Subgroups — C₁₃₇×#25915 #46875301

Order 35072.

1110 conjugacy classes of subgroups.

The subgroup lattice has 534 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, complemented
2	C₂×#490×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#486×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2619×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#2025×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2023×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	#1875×C₁₃₇	C₂	1	2	yes	maximal, complemented
10	#1874×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	#1750×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
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×#127×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×#121×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×#117×C₁₃₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
27	#490×C₁₃₇	—	2	4	no	—
28	#486×C₁₃₇	—	2	4	no	—
29	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
30	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
31	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
32	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	2	yes	—
34	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
36	#394×C₁₃₇	—	2	1	no	—
37	#394×C₁₃₇	—	2	1	no	—
38	#392×C₁₃₇	—	2	1	no	—
39	#392×C₁₃₇	—	2	1	no	—
40	#390×C₁₃₇	C₂²	1	2	yes	—
41	#382×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	#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×D₄×C₁₃₇	—	2	2	no	—
50	C₂²×D₄×C₁₃₇	—	2	1	no	—
51	C₂²×D₄×C₁₃₇	—	2	1	no	—
52	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₁₃₇	—	2	1	no	—
54	C₂³×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₁₃₇	—	2	8	no	—
58	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
59	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
60	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
61	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	complemented
62	#127×C₁₃₇	—	2	2	no	—
63	#127×C₁₃₇	—	2	2	no	—
64	#125×C₁₃₇	—	2	8	no	—
65	#125×C₁₃₇	—	2	2	no	—
66	#125×C₁₃₇	—	2	2	no	—
67	#125×C₁₃₇	—	2	2	no	—
68	#125×C₁₃₇	D₄	1	4	yes	complemented
69	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
70	C₂²⋊Q₈×C₁₃₇	—	2	8	no	—
71	#121×C₁₃₇	—	2	8	no	—
72	#117×C₁₃₇	—	2	2	no	—
73	#117×C₁₃₇	—	2	2	no	—
74	#117×C₁₃₇	—	2	2	no	—
75	#117×C₁₃₇	—	2	2	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	1	no	—
83	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
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	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₁₃₇	—	2	1	no	—
99	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
101	#102×C₁₃₇	—	2	4	no	—
102	#102×C₁₃₇	—	2	4	no	—
103	#102×C₁₃₇	—	2	4	no	—
104	#102×C₁₃₇	—	2	4	no	—
105	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
106	#97×C₁₃₇	—	2	2	no	—
107	C₄×C₈×C₁₃₇	—	2	2	no	—
108	#95×C₁₃₇	—	2	1	no	—
