Subgroups — C₁₃₇×#25845 #46875231

Order 35072.

1040 conjugacy classes of subgroups.

The subgroup lattice has 460 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×#501×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#478×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2584×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2059×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2059×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1988×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1985×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1864×C₁₃₇	C₂	1	2	yes	maximal
10	#1862×C₁₃₇	C₂	1	2	yes	maximal
11	#1762×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	#554×C₁₃₇	—	2	4	no	—
13	#554×C₁₃₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
19	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
23	#501×C₁₃₇	—	2	4	no	—
24	#478×C₁₃₇	—	2	4	no	—
25	C₂×#107×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
34	#398×C₁₃₇	—	2	1	no	—
35	#395×C₁₃₇	—	2	1	no	—
36	#389×C₁₃₇	—	2	2	no	—
37	#389×C₁₃₇	C₂²	1	2	yes	—
38	#389×C₁₃₇	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
40	#340×C₁₃₇	C₂²	1	2	yes	—
41	#340×C₁₃₇	C₂²	1	2	yes	—
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#336×C₁₃₇	C₂²	1	2	yes	—
44	#336×C₁₃₇	C₂²	1	2	yes	—
45	C₂²×Q₈×C₁₃₇	—	2	1	no	—
46	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₁₃₇	—	2	1	no	—
49	C₂³×C₄×C₁₃₇	—	2	2	no	—
50	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
54	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
55	#125×C₁₃₇	—	2	4	no	—
56	#125×C₁₃₇	—	2	4	no	—
57	#125×C₁₃₇	—	2	2	no	—
58	#125×C₁₃₇	—	2	2	no	—
59	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
60	C₂²⋊Q₈×C₁₃₇	—	2	16	no	—
61	C₄×Q₈×C₁₃₇	—	2	2	no	—
62	C₄×Q₈×C₁₃₇	—	2	2	no	—
63	C₄×D₄×C₁₃₇	—	2	4	no	—
64	C₄×D₄×C₁₃₇	—	2	4	no	—
65	C₄×D₄×C₁₃₇	—	2	2	no	—
66	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
67	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
69	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
84	C₂×C₄²×C₁₃₇	—	2	1	no	—
85	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
86	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
87	#107×C₁₃₇	—	2	2	no	—
88	#107×C₁₃₇	—	2	4	no	—
89	#106×C₁₃₇	—	2	2	no	—
90	#106×C₁₃₇	—	2	4	no	—
91	C₄⋊C₈×C₁₃₇	—	2	2	no	—
92	#102×C₁₃₇	—	2	2	no	—
93	#102×C₁₃₇	—	2	2	no	—
94	#102×C₁₃₇	Q₈	1	8	yes	complemented
95	#97×C₁₃₇	—	2	2	no	—
96	#95×C₁₃₇	—	2	1	no	—
97	#95×C₁₃₇	—	2	2	no	—
98	#95×C₁₃₇	—	2	2	no	—
99	C₂⁴×C₁₃₇	—	2	1	no	—
100	C₂×Q₈×C₁₃₇	—	2	2	no	—
101	C₂×Q₈×C₁₃₇	—	2	4	no	—
102	C₂×Q₈×C₁₃₇	—	4	2	no	—
103	C₂×Q₈×C₁₃₇	—	8	1	no	—
104	C₂×Q₈×C₁₃₇	—	8	1	no	—
105	C₂×D₄×C₁₃₇	—	2	1	no	—
106	C₂×D₄×C₁₃₇	—	2	2	no	—
107	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
109	C₂²×C₄×C₁₃₇	—	2	2	no	—
110	C₂²×C₄×C₁₃₇	—	2	1	no	—
111	C₂²×C₄×C₁₃₇	—	2	2	no	—
112	C₂²×C₄×C₁₃₇	—	2	8	no	—
113	C₂²×C₄×C₁₃₇	—	2	4	no	—
114	C₂²×C₄×C₁₃₇	—	2	1	no	—
115	C₂²×C₄×C₁₃₇	—	2	1	no	—
116	C₂²×C₄×C₁₃₇	—	4	4	no	—
