Subgroups — C₁₃₇×#25320 #46874706

Order 35072.

894 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#461×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#2542×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2365×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2072×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#2055×C₁₃₇	C₂	1	2	yes	maximal
7	#1988×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1939×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#1880×C₁₃₇	C₂	1	2	yes	maximal
10	#1803×C₁₃₇	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₁₃₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₁₃₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₁₃₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₁₃₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
19	#487×C₁₃₇	—	2	4	no	—
20	#461×C₁₃₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
23	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
24	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₃₇	C₂²	1	2	yes	—
26	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₁₃₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	#399×C₁₃₇	—	2	2	no	—
30	#391×C₁₃₇	C₂²	1	2	yes	—
31	#390×C₁₃₇	C₂²	1	2	yes	—
32	#389×C₁₃₇	—	2	1	no	—
33	#387×C₁₃₇	—	2	2	no	—
34	#384×C₁₃₇	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
36	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
38	#340×C₁₃₇	C₂²	1	1	yes	characteristic
39	#340×C₁₃₇	C₂²	1	1	yes	characteristic
40	#340×C₁₃₇	C₂²	1	2	yes	—
41	#340×C₁₃₇	C₂²	1	2	yes	—
42	#336×C₁₃₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₃₇	—	2	1	no	—
44	C₂²×Q₈×C₁₃₇	C₂³	1	2	yes	—
45	C₂²×D₄×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₁₃₇	—	2	1	no	—
47	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
48	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
49	C₂×SD₁₆×C₁₃₇	—	4	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	—	2	2	no	—
55	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
56	C₄⋊Q₈×C₁₃₇	D₄	1	4	yes	—
57	C₂²⋊Q₈×C₁₃₇	—	4	4	no	—
58	C₂²⋊Q₈×C₁₃₇	—	4	4	no	—
59	C₄×Q₈×C₁₃₇	—	2	4	no	—
60	C₄×Q₈×C₁₃₇	—	4	2	no	—
61	C₄×D₄×C₁₃₇	—	2	2	no	—
62	C₄×D₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₁₃₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₁₃₇	—	2	1	no	—
78	C₂×C₄²×C₁₃₇	—	2	1	no	—
79	C₂×C₄²×C₁₃₇	—	2	2	no	—
80	C₂×C₄²×C₁₃₇	C₂³	1	2	yes	—
81	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₁₃₇	—	2	2	no	—
83	C₄⋊C₈×C₁₃₇	—	4	2	no	—
84	#103×C₁₃₇	—	2	4	no	—
85	#103×C₁₃₇	—	2	4	no	—
86	#103×C₁₃₇	—	2	4	no	—
87	#102×C₁₃₇	—	2	2	no	—
88	#102×C₁₃₇	—	2	2	no	—
89	C₄×C₈×C₁₃₇	—	2	2	no	—
90	#95×C₁₃₇	—	2	2	no	—
91	#95×C₁₃₇	—	2	1	no	—
92	#95×C₁₃₇	—	2	2	no	—
93	#95×C₁₃₇	D₄	1	2	yes	—