109	#95×C₁₃₇	—	2	1	no	—
110	C₂⁴×C₁₃₇	—	2	2	no	—
111	C₂⁴×C₁₃₇	C₂×D₄	1	1	yes	characteristic
112	C₂×Q₈×C₁₃₇	—	2	4	no	—
113	C₂×Q₈×C₁₃₇	—	2	4	no	—
114	C₂×Q₈×C₁₃₇	—	8	2	no	—
115	C₂×Q₈×C₁₃₇	—	8	2	no	—
116	C₂×D₄×C₁₃₇	—	2	2	no	—
117	C₂×D₄×C₁₃₇	—	2	2	no	—
118	C₂×D₄×C₁₃₇	—	2	4	no	—
119	C₂×D₄×C₁₃₇	—	2	4	no	—
120	C₂×D₄×C₁₃₇	—	4	4	no	—
121	C₂×D₄×C₁₃₇	—	4	2	no	—
122	C₂×D₄×C₁₃₇	—	4	2	no	—
123	C₂×D₄×C₁₃₇	—	8	2	no	—
124	C₂×D₄×C₁₃₇	—	8	2	no	—
125	C₂×D₄×C₁₃₇	—	8	2	no	—
126	C₂×D₄×C₁₃₇	—	8	2	no	—
127	C₂²×C₄×C₁₃₇	—	2	1	no	—
128	C₂²×C₄×C₁₃₇	—	2	1	no	—
129	C₂²×C₄×C₁₃₇	—	2	2	no	—
130	C₂²×C₄×C₁₃₇	—	2	1	no	—
131	C₂²×C₄×C₁₃₇	—	2	1	no	—
132	C₂²×C₄×C₁₃₇	—	2	1	no	—
133	C₂²×C₄×C₁₃₇	—	2	1	no	—
134	C₂²×C₄×C₁₃₇	—	2	4	no	—
135	C₂²×C₄×C₁₃₇	—	2	2	no	—
136	C₂²×C₄×C₁₃₇	—	8	2	no	—
137	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₁₃₇	#41	1	4	yes	—
143	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
144	C₂×C₈×C₁₃₇	—	2	2	no	—
145	C₂×C₈×C₁₃₇	—	2	2	no	—
146	C₂×C₈×C₁₃₇	—	2	2	no	—
147	C₂×C₈×C₁₃₇	—	2	2	no	—
148	C₂×C₈×C₁₃₇	—	2	2	no	—
149	C₂×C₈×C₁₃₇	—	2	2	no	—
150	C₄⋊C₄×C₁₃₇	—	2	4	no	—
151	C₄⋊C₄×C₁₃₇	—	2	4	no	—
152	C₄⋊C₄×C₁₃₇	—	2	4	no	—
153	C₄⋊C₄×C₁₃₇	—	2	4	no	—
154	C₄⋊C₄×C₁₃₇	—	2	4	no	—
155	C₄⋊C₄×C₁₃₇	—	2	4	no	—
156	C₄⋊C₄×C₁₃₇	—	2	4	no	—
157	C₄⋊C₄×C₁₃₇	—	2	2	no	—
158	C₄⋊C₄×C₁₃₇	—	2	2	no	—
159	C₄⋊C₄×C₁₃₇	—	2	1	no	—
160	C₄⋊C₄×C₁₃₇	—	2	1	no	—
161	C₄⋊C₄×C₁₃₇	—	2	1	no	—
162	C₄⋊C₄×C₁₃₇	—	2	1	no	—
163	C₄⋊C₄×C₁₃₇	—	2	4	no	—
164	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
166	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
167	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
168	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
169	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
170	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
171	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
172	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
173	C₂²⋊C₄×C₁₃₇	—	8	2	no	—
174	C₄²×C₁₃₇	—	2	4	no	—
175	C₄²×C₁₃₇	—	2	1	no	—
176	C₄²×C₁₃₇	—	2	1	no	—
177	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
178	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
179	C₂³×C₁₃₇	—	2	1	no	—
180	C₂³×C₁₃₇	—	2	1	no	—
181	C₂³×C₁₃₇	—	2	2	no	—
182	C₂³×C₁₃₇	—	2	2	no	—
183	C₂³×C₁₃₇	—	4	2	no	—
184	C₂³×C₁₃₇	—	4	2	no	—
185	C₂³×C₁₃₇	—	4	2	no	—
186	C₂³×C₁₃₇	—	4	2	no	—
187	C₂³×C₁₃₇	—	4	2	no	—
188	C₂³×C₁₃₇	—	4	2	no	—
189	C₂³×C₁₃₇	—	4	2	no	—
190	C₂³×C₁₃₇	#124	1	2	yes	—
191	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
192	Q₈×C₁₃₇	—	8	4	no	—
193	Q₈×C₁₃₇	—	8	4	no	—
194	D₄×C₁₃₇	—	4	4	no	—
195	D₄×C₁₃₇	—	4	4	no	—
196	D₄×C₁₃₇	—	8	4	no	—
197	D₄×C₁₃₇	—	8	4	no	—
198	D₄×C₁₃₇	—	8	8	no	—
199	C₂×C₄×C₁₃₇	—	2	1	no	—
200	C₂×C₄×C₁₃₇	—	2	1	no	—
201	C₂×C₄×C₁₃₇	—	2	2	no	—
202	C₂×C₄×C₁₃₇	—	2	2	no	—
203	C₂×C₄×C₁₃₇	—	2	4	no	—
204	C₂×C₄×C₁₃₇	—	2	4	no	—
205	C₂×C₄×C₁₃₇	—	2	1	no	—
206	C₂×C₄×C₁₃₇	—	2	2	no	—
207	C₂×C₄×C₁₃₇	—	2	1	no	—
