Subgroups — C₁₃₇×#28810 #46878196

Order 35072.

734 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2550×C₁₃₇	C₂	1	2	yes	maximal
4	#2059×C₁₃₇	C₂	1	2	yes	maximal
5	#2057×C₁₃₇	C₂	1	2	yes	maximal
6	#1949×C₁₃₇	C₂	1	2	yes	maximal
7	#1941×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1941×C₁₃₇	C₂	1	1	yes	maximal, characteristic
9	#1846×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	C₄×#106×C₁₃₇	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
13	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
14	#499×C₁₃₇	—	2	4	no	—
15	#498×C₁₃₇	—	2	4	no	—
16	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
17	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
18	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	#398×C₁₃₇	—	2	2	no	—
28	#395×C₁₃₇	—	2	2	no	—
29	#391×C₁₃₇	—	2	2	no	—
30	#389×C₁₃₇	—	2	2	no	—
31	#389×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	C₂²	1	2	yes	—
33	#382×C₁₃₇	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
36	#340×C₁₃₇	C₂²	1	2	yes	—
37	#340×C₁₃₇	C₂²	1	2	yes	—
38	#340×C₁₃₇	C₂²	1	2	yes	—
39	#336×C₁₃₇	C₂²	1	1	yes	characteristic
40	#336×C₁₃₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃₇	—	2	2	no	—
42	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
47	C₄⋊Q₈×C₁₃₇	—	4	2	no	—
48	#125×C₁₃₇	—	2	4	no	—
49	#125×C₁₃₇	—	4	2	no	—
50	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₁₃₇	—	2	2	no	—
61	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
64	#107×C₁₃₇	—	2	4	no	—
65	#107×C₁₃₇	—	4	2	no	—
66	#106×C₁₃₇	—	2	2	no	—
67	#106×C₁₃₇	—	2	2	no	—
68	#106×C₁₃₇	—	2	4	no	—
69	#106×C₁₃₇	D₄	1	4	yes	—
70	#106×C₁₃₇	Q₈	1	4	yes	complemented
71	C₄⋊C₈×C₁₃₇	Q₈	1	4	yes	complemented
72	C₄×C₈×C₁₃₇	—	2	2	no	—
73	C₄×C₈×C₁₃₇	—	2	2	no	—
74	C₄×C₈×C₁₃₇	Q₈	1	4	yes	complemented
75	#95×C₁₃₇	—	2	2	no	—
76	#95×C₁₃₇	—	2	2	no	—
77	#95×C₁₃₇	—	2	2	no	—
78	#95×C₁₃₇	—	2	2	no	—
79	#95×C₁₃₇	—	2	2	no	—
80	#95×C₁₃₇	D₄	1	2	yes	—
81	C₂×Q₈×C₁₃₇	—	4	4	no	—
82	C₂×Q₈×C₁₃₇	—	4	4	no	—
83	C₂×Q₈×C₁₃₇	—	8	2	no	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	—	2	2	no	—
86	C₂²×C₄×C₁₃₇	—	2	2	no	—
87	C₂²×C₄×C₁₃₇	—	2	2	no	—
88	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	—	2	1	no	—
94	C₂×C₈×C₁₃₇	—	2	1	no	—
95	C₂×C₈×C₁₃₇	—	2	1	no	—
96	C₂×C₈×C₁₃₇	—	2	1	no	—
97	C₂×C₈×C₁₃₇	—	2	1	no	—
98	C₂×C₈×C₁₃₇	—	2	1	no	—
99	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
102	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
105	C₄⋊C₄×C₁₃₇	—	2	4	no	—
106	C₄⋊C₄×C₁₃₇	—	2	2	no	—
107	C₄⋊C₄×C₁₃₇	—	2	2	no	—
108	C₄⋊C₄×C₁₃₇	—	2	2	no	—
109	C₄⋊C₄×C₁₃₇	—	2	2	no	—
110	C₄⋊C₄×C₁₃₇	—	2	2	no	—
111	C₄⋊C₄×C₁₃₇	—	2	2	no	—
112	C₄⋊C₄×C₁₃₇	—	4	4	no	—
113	C₄⋊C₄×C₁₃₇	—	4	4	no	—
114	C₄⋊C₄×C₁₃₇	—	4	4	no	—
115	C₄⋊C₄×C₁₃₇	—	4	4	no	—
116	C₄⋊C₄×C₁₃₇	—	8	2	no	—
117	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
119	C₄²×C₁₃₇	—	2	1	no	—
120	C₄²×C₁₃₇	—	2	1	no	—
121	C₄²×C₁₃₇	—	2	1	no	—
122	C₄²×C₁₃₇	—	2	1	no	—
123	C₄²×C₁₃₇	—	8	2	no	—
124	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
125	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
126	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
127	Q₈×C₁₃₇	—	8	8	no	—
128	C₂×C₄×C₁₃₇	—	2	2	no	—
129	C₂×C₄×C₁₃₇	—	2	2	no	—
130	C₂×C₄×C₁₃₇	—	2	2	no	—
131	C₂×C₄×C₁₃₇	—	2	2	no	—
132	C₂×C₄×C₁₃₇	—	2	2	no	—
133	C₂×C₄×C₁₃₇	—	2	4	no	—
134	C₂×C₄×C₁₃₇	—	2	4	no	—
135	C₂×C₄×C₁₃₇	—	2	2	no	—