94	C₂⁴×C₁₃₇	—	2	1	no	—
95	C₂×Q₈×C₁₃₇	—	2	2	no	—
96	C₂×Q₈×C₁₃₇	—	2	2	no	—
97	C₂×Q₈×C₁₃₇	—	2	2	no	—
98	C₂×Q₈×C₁₃₇	—	2	2	no	—
99	C₂×Q₈×C₁₃₇	—	2	2	no	—
100	C₂×Q₈×C₁₃₇	—	4	2	no	—
101	C₂×Q₈×C₁₃₇	—	4	2	no	—
102	C₂×Q₈×C₁₃₇	—	4	2	no	—
103	C₂×Q₈×C₁₃₇	C₂×D₄	1	4	yes	—
104	C₂×D₄×C₁₃₇	—	2	1	no	—
105	C₂×D₄×C₁₃₇	—	2	1	no	—
106	C₂×D₄×C₁₃₇	—	2	1	no	—
107	C₂×D₄×C₁₃₇	—	2	1	no	—
108	C₂×D₄×C₁₃₇	—	2	1	no	—
109	C₂×D₄×C₁₃₇	#41	1	2	yes	—
110	C₂²×C₄×C₁₃₇	—	2	2	no	—
111	C₂²×C₄×C₁₃₇	—	2	1	no	—
112	C₂²×C₄×C₁₃₇	—	2	1	no	—
113	C₂²×C₄×C₁₃₇	—	2	2	no	—
114	C₂²×C₄×C₁₃₇	—	2	1	no	—
115	C₂²×C₄×C₁₃₇	—	4	2	no	—
116	C₂²×C₄×C₁₃₇	—	4	2	no	—
117	C₂²×C₄×C₁₃₇	—	4	2	no	—
118	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
123	SD₁₆×C₁₃₇	—	4	8	no	—
124	C₂×C₈×C₁₃₇	—	2	1	no	—
125	C₂×C₈×C₁₃₇	—	2	1	no	—
126	C₂×C₈×C₁₃₇	—	2	2	no	—
127	C₂×C₈×C₁₃₇	—	2	1	no	—
128	C₂×C₈×C₁₃₇	—	2	1	no	—
129	C₂×C₈×C₁₃₇	—	2	2	no	—
130	C₂×C₈×C₁₃₇	—	2	1	no	—
131	C₂×C₈×C₁₃₇	—	2	2	no	—
132	C₂×C₈×C₁₃₇	—	2	1	no	—
133	C₄⋊C₄×C₁₃₇	—	2	4	no	—
134	C₄⋊C₄×C₁₃₇	—	2	4	no	—
135	C₄⋊C₄×C₁₃₇	—	2	2	no	—
136	C₄⋊C₄×C₁₃₇	—	2	1	no	—
137	C₄⋊C₄×C₁₃₇	—	2	1	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	2	no	—
142	C₄⋊C₄×C₁₃₇	—	2	1	no	—
143	C₄⋊C₄×C₁₃₇	—	4	4	no	—
144	C₄⋊C₄×C₁₃₇	—	4	4	no	—
145	C₄⋊C₄×C₁₃₇	—	4	2	no	—
146	C₄⋊C₄×C₁₃₇	—	4	4	no	—
147	C₄⋊C₄×C₁₃₇	SD₁₆	1	4	yes	complemented
148	C₄⋊C₄×C₁₃₇	#41	1	2	yes	—
149	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
150	C₂²⋊C₄×C₁₃₇	—	4	2	no	—
151	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
152	C₄²×C₁₃₇	—	2	2	no	—
153	C₄²×C₁₃₇	—	2	2	no	—
154	C₄²×C₁₃₇	—	4	4	no	—
155	C₄²×C₁₃₇	—	4	2	no	—
156	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
157	C₄²×C₁₃₇	#41	1	2	yes	—
158	C₂³×C₁₃₇	—	2	2	no	—
159	C₂³×C₁₃₇	—	4	1	no	—
160	C₂³×C₁₃₇	—	4	1	no	—
161	C₂³×C₁₃₇	—	4	1	no	—
162	C₂³×C₁₃₇	—	4	1	no	—
163	C₂³×C₁₃₇	—	4	1	no	—
164	C₂³×C₁₃₇	—	4	1	no	—
165	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
166	Q₈×C₁₃₇	—	2	4	no	—
167	Q₈×C₁₃₇	—	2	4	no	—
168	Q₈×C₁₃₇	—	4	2	no	—
169	Q₈×C₁₃₇	—	4	2	no	—
170	Q₈×C₁₃₇	—	8	4	no	—
171	D₄×C₁₃₇	—	2	2	no	—
172	D₄×C₁₃₇	—	2	2	no	—
173	D₄×C₁₃₇	—	4	1	no	—
174	D₄×C₁₃₇	—	4	1	no	—
175	C₂×C₄×C₁₃₇	—	2	2	no	—
176	C₂×C₄×C₁₃₇	—	2	2	no	—
177	C₂×C₄×C₁₃₇	—	2	1	no	—
178	C₂×C₄×C₁₃₇	—	2	1	no	—
179	C₂×C₄×C₁₃₇	—	2	4	no	—
180	C₂×C₄×C₁₃₇	—	2	2	no	—
181	C₂×C₄×C₁₃₇	—	2	2	no	—
182	C₂×C₄×C₁₃₇	—	2	2	no	—
183	C₂×C₄×C₁₃₇	—	2	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	1	no	—
189	C₂×C₄×C₁₃₇	—	4	2	no	—
190	C₂×C₄×C₁₃₇	—	4	1	no	—
191	C₂×C₄×C₁₃₇	—	4	2	no	—
192	C₂×C₄×C₁₃₇	—	4	1	no	—
193	C₂×C₄×C₁₃₇	—	4	1	no	—
194	C₂×C₄×C₁₃₇	—	4	2	no	—
195	C₂×C₄×C₁₃₇	—	4	1	no	—
196	C₂×C₄×C₁₃₇	—	4	1	no	—
197	C₂×C₄×C₁₃₇	—	8	4	no	—
198	C₂×C₄×C₁₃₇	#121	1	2	yes	—