208	C₂×C₄×C₁₃₇	—	2	2	no	—
209	C₂×C₄×C₁₃₇	—	2	2	no	—
210	C₂×C₄×C₁₃₇	—	2	2	no	—
211	C₂×C₄×C₁₃₇	—	2	2	no	—
212	C₂×C₄×C₁₃₇	—	2	2	no	—
213	C₂×C₄×C₁₃₇	—	4	2	no	—
214	C₂×C₄×C₁₃₇	—	4	2	no	—
215	C₂×C₄×C₁₃₇	—	4	2	no	—
216	C₂×C₄×C₁₃₇	—	4	2	no	—
217	C₂×C₄×C₁₃₇	—	4	2	no	—
218	C₂×C₄×C₁₃₇	—	4	2	no	—
219	C₂×C₄×C₁₃₇	—	4	2	no	—
220	C₂×C₄×C₁₃₇	—	8	4	no	—
221	C₂×C₄×C₁₃₇	#120	1	2	yes	—
222	C₂×C₄×C₁₃₇	#120	1	2	yes	—
223	C₂×C₄×C₁₃₇	#124	1	2	yes	—
224	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
227	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
228	C₈×C₁₃₇	—	4	2	no	—
229	C₈×C₁₃₇	—	4	2	no	—
230	C₂²×C₁₃₇	—	2	4	no	—
231	C₂²×C₁₃₇	—	2	2	no	—
232	C₂²×C₁₃₇	—	4	2	no	—
233	C₂²×C₁₃₇	—	4	1	no	—
234	C₂²×C₁₃₇	—	4	1	no	—
235	C₂²×C₁₃₇	—	8	2	no	—
236	C₂²×C₁₃₇	—	8	2	no	—
237	C₂²×C₁₃₇	—	8	2	no	—
238	C₂²×C₁₃₇	—	8	2	no	—
239	C₂²×C₁₃₇	—	8	2	no	—
240	C₂²×C₁₃₇	—	8	2	no	—
241	C₂²×C₁₃₇	—	8	2	no	—
242	C₂²×C₁₃₇	C₂×#120	1	1	yes	characteristic
243	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
244	C₂²×C₁₃₇	#536	1	1	yes	characteristic
245	C₂²×C₁₃₇	#576	1	2	yes	—
246	C₂²×C₁₃₇	#577	1	2	yes	—
247	C₄×C₁₃₇	—	2	2	no	—
248	C₄×C₁₃₇	—	2	2	no	—
249	C₄×C₁₃₇	—	2	2	no	—
250	C₄×C₁₃₇	—	4	1	no	—
251	C₄×C₁₃₇	—	4	2	no	—
252	C₄×C₁₃₇	—	4	2	no	—
253	C₄×C₁₃₇	—	4	2	no	—
254	C₄×C₁₃₇	—	4	1	no	—
255	C₄×C₁₃₇	—	4	2	no	—
256	C₄×C₁₃₇	—	8	2	no	—
257	C₄×C₁₃₇	#548	1	2	yes	—
258	C₄×C₁₃₇	#548	1	2	yes	—
259	C₂×C₁₃₇	—	4	2	no	—
260	C₂×C₁₃₇	—	16	2	no	—
261	C₂×C₁₃₇	#2436	1	1	yes	characteristic
262	C₂×C₁₃₇	#3008	1	1	yes	characteristic
263	C₂×C₁₃₇	#3010	1	1	yes	characteristic
264	C₂×C₁₃₇	#3136	1	2	yes	—
265	C₂×C₁₃₇	#3241	1	2	yes	—
266	#25915	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₁₃₇	#25915	1	1	yes	characteristic, direct product factor, Sylow
268	C₂×#544	C₂×C₁₃₇	1	1	yes	characteristic, complemented
269	C₂×#490	C₂×C₁₃₇	1	1	yes	characteristic, complemented
270	C₂×#486	C₂×C₁₃₇	1	1	yes	characteristic, complemented
271	#2619	C₂×C₁₃₇	1	1	yes	characteristic
272	#2025	C₂×C₁₃₇	1	2	yes	complemented
273	#2023	C₂×C₁₃₇	1	2	yes	complemented
274	#1993	C₂×C₁₃₇	1	1	yes	characteristic, complemented
275	#1991	C₂×C₁₃₇	1	1	yes	characteristic, complemented
276	#1875	C₂×C₁₃₇	1	2	yes	complemented
277	#1874	C₂×C₁₃₇	1	2	yes	complemented
278	#1750	C₂×C₁₃₇	1	1	yes	characteristic, complemented
279	#544	—	2	4	no	—
280	#544	—	2	4	no	—
281	C₂×C₄⋊Q₈	—	2	1	no	—
282	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
284	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
285	C₂×#125	—	2	1	no	—
286	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
287	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
288	C₂×C₂²⋊Q₈	—	2	1	no	—
289	C₂×C₂²⋊Q₈	—	2	1	no	—
290	C₂×#121	C₂²×C₁₃₇	1	2	yes	—
291	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
292	C₂×#117	C₂²×C₁₃₇	1	1	yes	characteristic
293	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
294	#490	—	2	4	no	—
295	#486	—	2	4	no	—