117	C₂²×C₄×C₁₃₇	—	8	2	no	—
118	C₂²×C₄×C₁₃₇	—	8	2	no	—
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₁₃₇	—	2	2	no	—
125	C₂×C₈×C₁₃₇	—	2	2	no	—
126	C₂×C₈×C₁₃₇	—	2	1	no	—
127	C₂×C₈×C₁₃₇	—	2	1	no	—
128	C₂×C₈×C₁₃₇	—	2	2	no	—
129	C₂×C₈×C₁₃₇	—	2	2	no	—
130	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
131	C₄⋊C₄×C₁₃₇	—	2	8	no	—
132	C₄⋊C₄×C₁₃₇	—	2	8	no	—
133	C₄⋊C₄×C₁₃₇	—	2	2	no	—
134	C₄⋊C₄×C₁₃₇	—	2	4	no	—
135	C₄⋊C₄×C₁₃₇	—	2	4	no	—
136	C₄⋊C₄×C₁₃₇	—	2	1	no	—
137	C₄⋊C₄×C₁₃₇	—	2	2	no	—
138	C₄⋊C₄×C₁₃₇	—	2	2	no	—
139	C₄⋊C₄×C₁₃₇	—	2	1	no	—
140	C₄⋊C₄×C₁₃₇	—	2	1	no	—
141	C₄⋊C₄×C₁₃₇	—	2	1	no	—
142	C₄⋊C₄×C₁₃₇	—	2	1	no	—
143	C₄⋊C₄×C₁₃₇	—	4	4	no	—
144	C₄⋊C₄×C₁₃₇	—	4	2	no	—
145	C₄⋊C₄×C₁₃₇	—	4	2	no	—
146	C₄⋊C₄×C₁₃₇	—	8	2	no	—
147	C₄⋊C₄×C₁₃₇	C₂×D₄	1	4	yes	—
148	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
149	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
150	C₂²⋊C₄×C₁₃₇	—	2	8	no	—
151	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
152	C₄²×C₁₃₇	—	2	2	no	—
153	C₄²×C₁₃₇	—	2	2	no	—
154	C₄²×C₁₃₇	—	8	1	no	—
155	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
156	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
157	C₂³×C₁₃₇	—	2	2	no	—
158	C₂³×C₁₃₇	—	2	4	no	—
159	C₂³×C₁₃₇	—	4	2	no	—
160	C₂³×C₁₃₇	—	4	1	no	—
161	C₂³×C₁₃₇	—	4	1	no	—
162	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
163	Q₈×C₁₃₇	—	4	4	no	—
164	Q₈×C₁₃₇	—	8	4	no	—
165	D₄×C₁₃₇	—	2	2	no	—
166	D₄×C₁₃₇	—	2	4	no	—
167	D₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
168	C₂×C₄×C₁₃₇	—	2	2	no	—
169	C₂×C₄×C₁₃₇	—	2	1	no	—
170	C₂×C₄×C₁₃₇	—	2	1	no	—
171	C₂×C₄×C₁₃₇	—	2	2	no	—
172	C₂×C₄×C₁₃₇	—	2	2	no	—
173	C₂×C₄×C₁₃₇	—	2	2	no	—
174	C₂×C₄×C₁₃₇	—	2	4	no	—
175	C₂×C₄×C₁₃₇	—	2	4	no	—
176	C₂×C₄×C₁₃₇	—	2	2	no	—
177	C₂×C₄×C₁₃₇	—	2	2	no	—
178	C₂×C₄×C₁₃₇	—	2	2	no	—
179	C₂×C₄×C₁₃₇	—	2	2	no	—
180	C₂×C₄×C₁₃₇	—	4	8	no	—
181	C₂×C₄×C₁₃₇	—	4	2	no	—
182	C₂×C₄×C₁₃₇	—	4	2	no	—
183	C₂×C₄×C₁₃₇	—	4	2	no	—
184	C₂×C₄×C₁₃₇	—	4	2	no	—
185	C₂×C₄×C₁₃₇	—	4	2	no	—
186	C₂×C₄×C₁₃₇	—	4	1	no	—
187	C₂×C₄×C₁₃₇	—	4	1	no	—
188	C₂×C₄×C₁₃₇	—	4	2	no	—
189	C₂×C₄×C₁₃₇	—	4	1	no	—
190	C₂×C₄×C₁₃₇	—	4	1	no	—
191	C₂×C₄×C₁₃₇	—	8	8	no	—
192	C₂×C₄×C₁₃₇	#120	1	2	yes	—
193	C₂×C₄×C₁₃₇	#120	1	2	yes	—
194	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
197	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
198	C₈×C₁₃₇	—	2	4	no	—
199	C₈×C₁₃₇	—	4	1	no	—
200	C₈×C₁₃₇	—	4	1	no	—
201	C₂²×C₁₃₇	—	2	4	no	—
202	C₂²×C₁₃₇	—	4	4	no	—
203	C₂²×C₁₃₇	—	4	4	no	—
204	C₂²×C₁₃₇	—	4	2	no	—
205	C₂²×C₁₃₇	—	4	2	no	—
206	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
207	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
208	C₂²×C₁₃₇	#575	1	2	yes	—
209	C₂²×C₁₃₇	#576	1	1	yes	characteristic
210	C₂²×C₁₃₇	#577	1	1	yes	characteristic
211	C₄×C₁₃₇	—	2	2	no	—
212	C₄×C₁₃₇	—	2	2	no	—
213	C₄×C₁₃₇	—	4	4	no	—
214	C₄×C₁₃₇	—	4	2	no	—
215	C₄×C₁₃₇	—	4	2	no	—
216	C₄×C₁₃₇	—	4	2	no	—
217	C₄×C₁₃₇	—	4	4	no	—
218	C₄×C₁₃₇	—	8	1	no	—
219	C₄×C₁₃₇	—	8	1	no	—
220	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