199	C₂×C₄×C₁₃₇	#121	1	2	yes	—
200	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
201	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
202	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	1	yes	characteristic
208	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	1	yes	characteristic
209	C₈×C₁₃₇	—	4	1	no	—
210	C₈×C₁₃₇	—	4	2	no	—
211	C₈×C₁₃₇	—	4	1	no	—
212	C₂²×C₁₃₇	—	4	2	no	—
213	C₂²×C₁₃₇	—	4	2	no	—
214	C₂²×C₁₃₇	—	4	2	no	—
215	C₂²×C₁₃₇	—	8	1	no	—
216	C₂²×C₁₃₇	—	8	1	no	—
217	C₂²×C₁₃₇	—	8	1	no	—
218	C₂²×C₁₃₇	—	8	1	no	—
219	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
220	C₂²×C₁₃₇	#546	1	1	yes	characteristic
221	C₂²×C₁₃₇	#548	1	1	yes	characteristic
222	C₂²×C₁₃₇	C₂²×SD₁₆	1	1	yes	characteristic
223	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂²×C₁₃₇	#578	1	1	yes	characteristic
225	C₂²×C₁₃₇	#578	1	1	yes	characteristic
226	C₄×C₁₃₇	—	2	4	no	—
227	C₄×C₁₃₇	—	2	2	no	—
228	C₄×C₁₃₇	—	4	2	no	—
229	C₄×C₁₃₇	—	4	2	no	—
230	C₄×C₁₃₇	—	4	4	no	—
231	C₄×C₁₃₇	—	8	1	no	—
232	C₄×C₁₃₇	—	8	1	no	—
233	C₄×C₁₃₇	—	8	2	no	—
234	C₄×C₁₃₇	#460	1	4	yes	—
235	C₄×C₁₃₇	#529	1	1	yes	characteristic
236	C₄×C₁₃₇	#542	1	1	yes	characteristic
237	C₄×C₁₃₇	#546	1	1	yes	characteristic
238	C₄×C₁₃₇	#547	1	1	yes	characteristic
239	C₂×C₁₃₇	—	8	2	no	—
240	C₂×C₁₃₇	#2427	1	1	yes	characteristic
241	C₂×C₁₃₇	C₂×#460	1	1	yes	characteristic
242	C₂×C₁₃₇	#3044	1	1	yes	characteristic
243	C₂×C₁₃₇	#3298	1	1	yes	characteristic
244	C₂×C₁₃₇	#3330	1	1	yes	characteristic
245	C₂×C₁₃₇	#3359	1	1	yes	characteristic
246	C₂×C₁₃₇	#3367	1	1	yes	characteristic
247	#25320	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₁₃₇	#25320	1	1	yes	characteristic, direct product factor, Sylow
249	C₂×#487	C₂×C₁₃₇	1	1	yes	characteristic, complemented
250	C₂×#461	C₂×C₁₃₇	1	1	yes	characteristic
251	#2542	C₂×C₁₃₇	1	1	yes	characteristic
252	#2365	C₂×C₁₃₇	1	1	yes	characteristic, complemented
253	#2072	C₂×C₁₃₇	1	2	yes	complemented
254	#2055	C₂×C₁₃₇	1	2	yes	—
255	#1988	C₂×C₁₃₇	1	1	yes	characteristic, complemented
256	#1939	C₂×C₁₃₇	1	1	yes	characteristic, complemented
257	#1880	C₂×C₁₃₇	1	2	yes	—
258	#1803	C₂×C₁₃₇	1	2	yes	complemented
259	C₄×#102	C₂×C₁₃₇	1	1	yes	characteristic
260	C₂²×SD₁₆	C₂²×C₁₃₇	1	2	yes	—
261	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
262	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
263	C₂×C₂²⋊Q₈	—	2	1	no	—
264	C₂×C₂²⋊Q₈	—	2	1	no	—
265	C₂×C₄×Q₈	C₂²×C₁₃₇	1	2	yes	—
266	C₂×C₄×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
267	#487	—	2	4	no	—
268	#461	—	2	4	no	—
269	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
271	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
272	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
