Subgroups — C₁₃₇×#25835 #46875221

Order 35072.

1082 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#474×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2546×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2059×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1987×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1984×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1864×C₁₃₇	C₂	1	4	yes	maximal
9	C₄×#102×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	#554×C₁₃₇	—	2	4	no	—
11	#554×C₁₃₇	—	2	4	no	—
12	C₂×C₄⋊Q₈×C₁₃₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
18	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
22	#498×C₁₃₇	—	2	4	no	—
23	#474×C₁₃₇	—	2	4	no	—
24	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#106×C₁₃₇	C₂²	1	4	yes	—
26	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	#398×C₁₃₇	—	2	2	no	—
31	#391×C₁₃₇	—	2	1	no	—
32	#389×C₁₃₇	C₂²	1	4	yes	—
33	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
34	#340×C₁₃₇	C₂²	1	2	yes	—
35	#340×C₁₃₇	C₂²	1	4	yes	—
36	#336×C₁₃₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₁₃₇	—	2	1	no	—
38	C₂²×Q₈×C₁₃₇	—	2	1	no	—
39	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
40	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₁₃₇	—	2	1	no	—
42	C₂³×C₄×C₁₃₇	—	2	2	no	—
43	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
46	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
47	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
48	C₄⋊Q₈×C₁₃₇	—	4	4	no	—
49	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
50	#125×C₁₃₇	—	2	4	no	—
51	#125×C₁₃₇	—	2	4	no	—
52	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
53	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
54	C₄×Q₈×C₁₃₇	—	2	2	no	—
55	C₄×Q₈×C₁₃₇	—	2	2	no	—
56	C₄×D₄×C₁₃₇	—	2	4	no	—
57	C₄×D₄×C₁₃₇	—	2	4	no	—
58	C₄×D₄×C₁₃₇	—	2	2	no	—
59	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
61	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
68	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄²×C₁₃₇	—	2	1	no	—
75	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
77	#107×C₁₃₇	—	2	4	no	—
78	#106×C₁₃₇	—	2	8	no	—
79	C₄⋊C₈×C₁₃₇	—	2	2	no	—
80	#102×C₁₃₇	—	2	4	no	—
81	#102×C₁₃₇	Q₈	1	8	yes	complemented
82	C₄×C₈×C₁₃₇	—	2	2	no	—
83	#95×C₁₃₇	—	2	1	no	—
84	#95×C₁₃₇	—	2	4	no	—
85	C₂⁴×C₁₃₇	—	2	1	no	—
86	C₂×Q₈×C₁₃₇	—	2	2	no	—
87	C₂×Q₈×C₁₃₇	—	2	4	no	—
88	C₂×Q₈×C₁₃₇	—	2	2	no	—
89	C₂×Q₈×C₁₃₇	—	2	2	no	—
90	C₂×Q₈×C₁₃₇	—	4	2	no	—
91	C₂×Q₈×C₁₃₇	—	4	2	no	—
92	C₂×Q₈×C₁₃₇	—	4	2	no	—
93	C₂×Q₈×C₁₃₇	—	8	2	no	—
94	C₂×D₄×C₁₃₇	—	2	1	no	—
95	C₂×D₄×C₁₃₇	—	2	2	no	—
96	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
97	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
98	C₂²×C₄×C₁₃₇	—	2	2	no	—
99	C₂²×C₄×C₁₃₇	—	2	1	no	—
100	C₂²×C₄×C₁₃₇	—	2	2	no	—
101	C₂²×C₄×C₁₃₇	—	2	8	no	—
102	C₂²×C₄×C₁₃₇	—	2	4	no	—
103	C₂²×C₄×C₁₃₇	—	2	2	no	—
104	C₂²×C₄×C₁₃₇	—	4	2	no	—
105	C₂²×C₄×C₁₃₇	—	4	2	no	—
106	C₂²×C₄×C₁₃₇	—	8	4	no	—
107	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₈×C₁₃₇	—	2	2	no	—
113	C₂×C₈×C₁₃₇	—	2	2	no	—
114	C₂×C₈×C₁₃₇	—	2	2	no	—
115	C₂×C₈×C₁₃₇	—	2	4	no	—
116	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
117	C₄⋊C₄×C₁₃₇	—	2	8	no	—
118	C₄⋊C₄×C₁₃₇	—	2	8	no	—
119	C₄⋊C₄×C₁₃₇	—	2	2	no	—
120	C₄⋊C₄×C₁₃₇	—	2	4	no	—
121	C₄⋊C₄×C₁₃₇	—	2	4	no	—
122	C₄⋊C₄×C₁₃₇	—	2	1	no	—
123	C₄⋊C₄×C₁₃₇	—	2	2	no	—
124	C₄⋊C₄×C₁₃₇	—	2	2	no	—
125	C₄⋊C₄×C₁₃₇	—	2	2	no	—
126	C₄⋊C₄×C₁₃₇	—	2	2	no	—
127	C₄⋊C₄×C₁₃₇	—	4	8	no	—
128	C₄⋊C₄×C₁₃₇	—	4	2	no	—
129	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
130	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
132	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
133	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
134	C₄²×C₁₃₇	—	2	2	no	—
135	C₄²×C₁₃₇	—	2	2	no	—
136	C₄²×C₁₃₇	—	4	2	no	—
137	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
