Subgroups — C₁₃₇×#25850 #46875236

Order 35072.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 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×#474×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2567×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2059×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2057×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1989×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×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
21	C₂×C₄×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₃₇	—	2	1	no	—
26	#501×C₁₃₇	—	2	4	no	—
27	#474×C₁₃₇	—	2	4	no	—
28	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
31	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
34	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
35	#395×C₁₃₇	—	2	1	no	—
36	#395×C₁₃₇	—	2	1	no	—
37	#391×C₁₃₇	—	2	1	no	—
38	#389×C₁₃₇	C₂²	1	2	yes	—
39	#389×C₁₃₇	C₂²	1	2	yes	—
40	#384×C₁₃₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#340×C₁₃₇	C₂²	1	2	yes	—
44	#340×C₁₃₇	C₂²	1	2	yes	—
45	#336×C₁₃₇	C₂²	1	2	yes	—
46	#336×C₁₃₇	C₂²	1	2	yes	—
47	C₂²×Q₈×C₁₃₇	—	2	1	no	—
48	C₂²×Q₈×C₁₃₇	—	2	1	no	—
49	C₂²×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₁₃₇	—	2	1	no	—
52	C₂³×C₄×C₁₃₇	—	2	1	no	—
53	C₂³×C₄×C₁₃₇	—	2	1	no	—
54	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
57	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
58	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	#125×C₁₃₇	—	2	2	no	—
61	#125×C₁₃₇	—	2	2	no	—
62	#125×C₁₃₇	—	2	2	no	—
63	#125×C₁₃₇	D₄	1	4	yes	—
64	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
65	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
66	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
67	C₄×Q₈×C₁₃₇	—	2	2	no	—
68	C₄×Q₈×C₁₃₇	—	2	2	no	—
69	C₄×D₄×C₁₃₇	—	2	2	no	—
70	C₄×D₄×C₁₃₇	—	2	2	no	—
71	C₄×D₄×C₁₃₇	—	2	2	no	—
72	C₄×D₄×C₁₃₇	—	2	2	no	—
73	C₄×D₄×C₁₃₇	—	2	2	no	—
74	C₄×D₄×C₁₃₇	D₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₁₃₇	—	2	8	no	—
76	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
95	C₂×C₄²×C₁₃₇	—	2	1	no	—
96	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
99	#107×C₁₃₇	—	2	4	no	—
100	#107×C₁₃₇	—	2	4	no	—
101	#106×C₁₃₇	—	2	4	no	—
102	C₄⋊C₈×C₁₃₇	—	2	2	no	—
103	#102×C₁₃₇	—	2	4	no	—
104	#102×C₁₃₇	Q₈	1	8	yes	complemented
105	#97×C₁₃₇	—	2	2	no	—
106	#95×C₁₃₇	—	2	1	no	—
107	#95×C₁₃₇	—	2	2	no	—
108	#95×C₁₃₇	—	2	2	no	—
109	C₂⁴×C₁₃₇	—	2	1	no	—
110	C₂×Q₈×C₁₃₇	—	2	4	no	—
111	C₂×Q₈×C₁₃₇	—	2	2	no	—
112	C₂×Q₈×C₁₃₇	—	2	2	no	—
113	C₂×Q₈×C₁₃₇	—	4	1	no	—
114	C₂×Q₈×C₁₃₇	—	4	2	no	—
115	C₂×Q₈×C₁₃₇	—	8	2	no	—
116	C₂×Q₈×C₁₃₇	—	8	2	no	—
117	C₂×D₄×C₁₃₇	—	2	1	no	—
118	C₂×D₄×C₁₃₇	—	2	2	no	—
119	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₁₃₇	C₂×Q₈	1	4	yes	—
121	C₂²×C₄×C₁₃₇	—	2	4	no	—
122	C₂²×C₄×C₁₃₇	—	2	2	no	—
123	C₂²×C₄×C₁₃₇	—	2	1	no	—
124	C₂²×C₄×C₁₃₇	—	2	1	no	—
125	C₂²×C₄×C₁₃₇	—	2	1	no	—
126	C₂²×C₄×C₁₃₇	—	2	4	no	—
127	C₂²×C₄×C₁₃₇	—	2	4	no	—
128	C₂²×C₄×C₁₃₇	—	2	2	no	—
129	C₂²×C₄×C₁₃₇	—	4	2	no	—
130	C₂²×C₄×C₁₃₇	—	4	2	no	—
131	C₂²×C₄×C₁₃₇	—	8	2	no	—
132	C₂²×C₄×C₁₃₇	—	8	2	no	—