273	C₂×#103	C₂²×C₁₃₇	1	2	yes	—
274	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
275	C₂×#102	C₂²×C₁₃₇	1	1	yes	characteristic
276	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
277	#399	—	2	2	no	—
278	#391	C₂²×C₁₃₇	1	2	yes	—
279	#390	C₂²×C₁₃₇	1	2	yes	—
280	#389	—	2	1	no	—
281	#387	—	2	2	no	—
282	#384	C₂²×C₁₃₇	1	2	yes	—
283	C₄×C₄⋊C₄	—	2	1	no	—
284	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
285	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
286	#340	C₂²×C₁₃₇	1	2	yes	—
287	#340	C₂²×C₁₃₇	1	1	yes	characteristic
288	#340	C₂²×C₁₃₇	1	2	yes	—
289	#340	C₂²×C₁₃₇	1	1	yes	characteristic
290	#336	C₂²×C₁₃₇	1	2	yes	—
291	C₂²×Q₈	—	2	1	no	—
292	C₂²×Q₈	C₂³×C₁₃₇	1	2	yes	—
293	C₂²×D₄	C₂³×C₁₃₇	1	1	yes	characteristic
294	C₂³×C₄	—	2	1	no	—
295	C₂×SD₁₆	—	2	4	no	—
296	C₂×SD₁₆	—	2	4	no	—
297	C₂×SD₁₆	—	4	2	no	—
298	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
299	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
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	C₂²⋊Q₈	—	4	4	no	—
306	C₂²⋊Q₈	—	4	4	no	—
307	C₄×Q₈	—	2	4	no	—
308	C₄×Q₈	—	4	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
318	C₂×C₄⋊C₄	D₄×C₁₃₇	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₄²	—	2	2	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	C₂³×C₁₃₇	1	2	yes	—
329	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	—	4	2	no	—
332	#103	—	2	4	no	—
333	#103	—	2	4	no	—
334	#103	—	2	4	no	—
335	#102	—	2	2	no	—
336	#102	—	2	2	no	—
337	C₄×C₈	—	2	2	no	—
338	#95	—	2	2	no	—
339	#95	—	2	1	no	—
340	#95	—	2	2	no	—
341	#95	D₄×C₁₃₇	1	2	yes	—
342	C₂⁴	—	2	1	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	2	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	4	2	no	—
349	C₂×Q₈	—	4	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	C₂×D₄×C₁₃₇	1	4	yes	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	#41×C₁₃₇	1	2	yes	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	4	2	no	—
364	C₂²×C₄	—	4	2	no	—
365	C₂²×C₄	—	4	2	no	—
366	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
370	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
371	SD₁₆	—	4	8	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	—	2	1	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	4	4	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	4	no	—
395	C₄⋊C₄	SD₁₆×C₁₃₇	1	4	yes	complemented
396	C₄⋊C₄	#41×C₁₃₇	1	2	yes	—
397	C₂²⋊C₄	—	4	4	no	—
398	C₂²⋊C₄	—	4	2	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	4	4	no	—
403	C₄²	—	4	2	no	—
404	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
405	C₄²	#41×C₁₃₇	1	2	yes	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	1	no	—
412	C₂³	—	4	1	no	—
413	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
