—
94	#97×C₁₃₇	—	2	1	no	—
95	#97×C₁₃₇	—	2	2	no	—
96	C₄×C₈×C₁₃₇	—	2	1	no	—
97	C₄×C₈×C₁₃₇	—	2	1	no	—
98	C₂⁴×C₁₃₇	—	2	2	no	—
99	C₂×Q₈×C₁₃₇	—	4	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	—	2	2	no	—
105	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
106	C₂²×C₄×C₁₃₇	—	2	1	no	—
107	C₂²×C₄×C₁₃₇	—	2	2	no	—
108	C₂²×C₄×C₁₃₇	—	4	2	no	—
109	C₂²×C₄×C₁₃₇	—	4	2	no	—
110	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
111	D₈×C₁₃₇	—	2	4	no	—
112	D₈×C₁₃₇	—	2	4	no	—
113	M₁₆×C₁₃₇	—	2	1	no	—
114	M₁₆×C₁₃₇	—	2	2	no	—
115	M₁₆×C₁₃₇	—	2	2	no	—
116	M₁₆×C₁₃₇	—	2	1	no	—
117	M₁₆×C₁₃₇	—	2	1	no	—
118	M₁₆×C₁₃₇	#41	1	2	yes	—
119	C₂×C₈×C₁₃₇	—	2	1	no	—
120	C₂×C₈×C₁₃₇	—	2	1	no	—
121	C₂×C₈×C₁₃₇	—	2	1	no	—
122	C₂×C₈×C₁₃₇	—	2	1	no	—
123	C₂×C₈×C₁₃₇	—	2	2	no	—
124	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
129	C₂×C₈×C₁₃₇	#41	1	2	yes	—
130	C₄⋊C₄×C₁₃₇	—	2	2	no	—
131	C₄⋊C₄×C₁₃₇	—	2	2	no	—
132	C₄⋊C₄×C₁₃₇	—	4	2	no	—
133	C₄⋊C₄×C₁₃₇	—	4	2	no	—
134	C₄⋊C₄×C₁₃₇	—	4	2	no	—
135	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
138	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
140	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
141	C₂²⋊C₄×C₁₃₇	—	2	1	no	—
142	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
143	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
144	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
145	C₄²×C₁₃₇	—	2	1	no	—
146	C₄²×C₁₃₇	—	2	1	no	—
147	C₄²×C₁₃₇	—	2	2	no	—
148	C₄²×C₁₃₇	—	2	1	no	—
149	C₄²×C₁₃₇	—	2	1	no	—
150	C₂³×C₁₃₇	—	2	4	no	—
151	C₂³×C₁₃₇	—	4	2	no	—
152	C₂³×C₁₃₇	—	4	2	no	—
153	C₂³×C₁₃₇	—	8	2	no	—
154	C₂³×C₁₃₇	—	8	2	no	—
155	C₂³×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
156	Q₈×C₁₃₇	—	16	2	no	—
157	D₄×C₁₃₇	—	2	4	no	—
158	D₄×C₁₃₇	—	2	4	no	—
159	D₄×C₁₃₇	—	4	2	no	—
160	D₄×C₁₃₇	—	4	2	no	—
161	C₂×C₄×C₁₃₇	—	2	2	no	—
162	C₂×C₄×C₁₃₇	—	2	2	no	—
163	C₂×C₄×C₁₃₇	—	2	2	no	—
164	C₂×C₄×C₁₃₇	—	2	2	no	—
165	C₂×C₄×C₁₃₇	—	4	2	no	—
166	C₂×C₄×C₁₃₇	—	4	1	no	—
167	C₂×C₄×C₁₃₇	—	4	2	no	—
168	C₂×C₄×C₁₃₇	—	8	2	no	—
169	C₂×C₄×C₁₃₇	—	16	2	no	—
170	C₂×C₄×C₁₃₇	—	16	2	no	—
171	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
177	C₈×C₁₃₇	—	2	1	no	—
178	C₈×C₁₃₇	—	2	1	no	—
179	C₈×C₁₃₇	—	2	2	no	—
180	C₈×C₁₃₇	—	4	2	no	—
181	C₈×C₁₃₇	—	4	1	no	—
182	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
183	C₈×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
184	C₂²×C₁₃₇	—	2	2	no	—
185	C₂²×C₁₃₇	—	4	4	no	—
186	C₂²×C₁₃₇	—	8	4	no	—
187	C₂²×C₁₃₇	—	8	2	no	—
188	C₂²×C₁₃₇	—	8	2	no	—
189	C₂²×C₁₃₇	—	16	2	no	—
190	C₂²×C₁₃₇	#542	1	1	yes	characteristic
191	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
192	C₂²×C₁₃₇	#554	1	1	yes	characteristic
193	C₄×C₁₃₇	—	4	2	no	—
194	C₄×C₁₃₇	—	4	2	no	—
195	C₄×C₁₃₇	—	4	2	no	—
196	C₄×C₁₃₇	—	4	2	no	—
197	C₄×C₁₃₇	—	8	1	no	—
198	C₄×C₁₃₇	—	8	2	no	—
199	C₄×C₁₃₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
200	C₄×C₁₃₇	#532	1	1	yes	characteristic
201	C₄×C₁₃₇	#548	1	1	yes	characteristic
202	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
203	C₂×C₁₃₇	—	2	1	no	—
204	C₂×C₁₃₇	—	2	1	no	—
205	C₂×C₁₃₇	—	8	4	no	—
206	C₂×C₁₃₇	#2452	1	1	yes	characteristic
