Subgroups — C₁₃₇×#25820 #46875206

Order 35072.

990 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#501×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2628×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2059×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2058×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1993×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1991×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1854×C₁₃₇	C₂	1	2	yes	maximal
10	#1853×C₁₃₇	C₂	1	2	yes	maximal
11	#1750×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	#544×C₁₃₇	—	2	4	no	—
13	#544×C₁₃₇	—	2	4	no	—
14	C₂×C₄⋊Q₈×C₁₃₇	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
18	C₂×#125×C₁₃₇	—	2	1	no	—
19	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
22	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
23	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
26	#501×C₁₃₇	—	2	4	no	—
27	#498×C₁₃₇	—	2	4	no	—
28	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
29	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
35	#400×C₁₃₇	—	2	1	no	—
36	#399×C₁₃₇	—	2	1	no	—
37	#395×C₁₃₇	—	2	1	no	—
38	#393×C₁₃₇	—	2	1	no	—
39	#389×C₁₃₇	C₂²	1	2	yes	—
40	#382×C₁₃₇	C₂²	1	1	yes	characteristic
41	#380×C₁₃₇	C₂²	1	2	yes	—
42	#340×C₁₃₇	C₂²	1	2	yes	—
43	#340×C₁₃₇	C₂²	1	2	yes	—
44	#340×C₁₃₇	C₂²	1	2	yes	—
45	#340×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	—	2	1	no	—
50	C₂³×C₄×C₁₃₇	—	2	1	no	—
51	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
55	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
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	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
61	#125×C₁₃₇	—	2	8	no	—
62	#125×C₁₃₇	—	2	2	no	—
63	#125×C₁₃₇	—	2	2	no	—
64	#125×C₁₃₇	—	2	2	no	—
65	#125×C₁₃₇	D₄	1	4	yes	—
66	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
67	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
68	#117×C₁₃₇	—	2	2	no	—
69	#117×C₁₃₇	—	2	2	no	—
70	#117×C₁₃₇	—	2	2	no	—
71	#117×C₁₃₇	—	2	2	no	—
72	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₁₃₇	—	2	1	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	2	no	—
80	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
82	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
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	#107×C₁₃₇	—	2	4	no	—
99	#106×C₁₃₇	—	2	4	no	—
100	#106×C₁₃₇	—	2	4	no	—
101	#106×C₁₃₇	—	2	4	no	—
102	C₂²⋊C₈×C₁₃₇	Q₈	1	8	yes	complemented
103	#97×C₁₃₇	—	2	2	no	—
104	C₄×C₈×C₁₃₇	—	2	2	no	—
105	#95×C₁₃₇	—	2	2	no	—
106	#95×C₁₃₇	—	2	1	no	—
107	#95×C₁₃₇	—	2	1	no	—
108	#95×C₁₃₇	—	2	2	no	—
109	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
110	C₂×Q₈×C₁₃₇	—	2	4	no	—
111	C₂×Q₈×C₁₃₇	—	2	4	no	—
112	C₂×Q₈×C₁₃₇	—	4	2	no	—
113	C₂×Q₈×C₁₃₇	—	8	2	no	—
114	C₂×Q₈×C₁₃₇	—	8	2	no	—
115	C₂×Q₈×C₁₃₇	—	8	2	no	—
116	C₂²×C₄×C₁₃₇	—	2	1	no	—
117	C₂²×C₄×C₁₃₇	—	2	1	no	—
118	C₂²×C₄×C₁₃₇	—	2	2	no	—
119	C₂²×C₄×C₁₃₇	—	2	1	no	—
120	C₂²×C₄×C₁₃₇	—	2	1	no	—
121	C₂²×C₄×C₁₃₇	—	2	1	no	—
122	C₂²×C₄×C₁₃₇	—	2	1	no	—
123	C₂²×C₄×C₁₃₇	—	2	4	no	—
124	C₂²×C₄×C₁₃₇	—	2	2	no	—
125	C₂²×C₄×C₁₃₇	—	2	2	no	—
126	C₂²×C₄×C₁₃₇	—	8	2	no	—