138	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
139	C₂³×C₁₃₇	—	2	2	no	—
140	C₂³×C₁₃₇	—	2	4	no	—
141	C₂³×C₁₃₇	—	4	1	no	—
142	C₂³×C₁₃₇	—	4	1	no	—
143	C₂³×C₁₃₇	—	4	2	no	—
144	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₁₃₇	—	4	4	no	—
146	Q₈×C₁₃₇	—	8	4	no	—
147	Q₈×C₁₃₇	—	8	4	no	—
148	D₄×C₁₃₇	—	2	2	no	—
149	D₄×C₁₃₇	—	2	4	no	—
150	D₄×C₁₃₇	C₄⋊Q₈	1	4	yes	complemented
151	C₂×C₄×C₁₃₇	—	2	2	no	—
152	C₂×C₄×C₁₃₇	—	2	2	no	—
153	C₂×C₄×C₁₃₇	—	2	2	no	—
154	C₂×C₄×C₁₃₇	—	2	2	no	—
155	C₂×C₄×C₁₃₇	—	2	4	no	—
156	C₂×C₄×C₁₃₇	—	2	4	no	—
157	C₂×C₄×C₁₃₇	—	2	2	no	—
158	C₂×C₄×C₁₃₇	—	2	2	no	—
159	C₂×C₄×C₁₃₇	—	2	4	no	—
160	C₂×C₄×C₁₃₇	—	4	1	no	—
161	C₂×C₄×C₁₃₇	—	4	2	no	—
162	C₂×C₄×C₁₃₇	—	4	1	no	—
163	C₂×C₄×C₁₃₇	—	4	2	no	—
164	C₂×C₄×C₁₃₇	—	4	4	no	—
165	C₂×C₄×C₁₃₇	—	4	8	no	—
166	C₂×C₄×C₁₃₇	—	4	2	no	—
167	C₂×C₄×C₁₃₇	—	4	2	no	—
168	C₂×C₄×C₁₃₇	—	4	2	no	—
169	C₂×C₄×C₁₃₇	—	8	8	no	—
170	C₂×C₄×C₁₃₇	#120	1	2	yes	—
171	C₂×C₄×C₁₃₇	#120	1	2	yes	—
172	C₂×C₄×C₁₃₇	#135	1	2	yes	—
173	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
177	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
178	C₈×C₁₃₇	—	2	4	no	—
179	C₈×C₁₃₇	—	4	2	no	—
180	C₂²×C₁₃₇	—	2	4	no	—
181	C₂²×C₁₃₇	—	4	4	no	—
182	C₂²×C₁₃₇	—	4	4	no	—
183	C₂²×C₁₃₇	—	4	2	no	—
184	C₂²×C₁₃₇	—	4	2	no	—
185	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
186	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
187	C₂²×C₁₃₇	C₂×#135	1	1	yes	characteristic
188	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂²×C₁₃₇	#577	1	2	yes	—
190	C₄×C₁₃₇	—	2	2	no	—
191	C₄×C₁₃₇	—	4	4	no	—
192	C₄×C₁₃₇	—	4	2	no	—
193	C₄×C₁₃₇	—	4	2	no	—
194	C₄×C₁₃₇	—	4	2	no	—
195	C₄×C₁₃₇	—	4	4	no	—
196	C₄×C₁₃₇	—	8	2	no	—
197	C₄×C₁₃₇	#449	1	4	yes	—
198	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
199	C₄×C₁₃₇	#536	1	1	yes	characteristic
200	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
201	C₂×C₁₃₇	—	4	4	no	—
202	C₂×C₁₃₇	#2434	1	1	yes	characteristic
203	C₂×C₁₃₇	C₂×#449	1	1	yes	characteristic
204	C₂×C₁₃₇	#3009	1	1	yes	characteristic
205	C₂×C₁₃₇	#3381	1	2	yes	—
206	C₂×C₁₃₇	#3384	1	2	yes	—
207	#25835	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₁₃₇	#25835	1	1	yes	characteristic, direct product factor, Sylow
209	C₂×#554	C₂×C₁₃₇	1	1	yes	characteristic
210	C₂×#498	C₂×C₁₃₇	1	1	yes	characteristic, complemented
211	C₂×#474	C₂×C₁₃₇	1	1	yes	characteristic
212	#2546	C₂×C₁₃₇	1	1	yes	characteristic, complemented
213	#2059	C₂×C₁₃₇	1	4	yes	complemented
214	#1987	C₂×C₁₃₇	1	1	yes	characteristic, complemented
215	#1984	C₂×C₁₃₇	1	1	yes	characteristic, complemented
216	#1864	C₂×C₁₃₇	1	4	yes	—
217	C₄×#102	C₂×C₁₃₇	1	1	yes	characteristic
218	#554	—	2	4	no	—
219	#554	—	2	4	no	—
220	C₂×C₄⋊Q₈	—	2	1	no	—
221	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
223	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
224	C₂×C₂²⋊Q₈	—	2	1	no	—
225	C₂×C₂²⋊Q₈	—	2	2	no	—
226	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
229	C₂²×C₄⋊C₄	—	2	1	no	—
230	#498	—	2	4	no	—
231	#474	—	2	4	no	—
232	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
233	C₂×#106	C₂²×C₁₃₇	1	4	yes	—
234	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
235	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
236	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
238	#398	—	2	2	no	—
239	#391	—	2	1	no	—
240	#389	C₂²×C₁₃₇	1	4	yes	—
241	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
242	#340	C₂²×C₁₃₇	1	2	yes	—
243	#340	C₂²×C₁₃₇	1	4	yes	—
244	#336	C₂²×C₁₃₇	1	4	yes	—
245	C₂²×Q₈	—	2	1	no	—
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
248	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
249	C₂³×C₄	—	2	1	no	—
250	C₂³×C₄	—	2	2	no	—
251	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
252	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	—	2	2	no	—
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	—	4	4	no	—