207	C₂×C₁₃₇	#3288	1	2	yes	—
208	#26504	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₁₃₇	#26504	1	1	yes	characteristic, direct product factor, Sylow
210	#3163	C₂×C₁₃₇	1	1	yes	characteristic, complemented
211	#3081	C₂×C₁₃₇	1	2	yes	complemented
212	#2948	C₂×C₁₃₇	1	1	yes	characteristic
213	#2063	C₂×C₁₃₇	1	2	yes	complemented
214	#2027	C₂×C₁₃₇	1	2	yes	complemented
215	#1978	C₂×C₁₃₇	1	1	yes	characteristic
216	#1972	C₂×C₁₃₇	1	1	yes	characteristic
217	#1951	C₂×C₁₃₇	1	1	yes	characteristic, complemented
218	#1946	C₂×C₁₃₇	1	1	yes	characteristic, complemented
219	#1856	C₂×C₁₃₇	1	2	yes	complemented
220	#1773	C₂×C₁₃₇	1	1	yes	characteristic, complemented
221	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
222	#544	C₂²×C₁₃₇	1	2	yes	—
223	#518	C₂²×C₁₃₇	1	2	yes	—
224	#501	—	2	2	no	—
225	#499	—	2	1	no	—
226	#498	—	2	1	no	—
227	#478	—	2	2	no	—
228	#478	—	2	2	no	—
229	#476	—	2	2	no	—
230	#474	—	2	2	no	—
231	#442	—	2	1	no	—
232	#442	—	2	1	no	—
233	C₄×D₈	—	2	2	no	—
234	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
235	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
236	#428	C₂²×C₁₃₇	1	2	yes	—
237	#428	C₂²×C₁₃₇	1	2	yes	—
238	#427	C₂²×C₁₃₇	1	1	yes	characteristic
239	#427	C₂²×C₁₃₇	1	1	yes	characteristic
240	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
241	#424	C₂²×C₁₃₇	1	2	yes	—
242	#418	C₂²×C₁₃₇	1	2	yes	—
243	#418	C₂²×C₁₃₇	1	2	yes	—
244	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
245	C₂×#100	C₂²×C₁₃₇	1	2	yes	—
246	#405	C₂²×C₁₃₇	1	1	yes	characteristic
247	#405	C₂²×C₁₃₇	1	1	yes	characteristic
248	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
249	#390	C₂²×C₁₃₇	1	2	yes	—
250	#343	C₂²×C₁₃₇	1	2	yes	—
251	#339	C₂²×C₁₃₇	1	1	yes	characteristic
252	#339	C₂²×C₁₃₇	1	2	yes	—
253	#339	C₂²×C₁₃₇	1	1	yes	characteristic
254	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
255	C₂×D₈	—	2	1	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	2	no	—
258	C₂×D₈	Q₈×C₁₃₇	1	4	yes	complemented
259	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
260	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
261	C₂×M₁₆	C₂³×C₁₃₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	—	2	2	no	—
266	#125	—	2	2	no	—
267	#125	—	2	2	no	—
268	C₂²⋊Q₈	—	4	2	no	—
269	C₄×D₄	—	2	2	no	—
270	C₄×D₄	—	2	2	no	—
271	C₄×D₄	—	2	2	no	—
272	C₄×D₄	—	2	2	no	—
273	#117	C₂³×C₁₃₇	1	1	yes	characteristic
274	#117	C₂³×C₁₃₇	1	2	yes	—
275	#117	C₂³×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
277	C₂×C₂²⋊C₄	—	4	2	no	—
278	C₂×C₂²⋊C₄	—	4	2	no	—
279	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
280	#108	—	2	2	no	—
281	#108	—	2	2	no	—
282	#107	—	2	2	no	—
283	#107	—	2	2	no	—
284	#107	D₄×C₁₃₇	1	2	yes	—
285	#107	D₄×C₁₃₇	1	2	yes	—
286	#106	—	2	1	no	—
287	#106	—	2	1	no	—
288	#106	—	2	2	no	—
289	#106	—	2	2	no	—
290	#106	D₄×C₁₃₇	1	1	yes	characteristic
291	#106	D₄×C₁₃₇	1	1	yes	characteristic
292	#106	D₄×C₁₃₇	1	2	yes	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	—	2	2	no	—
299	#102	—	2	2	no	—
300	#102	—	2	2	no	—
301	#100	—	2	4	no	—
302	#97	—	2	1	no	—
303	#97	—	2	1	no	—
304	#97	—	2	2	no	—