133	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
140	C₂×C₈×C₁₃₇	—	2	2	no	—
141	C₂×C₈×C₁₃₇	—	2	2	no	—
142	C₂×C₈×C₁₃₇	—	2	2	no	—
143	C₂×C₈×C₁₃₇	—	2	4	no	—
144	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
145	C₄⋊C₄×C₁₃₇	—	2	4	no	—
146	C₄⋊C₄×C₁₃₇	—	2	4	no	—
147	C₄⋊C₄×C₁₃₇	—	2	4	no	—
148	C₄⋊C₄×C₁₃₇	—	2	2	no	—
149	C₄⋊C₄×C₁₃₇	—	2	4	no	—
150	C₄⋊C₄×C₁₃₇	—	2	4	no	—
151	C₄⋊C₄×C₁₃₇	—	2	2	no	—
152	C₄⋊C₄×C₁₃₇	—	2	2	no	—
153	C₄⋊C₄×C₁₃₇	—	2	1	no	—
154	C₄⋊C₄×C₁₃₇	—	2	1	no	—
155	C₄⋊C₄×C₁₃₇	—	2	1	no	—
156	C₄⋊C₄×C₁₃₇	—	2	2	no	—
157	C₄⋊C₄×C₁₃₇	—	2	2	no	—
158	C₄⋊C₄×C₁₃₇	—	2	2	no	—
159	C₄⋊C₄×C₁₃₇	—	4	2	no	—
160	C₄⋊C₄×C₁₃₇	—	8	2	no	—
161	C₄⋊C₄×C₁₃₇	—	8	2	no	—
162	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
166	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
167	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
168	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
169	C₄²×C₁₃₇	—	2	1	no	—
170	C₄²×C₁₃₇	—	2	1	no	—
171	C₄²×C₁₃₇	—	2	1	no	—
172	C₄²×C₁₃₇	—	2	1	no	—
173	C₄²×C₁₃₇	—	4	2	no	—
174	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
175	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
176	C₂³×C₁₃₇	—	2	2	no	—
177	C₂³×C₁₃₇	—	2	4	no	—
178	C₂³×C₁₃₇	—	4	1	no	—
179	C₂³×C₁₃₇	—	4	1	no	—
180	C₂³×C₁₃₇	—	4	2	no	—
181	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₁₃₇	—	4	4	no	—
183	Q₈×C₁₃₇	—	8	4	no	—
184	Q₈×C₁₃₇	—	8	4	no	—
185	D₄×C₁₃₇	—	2	2	no	—
186	D₄×C₁₃₇	—	2	4	no	—
187	D₄×C₁₃₇	C₄⋊Q₈	1	4	yes	—
188	C₂×C₄×C₁₃₇	—	2	1	no	—
189	C₂×C₄×C₁₃₇	—	2	1	no	—
190	C₂×C₄×C₁₃₇	—	2	1	no	—
191	C₂×C₄×C₁₃₇	—	2	1	no	—
192	C₂×C₄×C₁₃₇	—	2	2	no	—
193	C₂×C₄×C₁₃₇	—	2	1	no	—
194	C₂×C₄×C₁₃₇	—	2	1	no	—
195	C₂×C₄×C₁₃₇	—	2	1	no	—
196	C₂×C₄×C₁₃₇	—	2	2	no	—
197	C₂×C₄×C₁₃₇	—	2	2	no	—
198	C₂×C₄×C₁₃₇	—	2	2	no	—
199	C₂×C₄×C₁₃₇	—	2	4	no	—
200	C₂×C₄×C₁₃₇	—	2	1	no	—
201	C₂×C₄×C₁₃₇	—	2	1	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	2	2	no	—
204	C₂×C₄×C₁₃₇	—	4	8	no	—
205	C₂×C₄×C₁₃₇	—	4	1	no	—
206	C₂×C₄×C₁₃₇	—	4	2	no	—
207	C₂×C₄×C₁₃₇	—	4	1	no	—
208	C₂×C₄×C₁₃₇	—	4	2	no	—
209	C₂×C₄×C₁₃₇	—	4	2	no	—
210	C₂×C₄×C₁₃₇	—	4	2	no	—
211	C₂×C₄×C₁₃₇	—	4	2	no	—
212	C₂×C₄×C₁₃₇	—	4	2	no	—
213	C₂×C₄×C₁₃₇	—	4	2	no	—
214	C₂×C₄×C₁₃₇	—	8	4	no	—
215	C₂×C₄×C₁₃₇	—	8	4	no	—
216	C₂×C₄×C₁₃₇	#120	1	2	yes	—
217	C₂×C₄×C₁₃₇	#120	1	2	yes	—
218	C₂×C₄×C₁₃₇	#137	1	2	yes	—
219	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
222	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
223	C₈×C₁₃₇	—	2	4	no	—
224	C₈×C₁₃₇	—	4	2	no	—
225	C₂²×C₁₃₇	—	2	4	no	—
226	C₂²×C₁₃₇	—	4	4	no	—
227	C₂²×C₁₃₇	—	4	4	no	—
228	C₂²×C₁₃₇	—	4	2	no	—
229	C₂²×C₁₃₇	—	4	2	no	—
230	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
231	C₂²×C₁₃₇	D₄×Q₈	1	2	yes	—
232	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
233	C₂²×C₁₃₇	#575	1	1	yes	characteristic
234	C₂²×C₁₃₇	#576	1	2	yes	—
235	C₄×C₁₃₇	—	2	2	no	—
236	C₄×C₁₃₇	—	2	2	no	—
237	C₄×C₁₃₇	—	2	2	no	—
238	C₄×C₁₃₇	—	4	4	no	—
239	C₄×C₁₃₇	—	4	1	no	—
240	C₄×C₁₃₇	—	4	2	no	—
241	C₄×C₁₃₇	—	4	1	no	—
242	C₄×C₁₃₇	—	4	1	no	—
243	C₄×C₁₃₇	—	4	2	no	—
244	C₄×C₁₃₇	—	4	2	no	—
245	C₄×C₁₃₇	—	8	2	no	—
246	C₄×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