414	Q₈	—	2	4	no	—
415	Q₈	—	2	4	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	8	4	no	—
419	D₄	—	2	2	no	—
420	D₄	—	2	2	no	—
421	D₄	—	4	1	no	—
422	D₄	—	4	1	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	8	4	no	—
446	C₂×C₄	#121×C₁₃₇	1	2	yes	—
447	C₂×C₄	#121×C₁₃₇	1	2	yes	—
448	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
449	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
450	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
454	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
455	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	1	yes	characteristic
456	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	1	yes	characteristic
457	C₈	—	4	2	no	—
458	C₈	—	4	1	no	—
459	C₈	—	4	1	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	1	no	—
465	C₂²	—	8	1	no	—
466	C₂²	—	8	1	no	—
467	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
468	C₂²	#546×C₁₃₇	1	1	yes	characteristic
469	C₂²	#548×C₁₃₇	1	1	yes	characteristic
470	C₂²	C₂²×SD₁₆×C₁₃₇	1	1	yes	characteristic
471	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
472	C₂²	#578×C₁₃₇	1	1	yes	characteristic
473	C₂²	#578×C₁₃₇	1	1	yes	characteristic
474	C₄	—	2	4	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	4	no	—
479	C₄	—	8	1	no	—
480	C₄	—	8	1	no	—
481	C₄	—	8	2	no	—
482	C₄	#460×C₁₃₇	1	4	yes	—
483	C₄	#529×C₁₃₇	1	1	yes	characteristic
484	C₄	#542×C₁₃₇	1	1	yes	characteristic
485	C₄	#546×C₁₃₇	1	1	yes	characteristic
486	C₄	#547×C₁₃₇	1	1	yes	characteristic
487	C₂	—	8	2	no	—
488	C₂	#2427×C₁₃₇	1	1	yes	characteristic
489	C₂	C₂×#460×C₁₃₇	1	1	yes	characteristic
490	C₂	#3044×C₁₃₇	1	1	yes	characteristic
491	C₂	#3298×C₁₃₇	1	1	yes	characteristic
492	C₂	#3330×C₁₃₇	1	1	yes	characteristic
493	C₂	#3359×C₁₃₇	1	1	yes	characteristic
494	C₂	#3367×C₁₃₇	1	1	yes	characteristic

	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 ◁ C₄×C₄⋊C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#487×C₁₃₇	C₂	semidirect
4. #2365×C₁₃₇	C₂	semidirect
5. #2072×C₁₃₇	C₂	semidirect
7. #1988×C₁₃₇	C₂	semidirect
8. #1939×C₁₃₇	C₂	semidirect
10. #1803×C₁₃₇	C₂	semidirect
147. C₄⋊C₄×C₁₃₇	SD₁₆	semidirect
247. #25320	C₁₃₇	direct (N × H)
248. C₁₃₇	#25320	direct (N × H)
249. C₂×#487	C₂×C₁₃₇	semidirect
252. #2365	C₂×C₁₃₇	semidirect
253. #2072	C₂×C₁₃₇	semidirect
255. #1988	C₂×C₁₃₇	semidirect
256. #1939	C₂×C₁₃₇	semidirect
258. #1803	C₂×C₁₃₇	semidirect
395. C₄⋊C₄	SD₁₆×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#487×C₁₃₇	2
C₂×#461×C₁₃₇	2
#2542×C₁₃₇	2
#2365×C₁₃₇	2
#2072×C₁₃₇	2
#2055×C₁₃₇	2
#1988×C₁₃₇	2
#1939×C₁₃₇	2
#1880×C₁₃₇	2
#1803×C₁₃₇	2
C₄×#102×C₁₃₇	2
#25320	137

Subgroups — C137×#25320 #46874706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#25320 #46874706