221	C₄×C₁₃₇	#536	1	1	yes	characteristic
222	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
223	C₂×C₁₃₇	—	4	4	no	—
224	C₂×C₁₃₇	#2434	1	1	yes	characteristic
225	C₂×C₁₃₇	#3005	1	1	yes	characteristic
226	C₂×C₁₃₇	#3012	1	1	yes	characteristic
227	C₂×C₁₃₇	#3388	1	2	yes	—
228	C₂×C₁₃₇	#3389	1	2	yes	—
229	#25845	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₁₃₇	#25845	1	1	yes	characteristic, direct product factor, Sylow
231	C₂×#554	C₂×C₁₃₇	1	1	yes	characteristic
232	C₂×#501	C₂×C₁₃₇	1	1	yes	characteristic, complemented
233	C₂×#478	C₂×C₁₃₇	1	1	yes	characteristic
234	#2584	C₂×C₁₃₇	1	1	yes	characteristic, complemented
235	#2059	C₂×C₁₃₇	1	2	yes	complemented
236	#2059	C₂×C₁₃₇	1	2	yes	complemented
237	#1988	C₂×C₁₃₇	1	1	yes	characteristic, complemented
238	#1985	C₂×C₁₃₇	1	1	yes	characteristic, complemented
239	#1864	C₂×C₁₃₇	1	2	yes	—
240	#1862	C₂×C₁₃₇	1	2	yes	—
241	#1762	C₂×C₁₃₇	1	1	yes	characteristic
242	#554	—	2	4	no	—
243	#554	—	2	4	no	—
244	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
245	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
246	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
247	C₂×C₂²⋊Q₈	—	2	1	no	—
248	C₂×C₂²⋊Q₈	—	2	2	no	—
249	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
250	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
251	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
252	C₂²×C₄⋊C₄	—	2	1	no	—
253	#501	—	2	4	no	—
254	#478	—	2	4	no	—
255	C₂×#107	C₂²×C₁₃₇	1	1	yes	characteristic
256	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
257	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
258	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
259	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
260	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
261	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
262	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
263	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
264	#398	—	2	1	no	—
265	#395	—	2	1	no	—
266	#389	—	2	2	no	—
267	#389	C₂²×C₁₃₇	1	2	yes	—
268	#389	C₂²×C₁₃₇	1	2	yes	—
269	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
270	#340	C₂²×C₁₃₇	1	2	yes	—
271	#340	C₂²×C₁₃₇	1	2	yes	—
272	#340	C₂²×C₁₃₇	1	2	yes	—
273	#336	C₂²×C₁₃₇	1	2	yes	—
274	#336	C₂²×C₁₃₇	1	2	yes	—
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
277	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	2	no	—
280	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
283	C₄⋊Q₈	—	2	2	no	—
284	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
285	#125	—	2	4	no	—
286	#125	—	2	4	no	—
287	#125	—	2	2	no	—
288	#125	—	2	2	no	—
289	C₂²⋊Q₈	—	2	8	no	—
290	C₂²⋊Q₈	—	2	16	no	—
291	C₄×Q₈	—	2	2	no	—
292	C₄×Q₈	—	2	2	no	—
293	C₄×D₄	—	2	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
297	C₂×C₄⋊C₄	—	2	8	no	—
298	C₂×C₄⋊C₄	—	2	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	2	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