136	C₂×C₄×C₁₃₇	—	2	2	no	—
137	C₂×C₄×C₁₃₇	—	4	2	no	—
138	C₂×C₄×C₁₃₇	—	4	2	no	—
139	C₂×C₄×C₁₃₇	—	4	2	no	—
140	C₂×C₄×C₁₃₇	—	4	2	no	—
141	C₂×C₄×C₁₃₇	—	4	2	no	—
142	C₂×C₄×C₁₃₇	—	4	2	no	—
143	C₂×C₄×C₁₃₇	—	4	2	no	—
144	C₂×C₄×C₁₃₇	—	4	2	no	—
145	C₂×C₄×C₁₃₇	—	4	2	no	—
146	C₂×C₄×C₁₃₇	—	4	2	no	—
147	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
148	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₁₃₇	#135	1	4	yes	—
150	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃₇	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈×C₁₃₇	—	2	2	no	—
157	C₈×C₁₃₇	—	2	2	no	—
158	C₈×C₁₃₇	—	2	2	no	—
159	C₈×C₁₃₇	C₄⋊Q₈	1	4	yes	—
160	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₁₃₇	D₄²	1	1	yes	characteristic
162	C₂²×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
163	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
164	C₂²×C₁₃₇	#577	1	2	yes	—
165	C₄×C₁₃₇	—	2	2	no	—
166	C₄×C₁₃₇	—	2	2	no	—
167	C₄×C₁₃₇	—	2	4	no	—
168	C₄×C₁₃₇	—	4	4	no	—
169	C₄×C₁₃₇	—	4	2	no	—
170	C₄×C₁₃₇	—	4	4	no	—
171	C₄×C₁₃₇	—	8	2	no	—
172	C₄×C₁₃₇	—	8	2	no	—
173	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₄×C₁₃₇	#544	1	1	yes	characteristic
175	C₄×C₁₃₇	D₄×Q₈	1	1	yes	characteristic
176	C₄×C₁₃₇	Q₈²	1	1	yes	characteristic
177	C₂×C₁₃₇	#2551	1	1	yes	characteristic
178	C₂×C₁₃₇	#3160	1	1	yes	characteristic
179	C₂×C₁₃₇	#3163	1	1	yes	characteristic
180	C₂×C₁₃₇	#3282	1	2	yes	—
181	C₂×C₁₃₇	#3381	1	2	yes	—
182	#28810	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₁₃₇	#28810	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic
185	C₂×#498	C₂×C₁₃₇	1	1	yes	characteristic
186	#2550	C₂×C₁₃₇	1	2	yes	—
187	#2059	C₂×C₁₃₇	1	2	yes	—
188	#2057	C₂×C₁₃₇	1	2	yes	—
189	#1949	C₂×C₁₃₇	1	2	yes	—
190	#1941	C₂×C₁₃₇	1	1	yes	characteristic
191	#1941	C₂×C₁₃₇	1	1	yes	characteristic
192	#1846	C₂×C₁₃₇	1	1	yes	characteristic
193	C₄×#106	C₂×C₁₃₇	1	1	yes	characteristic
194	C₈×C₄⋊C₄	C₂×C₁₃₇	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
196	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
197	#499	—	2	4	no	—
198	#498	—	2	4	no	—
199	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
200	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
201	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
202	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
203	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
204	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
205	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
206	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
207	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
210	#398	—	2	2	no	—
211	#395	—	2	2	no	—
212	#391	—	2	2	no	—
213	#389	—	2	2	no	—
214	#389	C₂²×C₁₃₇	1	2	yes	—
215	#389	C₂²×C₁₃₇	1	2	yes	—
216	#382	C₂²×C₁₃₇	1	2	yes	—
217	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
218	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
219	#340	C₂²×C₁₃₇	1	2	yes	—
220	#340	C₂²×C₁₃₇	1	2	yes	—
221	#340	C₂²×C₁₃₇	1	2	yes	—
222	#336	C₂²×C₁₃₇	1	1	yes	characteristic
223	#336	C₂²×C₁₃₇	1	1	yes	characteristic
224	C₂²×Q₈	—	2	2	no	—
225	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
229	C₄⋊Q₈	—	2	4	no	—
230	C₄⋊Q₈	—	4	2	no	—
231	#125	—	2	4	no	—
232	#125	—	4	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	2	no	—
238	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
239	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