127	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₃₇	C₂×Q₈	1	4	yes	—
133	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
134	C₂×C₈×C₁₃₇	—	2	2	no	—
135	C₂×C₈×C₁₃₇	—	2	2	no	—
136	C₂×C₈×C₁₃₇	—	2	2	no	—
137	C₂×C₈×C₁₃₇	—	2	2	no	—
138	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
139	C₂×C₈×C₁₃₇	C₂×Q₈	1	4	yes	—
140	C₄⋊C₄×C₁₃₇	—	2	4	no	—
141	C₄⋊C₄×C₁₃₇	—	2	4	no	—
142	C₄⋊C₄×C₁₃₇	—	2	4	no	—
143	C₄⋊C₄×C₁₃₇	—	2	4	no	—
144	C₄⋊C₄×C₁₃₇	—	2	4	no	—
145	C₄⋊C₄×C₁₃₇	—	2	4	no	—
146	C₄⋊C₄×C₁₃₇	—	2	4	no	—
147	C₄⋊C₄×C₁₃₇	—	2	2	no	—
148	C₄⋊C₄×C₁₃₇	—	2	2	no	—
149	C₄⋊C₄×C₁₃₇	—	2	1	no	—
150	C₄⋊C₄×C₁₃₇	—	2	1	no	—
151	C₄⋊C₄×C₁₃₇	—	2	1	no	—
152	C₄⋊C₄×C₁₃₇	—	2	1	no	—
153	C₄⋊C₄×C₁₃₇	—	2	2	no	—
154	C₄⋊C₄×C₁₃₇	—	2	2	no	—
155	C₄⋊C₄×C₁₃₇	—	2	4	no	—
156	C₄⋊C₄×C₁₃₇	—	4	2	no	—
157	C₄⋊C₄×C₁₃₇	—	8	2	no	—
158	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
163	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
164	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
165	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
166	C₄²×C₁₃₇	—	2	4	no	—
167	C₄²×C₁₃₇	—	2	1	no	—
168	C₄²×C₁₃₇	—	2	1	no	—
169	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
170	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
171	C₂³×C₁₃₇	—	2	1	no	—
172	C₂³×C₁₃₇	—	2	1	no	—
173	C₂³×C₁₃₇	—	2	2	no	—
174	C₂³×C₁₃₇	—	2	2	no	—
175	C₂³×C₁₃₇	C₄⋊Q₈	1	2	yes	—
176	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
177	Q₈×C₁₃₇	—	8	4	no	—
178	Q₈×C₁₃₇	—	8	4	no	—
179	Q₈×C₁₃₇	—	8	4	no	—
180	C₂×C₄×C₁₃₇	—	2	1	no	—
181	C₂×C₄×C₁₃₇	—	2	1	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	2	no	—
184	C₂×C₄×C₁₃₇	—	2	4	no	—
185	C₂×C₄×C₁₃₇	—	2	4	no	—
186	C₂×C₄×C₁₃₇	—	2	1	no	—
187	C₂×C₄×C₁₃₇	—	2	2	no	—
188	C₂×C₄×C₁₃₇	—	2	1	no	—
189	C₂×C₄×C₁₃₇	—	2	2	no	—
190	C₂×C₄×C₁₃₇	—	2	2	no	—
191	C₂×C₄×C₁₃₇	—	2	2	no	—
192	C₂×C₄×C₁₃₇	—	2	2	no	—
193	C₂×C₄×C₁₃₇	—	2	2	no	—
194	C₂×C₄×C₁₃₇	—	4	2	no	—
195	C₂×C₄×C₁₃₇	—	4	2	no	—
196	C₂×C₄×C₁₃₇	—	4	2	no	—
197	C₂×C₄×C₁₃₇	—	4	2	no	—
198	C₂×C₄×C₁₃₇	—	4	2	no	—
199	C₂×C₄×C₁₃₇	—	4	2	no	—
200	C₂×C₄×C₁₃₇	—	4	2	no	—
201	C₂×C₄×C₁₃₇	—	4	2	no	—
202	C₂×C₄×C₁₃₇	—	4	2	no	—
203	C₂×C₄×C₁₃₇	—	4	2	no	—
204	C₂×C₄×C₁₃₇	—	8	4	no	—
205	C₂×C₄×C₁₃₇	#120	1	2	yes	—
206	C₂×C₄×C₁₃₇	#120	1	2	yes	—
207	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
208	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₁₃₇	C₂²×Q₈	1	2	yes	—
211	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
212	C₈×C₁₃₇	—	2	4	no	—
213	C₈×C₁₃₇	—	2	4	no	—
214	C₂²×C₁₃₇	—	2	4	no	—
215	C₂²×C₁₃₇	—	2	2	no	—
216	C₂²×C₁₃₇	—	4	2	no	—
217	C₂²×C₁₃₇	—	4	1	no	—
218	C₂²×C₁₃₇	—	4	1	no	—
219	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
220	C₂²×C₁₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
221	C₂²×C₁₃₇	#536	1	1	yes	characteristic
222	C₂²×C₁₃₇	#577	1	2	yes	—
223	C₂²×C₁₃₇	#577	1	2	yes	—
224	C₄×C₁₃₇	—	2	2	no	—
225	C₄×C₁₃₇	—	2	2	no	—
226	C₄×C₁₃₇	—	2	2	no	—
227	C₄×C₁₃₇	—	4	1	no	—
228	C₄×C₁₃₇	—	4	2	no	—
229	C₄×C₁₃₇	—	4	2	no	—
230	C₄×C₁₃₇	—	4	2	no	—
231	C₄×C₁₃₇	—	4	1	no	—
232	C₄×C₁₃₇	—	4	2	no	—
233	C₄×C₁₃₇	—	8	2	no	—
234	C₄×C₁₃₇	—	8	2	no	—
235	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
236	C₄×C₁₃₇	D₄×Q₈	1	2	yes	—
237	C₂×C₁₃₇	—	4	2	no	—