247	C₄×C₁₃₇	#536	1	1	yes	characteristic
248	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
249	C₂×C₁₃₇	—	4	4	no	—
250	C₂×C₁₃₇	#2434	1	1	yes	characteristic
251	C₂×C₁₃₇	#3007	1	1	yes	characteristic
252	C₂×C₁₃₇	#3008	1	1	yes	characteristic
253	C₂×C₁₃₇	#3385	1	2	yes	—
254	C₂×C₁₃₇	#3389	1	2	yes	—
255	#25850	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₁₃₇	#25850	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#554	C₂×C₁₃₇	1	1	yes	characteristic
258	C₂×#501	C₂×C₁₃₇	1	1	yes	characteristic, complemented
259	C₂×#474	C₂×C₁₃₇	1	1	yes	characteristic
260	#2567	C₂×C₁₃₇	1	1	yes	characteristic, complemented
261	#2059	C₂×C₁₃₇	1	2	yes	complemented
262	#2057	C₂×C₁₃₇	1	2	yes	complemented
263	#1989	C₂×C₁₃₇	1	1	yes	characteristic, complemented
264	#1985	C₂×C₁₃₇	1	1	yes	characteristic, complemented
265	#1864	C₂×C₁₃₇	1	2	yes	—
266	#1862	C₂×C₁₃₇	1	2	yes	—
267	#1762	C₂×C₁₃₇	1	1	yes	characteristic
268	#554	—	2	4	no	—
269	#554	—	2	4	no	—
270	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
272	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
273	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
274	C₂×C₂²⋊Q₈	—	2	1	no	—
275	C₂×C₂²⋊Q₈	—	2	1	no	—
276	C₂×C₂²⋊Q₈	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
279	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	—	2	1	no	—
282	#501	—	2	4	no	—
283	#474	—	2	4	no	—
284	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
285	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
286	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
287	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
288	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
289	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
290	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
291	#395	—	2	1	no	—
292	#395	—	2	1	no	—
293	#391	—	2	1	no	—
294	#389	C₂²×C₁₃₇	1	2	yes	—
295	#389	C₂²×C₁₃₇	1	2	yes	—
296	#384	—	2	1	no	—
297	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
298	#340	C₂²×C₁₃₇	1	2	yes	—
299	#340	C₂²×C₁₃₇	1	2	yes	—
300	#340	C₂²×C₁₃₇	1	2	yes	—
301	#336	C₂²×C₁₃₇	1	2	yes	—
302	#336	C₂²×C₁₃₇	1	2	yes	—
303	C₂²×Q₈	—	2	1	no	—
304	C₂²×Q₈	—	2	1	no	—
305	C₂²×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
306	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
307	C₂³×C₄	—	2	1	no	—
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	—	2	1	no	—
310	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	#125	—	2	2	no	—
317	#125	—	2	2	no	—
318	#125	—	2	2	no	—
319	#125	D₄×C₁₃₇	1	4	yes	—
320	C₂²⋊Q₈	—	2	8	no	—
321	C₂²⋊Q₈	—	2	8	no	—
322	C₂²⋊Q₈	—	2	8	no	—
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	2	no	—
328	C₄×D₄	—	2	2	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	D₄×C₁₃₇	1	4	yes	—
331	C₂×C₄⋊C₄	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
341	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
347	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₄²	—	2	1	no	—
352	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
355	#107	—	2	4	no	—
356	#107	—	2	4	no	—
357	#106	—	2	4	no	—