296	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
297	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
298	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
299	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
300	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	2	yes	—
301	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
302	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
303	#394	—	2	1	no	—
304	#394	—	2	1	no	—
305	#392	—	2	1	no	—
306	#392	—	2	1	no	—
307	#390	C₂²×C₁₃₇	1	2	yes	—
308	#382	C₂²×C₁₃₇	1	1	yes	characteristic
309	#340	C₂²×C₁₃₇	1	2	yes	—
310	#340	C₂²×C₁₃₇	1	2	yes	—
311	#340	C₂²×C₁₃₇	1	2	yes	—
312	#340	C₂²×C₁₃₇	1	2	yes	—
313	#336	C₂²×C₁₃₇	1	2	yes	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×D₄	—	2	2	no	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
320	C₂³×C₄	—	2	1	no	—
321	C₂³×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
322	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
323	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
324	C₄⋊Q₈	—	2	8	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	—	2	2	no	—
328	C₄⋊Q₈	D₄×C₁₃₇	1	4	yes	complemented
329	#127	—	2	2	no	—
330	#127	—	2	2	no	—
331	#125	—	2	8	no	—
332	#125	—	2	2	no	—
333	#125	—	2	2	no	—
334	#125	—	2	2	no	—
335	#125	D₄×C₁₃₇	1	4	yes	complemented
336	C₂²⋊Q₈	—	2	8	no	—
337	C₂²⋊Q₈	—	2	8	no	—
338	#121	—	2	8	no	—
339	#117	—	2	2	no	—
340	#117	—	2	2	no	—
341	#117	—	2	2	no	—
342	#117	—	2	2	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	1	no	—
349	C₂×C₄⋊C₄	—	2	1	no	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
353	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
355	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
356	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
357	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
358	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
359	C₂×C₂²⋊C₄	—	2	1	no	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	2	no	—
362	C₂×C₂²⋊C₄	—	2	2	no	—
363	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
364	C₂×C₂²⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
368	#102	—	2	4	no	—
369	#102	—	2	4	no	—
370	#102	—	2	4	no	—
371	#102	—	2	4	no	—
372	C₂²⋊C₈	—	2	4	no	—
373	#97	—	2	2	no	—
374	C₄×C₈	—	2	2	no	—
375	#95	—	2	1	no	—
376	#95	—	2	1	no	—
377	C₂⁴	—	2	2	no	—
378	C₂⁴	C₂×D₄×C₁₃₇	1	1	yes	characteristic
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	8	2	no	—
382	C₂×Q₈	—	8	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	4	4	no	—
388	C₂×D₄	—	4	2	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	8	2	no	—
391	C₂×D₄	—	8	2	no	—
392	C₂×D₄	—	8	2	no	—
393	C₂×D₄	—	8	2	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	8	2	no	—
404	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
409	C₂²×C₄	#41×C₁₃₇	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	4	no	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