238	C₂×C₁₃₇	#2434	1	1	yes	characteristic
239	C₂×C₁₃₇	#3009	1	1	yes	characteristic
240	C₂×C₁₃₇	#3010	1	1	yes	characteristic
241	C₂×C₁₃₇	#3163	1	2	yes	—
242	C₂×C₁₃₇	#3244	1	2	yes	—
243	#25820	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₁₃₇	#25820	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×#544	C₂×C₁₃₇	1	1	yes	characteristic
246	C₂×#501	C₂×C₁₃₇	1	1	yes	characteristic, complemented
247	C₂×#498	C₂×C₁₃₇	1	1	yes	characteristic, complemented
248	#2628	C₂×C₁₃₇	1	1	yes	characteristic
249	#2059	C₂×C₁₃₇	1	2	yes	complemented
250	#2058	C₂×C₁₃₇	1	2	yes	complemented
251	#1993	C₂×C₁₃₇	1	1	yes	characteristic, complemented
252	#1991	C₂×C₁₃₇	1	1	yes	characteristic, complemented
253	#1854	C₂×C₁₃₇	1	2	yes	—
254	#1853	C₂×C₁₃₇	1	2	yes	—
255	#1750	C₂×C₁₃₇	1	1	yes	characteristic
256	#544	—	2	4	no	—
257	#544	—	2	4	no	—
258	C₂×C₄⋊Q₈	—	2	1	no	—
259	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
260	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
261	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
262	C₂×#125	—	2	1	no	—
263	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
264	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
268	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
269	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
270	#501	—	2	4	no	—
271	#498	—	2	4	no	—
272	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
273	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
274	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
275	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
276	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
277	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
279	#400	—	2	1	no	—
280	#399	—	2	1	no	—
281	#395	—	2	1	no	—
282	#393	—	2	1	no	—
283	#389	C₂²×C₁₃₇	1	2	yes	—
284	#382	C₂²×C₁₃₇	1	1	yes	characteristic
285	#380	C₂²×C₁₃₇	1	2	yes	—
286	#340	C₂²×C₁₃₇	1	2	yes	—
287	#340	C₂²×C₁₃₇	1	2	yes	—
288	#340	C₂²×C₁₃₇	1	2	yes	—
289	#340	C₂²×C₁₃₇	1	2	yes	—
290	#336	C₂²×C₁₃₇	1	2	yes	—
291	C₂²×Q₈	—	2	1	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	—	2	1	no	—
294	C₂³×C₄	—	2	1	no	—
295	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
297	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
298	C₄⋊Q₈	—	2	8	no	—
299	C₄⋊Q₈	—	2	2	no	—
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₄⋊Q₈	—	2	2	no	—
303	C₄⋊Q₈	—	2	2	no	—
304	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	—
305	#125	—	2	8	no	—
306	#125	—	2	2	no	—
307	#125	—	2	2	no	—
308	#125	—	2	2	no	—
309	#125	D₄×C₁₃₇	1	4	yes	—
310	C₂²⋊Q₈	—	2	8	no	—
311	C₂²⋊Q₈	—	2	8	no	—
312	#117	—	2	2	no	—
313	#117	—	2	2	no	—
314	#117	—	2	2	no	—
315	#117	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
328	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
334	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
338	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
342	#107	—	2	4	no	—
343	#106	—	2	4	no	—
344	#106	—	2	4	no	—
345	#106	—	2	4	no	—
346	C₂²⋊C₈	Q₈×C₁₃₇	1	8	yes	complemented
347	#97	—	2	2	no	—
348	C₄×C₈	—	2	2	no	—
349	#95	—	2	2	no	—
350	#95	—	2	1	no	—
351	#95	—	2	1	no	—
352	#95	—	2	2	no	—