316	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
317	#107	—	2	2	no	—
318	#107	—	2	4	no	—
319	#106	—	2	2	no	—
320	#106	—	2	4	no	—
321	C₄⋊C₈	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	Q₈×C₁₃₇	1	8	yes	complemented
325	#97	—	2	2	no	—
326	#95	—	2	1	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	C₂⁴	—	2	1	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	8	1	no	—
334	C₂×Q₈	—	8	1	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
338	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	8	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	4	4	no	—
347	C₂²×C₄	—	8	2	no	—
348	C₂²×C₄	—	8	2	no	—
349	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	8	2	no	—
377	C₄⋊C₄	C₂×D₄×C₁₃₇	1	4	yes	—
378	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
379	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	8	1	no	—
385	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
386	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
393	Q₈	—	4	4	no	—
394	Q₈	—	8	4	no	—
395	D₄	—	2	2	no	—
396	D₄	—	2	4	no	—
397	D₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	8	8	no	—
422	C₂×C₄	#120×C₁₃₇	1	2	yes	—
423	C₂×C₄	#120×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	C₂²×Q₈×C₁₃₇	1	2	yes	—
427	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
428	C₈	—	2	4	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	1	no	—
431	C₂²	—	2	4	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	4	2	no	—
436	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
437	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
438	C₂²	#575×C₁₃₇	1	2	yes	—
439	C₂²	#576×C₁₃₇	1	1	yes	characteristic
440	C₂²	#577×C₁₃₇	1	1	yes	characteristic
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	4	4	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	4	no	—
448	C₄	—	8	1	no	—
449	C₄	—	8	1	no	—
450	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
451	C₄	#536×C₁₃₇	1	1	yes	characteristic
452	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
453	C₂	—	4	4	no	—
454	C₂	#2434×C₁₃₇	1	1	yes	characteristic
455	C₂	#3005×C₁₃₇	1	1	yes	characteristic
456	C₂	#3012×C₁₃₇	1	1	yes	characteristic
457	C₂	#3388×C₁₃₇	1	2	yes	—
458	C₂	#3389×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×#501×C₁₃₇	C₂	semidirect
4. #2584×C₁₃₇	C₂	semidirect
5. #2059×C₁₃₇	C₂	semidirect
6. #2059×C₁₃₇	C₂	semidirect
7. #1988×C₁₃₇	C₂	semidirect
8. #1985×C₁₃₇	C₂	semidirect
94. #102×C₁₃₇	Q₈	semidirect
229. #25845	C₁₃₇	direct (N × H)
230. C₁₃₇	#25845	direct (N × H)
232. C₂×#501	C₂×C₁₃₇	semidirect
234. #2584	C₂×C₁₃₇	semidirect
235. #2059	C₂×C₁₃₇	semidirect
236. #2059	C₂×C₁₃₇	semidirect
237. #1988	C₂×C₁₃₇	semidirect
238. #1985	C₂×C₁₃₇	semidirect
324. #102	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×#501×C₁₃₇	2
C₂×#478×C₁₃₇	2
#2584×C₁₃₇	2
#2059×C₁₃₇	2
#1988×C₁₃₇	2
#1985×C₁₃₇	2
#1864×C₁₃₇	2
#1862×C₁₃₇	2
#1762×C₁₃₇	2
#25845	137

Subgroups — C137×#25845 #46875231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25845 #46875231