358	C₄⋊C₈	—	2	2	no	—
359	#102	—	2	4	no	—
360	#102	Q₈×C₁₃₇	1	8	yes	complemented
361	#97	—	2	2	no	—
362	#95	—	2	1	no	—
363	#95	—	2	2	no	—
364	#95	—	2	2	no	—
365	C₂⁴	—	2	1	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	4	1	no	—
370	C₂×Q₈	—	4	2	no	—
371	C₂×Q₈	—	8	2	no	—
372	C₂×Q₈	—	8	2	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
376	C₂×D₄	C₂×Q₈×C₁₃₇	1	4	yes	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	4	2	no	—
386	C₂²×C₄	—	4	2	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	—	8	2	no	—
389	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
392	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
394	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	—	8	2	no	—
417	C₄⋊C₄	—	8	2	no	—
418	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
419	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
421	C₄⋊C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	4	2	no	—
430	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
431	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	4	no	—
434	C₂³	—	4	1	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	2	no	—
437	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
438	Q₈	—	4	4	no	—
439	Q₈	—	8	4	no	—
440	Q₈	—	8	4	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	4	no	—
443	D₄	C₄⋊Q₈×C₁₃₇	1	4	yes	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	8	4	no	—
471	C₂×C₄	—	8	4	no	—
472	C₂×C₄	#120×C₁₃₇	1	2	yes	—
473	C₂×C₄	#120×C₁₃₇	1	2	yes	—
474	C₂×C₄	#137×C₁₃₇	1	2	yes	—
475	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
477	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
478	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
479	C₈	—	2	4	no	—
480	C₈	—	4	2	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
487	C₂²	D₄×Q₈×C₁₃₇	1	2	yes	—
488	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
489	C₂²	#575×C₁₃₇	1	1	yes	characteristic
490	C₂²	#576×C₁₃₇	1	2	yes	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	8	2	no	—
502	C₄	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
503	C₄	#536×C₁₃₇	1	1	yes	characteristic
504	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
505	C₂	—	4	4	no	—
506	C₂	#2434×C₁₃₇	1	1	yes	characteristic
507	C₂	#3007×C₁₃₇	1	1	yes	characteristic
508	C₂	#3008×C₁₃₇	1	1	yes	characteristic
509	C₂	#3385×C₁₃₇	1	2	yes	—
510	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. #2567×C₁₃₇	C₂	semidirect
5. #2059×C₁₃₇	C₂	semidirect
6. #2057×C₁₃₇	C₂	semidirect
7. #1989×C₁₃₇	C₂	semidirect
8. #1985×C₁₃₇	C₂	semidirect
104. #102×C₁₃₇	Q₈	semidirect
255. #25850	C₁₃₇	direct (N × H)
256. C₁₃₇	#25850	direct (N × H)
258. C₂×#501	C₂×C₁₃₇	semidirect
260. #2567	C₂×C₁₃₇	semidirect
261. #2059	C₂×C₁₃₇	semidirect
262. #2057	C₂×C₁₃₇	semidirect
263. #1989	C₂×C₁₃₇	semidirect
264. #1985	C₂×C₁₃₇	semidirect
360. #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×#474×C₁₃₇	2
#2567×C₁₃₇	2
#2059×C₁₃₇	2
#2057×C₁₃₇	2
#1989×C₁₃₇	2
#1985×C₁₃₇	2
#1864×C₁₃₇	2
#1862×C₁₃₇	2
#1762×C₁₃₇	2
#25850	137

Subgroups — C137×#25850 #46875236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25850 #46875236