353	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	2	4	no	—
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	—	8	2	no	—
358	C₂×Q₈	—	8	2	no	—
359	C₂×Q₈	—	8	2	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	8	2	no	—
371	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
376	C₂²×C₄	C₂×Q₈×C₁₃₇	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
383	C₂×C₈	C₂×Q₈×C₁₃₇	1	4	yes	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	4	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	4	2	no	—
401	C₄⋊C₄	—	8	2	no	—
402	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
403	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
404	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
405	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	—	2	2	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	8	2	no	—
410	C₄²	—	2	4	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
414	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	C₄⋊Q₈×C₁₃₇	1	2	yes	—
420	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
421	Q₈	—	8	4	no	—
422	Q₈	—	8	4	no	—
423	Q₈	—	8	4	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	8	4	no	—
449	C₂×C₄	#120×C₁₃₇	1	2	yes	—
450	C₂×C₄	#120×C₁₃₇	1	2	yes	—
451	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
452	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
454	C₂×C₄	C₂²×Q₈×C₁₃₇	1	2	yes	—
455	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
456	C₈	—	2	4	no	—
457	C₈	—	2	4	no	—
458	C₂²	—	2	4	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	1	no	—
462	C₂²	—	4	1	no	—
463	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
464	C₂²	C₂×C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
465	C₂²	#536×C₁₃₇	1	1	yes	characteristic
466	C₂²	#577×C₁₃₇	1	2	yes	—
467	C₂²	#577×C₁₃₇	1	2	yes	—
468	C₄	—	2	2	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	1	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	4	1	no	—
476	C₄	—	4	2	no	—
477	C₄	—	8	2	no	—
478	C₄	—	8	2	no	—
479	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
480	C₄	D₄×Q₈×C₁₃₇	1	2	yes	—
481	C₂	—	4	2	no	—
482	C₂	#2434×C₁₃₇	1	1	yes	characteristic
483	C₂	#3009×C₁₃₇	1	1	yes	characteristic
484	C₂	#3010×C₁₃₇	1	1	yes	characteristic
485	C₂	#3163×C₁₃₇	1	2	yes	—
486	C₂	#3244×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₂×#117×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#501×C₁₃₇	C₂	semidirect
3. C₂×#498×C₁₃₇	C₂	semidirect
5. #2059×C₁₃₇	C₂	semidirect
6. #2058×C₁₃₇	C₂	semidirect
7. #1993×C₁₃₇	C₂	semidirect
8. #1991×C₁₃₇	C₂	semidirect
102. C₂²⋊C₈×C₁₃₇	Q₈	semidirect
243. #25820	C₁₃₇	direct (N × H)
244. C₁₃₇	#25820	direct (N × H)
246. C₂×#501	C₂×C₁₃₇	semidirect
247. C₂×#498	C₂×C₁₃₇	semidirect
249. #2059	C₂×C₁₃₇	semidirect
250. #2058	C₂×C₁₃₇	semidirect
251. #1993	C₂×C₁₃₇	semidirect
252. #1991	C₂×C₁₃₇	semidirect
346. C₂²⋊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×#544×C₁₃₇	2
C₂×#501×C₁₃₇	2
C₂×#498×C₁₃₇	2
#2628×C₁₃₇	2
#2059×C₁₃₇	2
#2058×C₁₃₇	2
#1993×C₁₃₇	2
#1991×C₁₃₇	2
#1854×C₁₃₇	2
#1853×C₁₃₇	2
#1750×C₁₃₇	2
#25820	137

Subgroups — C137×#25820 #46875206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25820 #46875206