257	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
258	#125	—	2	4	no	—
259	#125	—	2	4	no	—
260	C₂²⋊Q₈	—	2	8	no	—
261	C₂²⋊Q₈	—	2	16	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×D₄	—	2	4	no	—
265	C₄×D₄	—	2	4	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
268	C₂×C₄⋊C₄	—	2	8	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	4	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
284	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
285	#107	—	2	4	no	—
286	#106	—	2	8	no	—
287	C₄⋊C₈	—	2	2	no	—
288	#102	—	2	4	no	—
289	#102	Q₈×C₁₃₇	1	8	yes	complemented
290	C₄×C₈	—	2	2	no	—
291	#95	—	2	1	no	—
292	#95	—	2	4	no	—
293	C₂⁴	—	2	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	4	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	4	2	no	—
301	C₂×Q₈	—	8	2	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
305	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	2	no	—
314	C₂²×C₄	—	8	4	no	—
315	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
319	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
325	C₄⋊C₄	—	2	8	no	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	4	8	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
338	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
339	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	4	2	no	—
345	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
346	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	4	1	no	—
350	C₂³	—	4	1	no	—
351	C₂³	—	4	2	no	—
352	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
353	Q₈	—	4	4	no	—
354	Q₈	—	8	4	no	—
355	Q₈	—	8	4	no	—
356	D₄	—	2	2	no	—
357	D₄	—	2	4	no	—
358	D₄	C₄⋊Q₈×C₁₃₇	1	4	yes	complemented
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	8	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	8	no	—
378	C₂×C₄	#120×C₁₃₇	1	2	yes	—
379	C₂×C₄	#120×C₁₃₇	1	2	yes	—
380	C₂×C₄	#135×C₁₃₇	1	2	yes	—
381	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
382	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
383	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
384	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
385	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
386	C₈	—	2	4	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	4	4	no	—
390	C₂²	—	4	4	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
394	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
395	C₂²	C₂×#135×C₁₃₇	1	1	yes	characteristic
396	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
397	C₂²	#577×C₁₃₇	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	4	4	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	4	no	—
404	C₄	—	8	2	no	—
405	C₄	#449×C₁₃₇	1	4	yes	—
406	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
407	C₄	#536×C₁₃₇	1	1	yes	characteristic
408	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
409	C₂	—	4	4	no	—
410	C₂	#2434×C₁₃₇	1	1	yes	characteristic
411	C₂	C₂×#449×C₁₃₇	1	1	yes	characteristic
412	C₂	#3009×C₁₃₇	1	1	yes	characteristic
413	C₂	#3381×C₁₃₇	1	2	yes	—
414	C₂	#3384×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×C₄×Q₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#498×C₁₃₇	C₂	semidirect
4. #2546×C₁₃₇	C₂	semidirect
5. #2059×C₁₃₇	C₂	semidirect
6. #1987×C₁₃₇	C₂	semidirect
7. #1984×C₁₃₇	C₂	semidirect
81. #102×C₁₃₇	Q₈	semidirect
150. D₄×C₁₃₇	C₄⋊Q₈	semidirect
207. #25835	C₁₃₇	direct (N × H)
208. C₁₃₇	#25835	direct (N × H)
210. C₂×#498	C₂×C₁₃₇	semidirect
212. #2546	C₂×C₁₃₇	semidirect
213. #2059	C₂×C₁₃₇	semidirect
214. #1987	C₂×C₁₃₇	semidirect
215. #1984	C₂×C₁₃₇	semidirect
289. #102	Q₈×C₁₃₇	semidirect
358. D₄	C₄⋊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×#554×C₁₃₇	2
C₂×#498×C₁₃₇	2
C₂×#474×C₁₃₇	2
#2546×C₁₃₇	2
#2059×C₁₃₇	2
#1987×C₁₃₇	2
#1984×C₁₃₇	2
#1864×C₁₃₇	2
C₄×#102×C₁₃₇	2
#25835	137

Subgroups — C137×#25835 #46875221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25835 #46875221