240	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
242	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
243	C₂×C₄²	—	2	2	no	—
244	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
246	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
247	#107	—	2	4	no	—
248	#107	—	4	2	no	—
249	#106	—	2	2	no	—
250	#106	—	2	2	no	—
251	#106	—	2	4	no	—
252	#106	D₄×C₁₃₇	1	4	yes	—
253	#106	Q₈×C₁₃₇	1	4	yes	complemented
254	C₄⋊C₈	Q₈×C₁₃₇	1	4	yes	complemented
255	C₄×C₈	—	2	2	no	—
256	C₄×C₈	—	2	2	no	—
257	C₄×C₈	Q₈×C₁₃₇	1	4	yes	complemented
258	#95	—	2	2	no	—
259	#95	—	2	2	no	—
260	#95	—	2	2	no	—
261	#95	—	2	2	no	—
262	#95	—	2	2	no	—
263	#95	D₄×C₁₃₇	1	2	yes	—
264	C₂×Q₈	—	4	4	no	—
265	C₂×Q₈	—	4	4	no	—
266	C₂×Q₈	—	8	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
272	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
275	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
283	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
284	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
285	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
286	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
287	C₂×C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	4	no	—
296	C₄⋊C₄	—	4	4	no	—
297	C₄⋊C₄	—	4	4	no	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄⋊C₄	—	8	2	no	—
300	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
301	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	8	2	no	—
307	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
308	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
309	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
310	Q₈	—	8	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
331	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
332	C₂×C₄	#135×C₁₃₇	1	4	yes	—
333	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
335	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
336	C₂×C₄	C₂²×Q₈×C₁₃₇	1	1	yes	characteristic
337	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
338	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	C₄⋊Q₈×C₁₃₇	1	4	yes	—
343	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
344	C₂²	D₄²×C₁₃₇	1	1	yes	characteristic
345	C₂²	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
346	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
347	C₂²	#577×C₁₃₇	1	2	yes	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	4	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
357	C₄	#544×C₁₃₇	1	1	yes	characteristic
358	C₄	D₄×Q₈×C₁₃₇	1	1	yes	characteristic
359	C₄	Q₈²×C₁₃₇	1	1	yes	characteristic
360	C₂	#2551×C₁₃₇	1	1	yes	characteristic
361	C₂	#3160×C₁₃₇	1	1	yes	characteristic
362	C₂	#3163×C₁₃₇	1	1	yes	characteristic
363	C₂	#3282×C₁₃₇	1	2	yes	—
364	C₂	#3381×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
70. #106×C₁₃₇	Q₈	semidirect
71. C₄⋊C₈×C₁₃₇	Q₈	semidirect
74. C₄×C₈×C₁₃₇	Q₈	semidirect
182. #28810	C₁₃₇	direct (N × H)
183. C₁₃₇	#28810	direct (N × H)
253. #106	Q₈×C₁₃₇	semidirect
254. C₄⋊C₈	Q₈×C₁₃₇	semidirect
257. C₄×C₈	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499×C₁₃₇	2
C₂×#498×C₁₃₇	2
#2550×C₁₃₇	2
#2059×C₁₃₇	2
#2057×C₁₃₇	2
#1949×C₁₃₇	2
#1941×C₁₃₇	2
#1846×C₁₃₇	2
C₄×#106×C₁₃₇	2
C₈×C₄⋊C₄×C₁₃₇	2
#28810	137

Subgroups — C137×#28810 #46878196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28810 #46878196