432	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
433	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
434	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	4	4	no	—
439	C₂²⋊C₄	—	8	2	no	—
440	C₂²⋊C₄	—	8	2	no	—
441	C₄²	—	2	4	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
445	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	1	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	#124×C₁₃₇	1	2	yes	—
458	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
459	Q₈	—	8	4	no	—
460	Q₈	—	8	4	no	—
461	D₄	—	4	4	no	—
462	D₄	—	4	4	no	—
463	D₄	—	8	4	no	—
464	D₄	—	8	4	no	—
465	D₄	—	8	8	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	8	4	no	—
488	C₂×C₄	#120×C₁₃₇	1	2	yes	—
489	C₂×C₄	#120×C₁₃₇	1	2	yes	—
490	C₂×C₄	#124×C₁₃₇	1	2	yes	—
491	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
494	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
495	C₈	—	4	2	no	—
496	C₈	—	4	2	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	—	8	2	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	C₂×#120×C₁₃₇	1	1	yes	characteristic
510	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
511	C₂²	#536×C₁₃₇	1	1	yes	characteristic
512	C₂²	#576×C₁₃₇	1	2	yes	—
513	C₂²	#577×C₁₃₇	1	2	yes	—
514	C₄	—	2	2	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	4	1	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	2	no	—
523	C₄	—	8	2	no	—
524	C₄	#548×C₁₃₇	1	2	yes	—
525	C₄	#548×C₁₃₇	1	2	yes	—
526	C₂	—	4	2	no	—
527	C₂	—	16	2	no	—
528	C₂	#2436×C₁₃₇	1	1	yes	characteristic
529	C₂	#3008×C₁₃₇	1	1	yes	characteristic
530	C₂	#3010×C₁₃₇	1	1	yes	characteristic
531	C₂	#3136×C₁₃₇	1	2	yes	—
532	C₂	#3241×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#2619×C₁₃₇	C₂
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
1. C₂×#544×C₁₃₇	C₂	semidirect
2. C₂×#490×C₁₃₇	C₂	semidirect
3. C₂×#486×C₁₃₇	C₂	semidirect
5. #2025×C₁₃₇	C₂	semidirect
6. #2023×C₁₃₇	C₂	semidirect
7. #1993×C₁₃₇	C₂	semidirect
8. #1991×C₁₃₇	C₂	semidirect
9. #1875×C₁₃₇	C₂	semidirect
10. #1874×C₁₃₇	C₂	semidirect
11. #1750×C₁₃₇	C₂	semidirect
61. C₄⋊Q₈×C₁₃₇	D₄	semidirect
68. #125×C₁₃₇	D₄	semidirect
266. #25915	C₁₃₇	direct (N × H)
267. C₁₃₇	#25915	direct (N × H)
268. C₂×#544	C₂×C₁₃₇	semidirect
269. C₂×#490	C₂×C₁₃₇	semidirect
270. C₂×#486	C₂×C₁₃₇	semidirect
272. #2025	C₂×C₁₃₇	semidirect
273. #2023	C₂×C₁₃₇	semidirect
274. #1993	C₂×C₁₃₇	semidirect
275. #1991	C₂×C₁₃₇	semidirect
276. #1875	C₂×C₁₃₇	semidirect
277. #1874	C₂×C₁₃₇	semidirect
278. #1750	C₂×C₁₃₇	semidirect
328. C₄⋊Q₈	D₄×C₁₃₇	semidirect
335. #125	D₄×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×#490×C₁₃₇	2
C₂×#486×C₁₃₇	2
#2619×C₁₃₇	2
#2025×C₁₃₇	2
#2023×C₁₃₇	2
#1993×C₁₃₇	2
#1991×C₁₃₇	2
#1875×C₁₃₇	2
#1874×C₁₃₇	2
#1750×C₁₃₇	2
#25915	137

Subgroups — C137×#25915 #46875301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25915 #46875301