305	C₄×C₈	—	2	1	no	—
306	C₄×C₈	—	2	1	no	—
307	C₂⁴	—	2	2	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	4	2	no	—
319	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
320	D₈	—	2	4	no	—
321	D₈	—	2	4	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	2	1	no	—
327	M₁₆	#41×C₁₃₇	1	2	yes	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
334	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
336	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
337	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
338	C₂×C₈	#41×C₁₃₇	1	2	yes	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
345	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
346	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
347	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	2	2	no	—
350	C₂²⋊C₄	—	2	1	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₂²⋊C₄	—	8	4	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	1	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	8	2	no	—
363	C₂³	—	8	2	no	—
364	C₂³	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
365	Q₈	—	16	2	no	—
366	D₄	—	2	4	no	—
367	D₄	—	2	4	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	2	no	—
378	C₂×C₄	—	16	2	no	—
379	C₂×C₄	—	16	2	no	—
380	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
385	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
386	C₈	—	2	1	no	—
387	C₈	—	2	1	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	2	no	—
390	C₈	—	4	1	no	—
391	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
392	C₈	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	8	4	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	16	2	no	—
399	C₂²	#542×C₁₃₇	1	1	yes	characteristic
400	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
401	C₂²	#554×C₁₃₇	1	1	yes	characteristic
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	—	8	1	no	—
407	C₄	—	8	2	no	—
408	C₄	C₂×C₂²⋊Q₈×C₁₃₇	1	1	yes	characteristic
409	C₄	#532×C₁₃₇	1	1	yes	characteristic
410	C₄	#548×C₁₃₇	1	1	yes	characteristic
411	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	4	no	—
415	C₂	#2452×C₁₃₇	1	1	yes	characteristic
416	C₂	#3288×C₁₃₇	1	2	yes	—

	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. #3163×C₁₃₇	C₂	semidirect
2. #3081×C₁₃₇	C₂	semidirect
4. #2063×C₁₃₇	C₂	semidirect
5. #2027×C₁₃₇	C₂	semidirect
8. #1951×C₁₃₇	C₂	semidirect
9. #1946×C₁₃₇	C₂	semidirect
10. #1856×C₁₃₇	C₂	semidirect
11. #1773×C₁₃₇	C₂	semidirect
49. C₂×D₈×C₁₃₇	Q₈	semidirect
208. #26504	C₁₃₇	direct (N × H)
209. C₁₃₇	#26504	direct (N × H)
210. #3163	C₂×C₁₃₇	semidirect
211. #3081	C₂×C₁₃₇	semidirect
213. #2063	C₂×C₁₃₇	semidirect
214. #2027	C₂×C₁₃₇	semidirect
217. #1951	C₂×C₁₃₇	semidirect
218. #1946	C₂×C₁₃₇	semidirect
219. #1856	C₂×C₁₃₇	semidirect
220. #1773	C₂×C₁₃₇	semidirect
258. 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
#3163×C₁₃₇	2
#3081×C₁₃₇	2
#2948×C₁₃₇	2
#2063×C₁₃₇	2
#2027×C₁₃₇	2
#1978×C₁₃₇	2
#1972×C₁₃₇	2
#1951×C₁₃₇	2
#1946×C₁₃₇	2
#1856×C₁₃₇	2
#1773×C₁₃₇	2
#26504	137

Subgroups — C137×#26504 #46875890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#26504 #46875890