Subgroups — C₁₃₇×#21646 #46871032

Order 35072.

1022 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1990×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#1957×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	#1936×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	#1834×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1762×C₁₃₇	C₂	1	4	yes	maximal, complemented
9	C₄×#97×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₇	—	2	1	no	—
13	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
14	C₂×C₄×D₄×C₁₃₇	—	2	1	no	—
15	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
16	#486×C₁₃₇	C₄	1	8	yes	complemented
17	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
18	C₂×#102×C₁₃₇	C₂²	1	4	yes	—
19	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₃₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
23	#391×C₁₃₇	—	2	1	no	—
24	#390×C₁₃₇	—	2	1	no	—
25	#390×C₁₃₇	—	2	1	no	—
26	#390×C₁₃₇	C₂²	1	2	yes	—
27	#389×C₁₃₇	—	2	1	no	—
28	#389×C₁₃₇	C₂²	1	2	yes	—
29	#384×C₁₃₇	C₂²	1	1	yes	characteristic
30	#384×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₁₃₇	—	2	1	no	—
32	C₄×C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
33	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
34	#340×C₁₃₇	C₂²	1	4	yes	—
35	#340×C₁₃₇	C₂²	1	4	yes	—
36	C₂²×Q₈×C₁₃₇	—	2	1	no	—
37	C₂²×D₄×C₁₃₇	—	2	1	no	—
38	C₂²×D₄×C₁₃₇	—	2	1	no	—
39	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₁₃₇	—	2	2	no	—
41	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃₇	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
45	#127×C₁₃₇	—	2	2	no	—
46	#127×C₁₃₇	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₁₃₇	—	4	4	no	—
48	C₄×D₄×C₁₃₇	—	2	8	no	—
49	C₄×D₄×C₁₃₇	—	4	4	no	—
50	C₄×D₄×C₁₃₇	—	4	4	no	—
51	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
63	C₂×C₄²×C₁₃₇	—	2	2	no	—
64	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
71	#102×C₁₃₇	—	2	8	no	—
72	#102×C₁₃₇	C₂×C₄	1	16	yes	complemented
73	#97×C₁₃₇	—	2	4	no	—
74	#97×C₁₃₇	—	2	4	no	—
75	C₄×C₈×C₁₃₇	—	2	2	no	—
76	C₄×C₈×C₁₃₇	C₂×C₄	1	4	yes	complemented
77	#95×C₁₃₇	—	2	2	no	—
78	#95×C₁₃₇	—	2	2	no	—
79	C₂⁴×C₁₃₇	—	2	2	no	—
80	C₂×Q₈×C₁₃₇	—	2	4	no	—
81	C₂×Q₈×C₁₃₇	—	4	2	no	—
82	C₂×Q₈×C₁₃₇	—	4	2	no	—
83	C₂×D₄×C₁₃₇	—	2	4	no	—
84	C₂×D₄×C₁₃₇	—	2	4	no	—
85	C₂×D₄×C₁₃₇	—	2	4	no	—
86	C₂×D₄×C₁₃₇	—	2	4	no	—
87	C₂×D₄×C₁₃₇	—	2	2	no	—
88	C₂×D₄×C₁₃₇	—	4	2	no	—
89	C₂×D₄×C₁₃₇	—	4	2	no	—
90	C₂×D₄×C₁₃₇	—	4	2	no	—
91	C₂×D₄×C₁₃₇	—	4	2	no	—
92	C₂×D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₁₃₇	—	2	2	no	—
94	C₂²×C₄×C₁₃₇	—	2	2	no	—
95	C₂²×C₄×C₁₃₇	—	2	2	no	—
96	C₂²×C₄×C₁₃₇	—	4	4	no	—
97	C₂²×C₄×C₁₃₇	—	4	4	no	—
98	C₂²×C₄×C₁₃₇	—	4	4	no	—
99	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₁₃₇	—	2	2	no	—
107	C₂×C₈×C₁₃₇	—	2	2	no	—
108	C₂×C₈×C₁₃₇	—	2	2	no	—
109	C₂×C₈×C₁₃₇	—	2	2	no	—
110	C₂×C₈×C₁₃₇	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₁₃₇	#41	1	4	yes	—
112	C₄⋊C₄×C₁₃₇	—	2	4	no	—
113	C₄⋊C₄×C₁₃₇	—	2	2	no	—
114	C₄⋊C₄×C₁₃₇	—	2	4	no	—
115	C₄⋊C₄×C₁₃₇	—	2	4	no	—
116	C₄⋊C₄×C₁₃₇	—	4	4	no	—
117	C₄⋊C₄×C₁₃₇	—	4	2	no	—
118	C₄⋊C₄×C₁₃₇	—	4	2	no	—
119	C₄⋊C₄×C₁₃₇	—	4	2	no	—
120	C₄⋊C₄×C₁₃₇	—	4	2	no	—
121	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
123	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
124	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
125	C₄²×C₁₃₇	—	2	2	no	—
126	C₄²×C₁₃₇	—	2	2	no	—
127	C₄²×C₁₃₇	—	2	2	no	—
128	C₄²×C₁₃₇	—	2	2	no	—
129	C₄²×C₁₃₇	—	2	2	no	—
130	C₄²×C₁₃₇	—	4	4	no	—
131	C₄²×C₁₃₇	C₂²×C₄	1	2	yes	—
132	C₄²×C₁₃₇	#41	1	2	yes	—
133	C₄²×C₁₃₇	#41	1	4	yes	—
134	C₂³×C₁₃₇	—	2	4	no	—
135	C₂³×C₁₃₇	—	4	2	no	—
136	C₂³×C₁₃₇	—	4	2	no	—
137	C₂³×C₁₃₇	—	4	2	no	—
138	C₂³×C₁₃₇	—	4	2	no	—
139	C₂³×C₁₃₇	—	4	2	no	—
140	C₂³×C₁₃₇	—	4	2	no	—
141	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
142	Q₈×C₁₃₇	—	4	8	no	—
143	D₄×C₁₃₇	—	2	8	no	—
144	D₄×C₁₃₇	—	4	4	no	—
145	D₄×C₁₃₇	—	4	8	no	—
146	D₄×C₁₃₇	—	4	8	no	—
147	C₂×C₄×C₁₃₇	—	2	1	no	—
148	C₂×C₄×C₁₃₇	—	2	1	no	—
149	C₂×C₄×C₁₃₇	—	2	1	no	—
150	C₂×C₄×C₁₃₇	—	2	1	no	—
151	C₂×C₄×C₁₃₇	—	2	4	no	—
152	C₂×C₄×C₁₃₇	—	2	1	no	—
153	C₂×C₄×C₁₃₇	—	2	1	no	—
154	C₂×C₄×C₁₃₇	—	2	1	no	—
155	C₂×C₄×C₁₃₇	—	2	1	no	—
156	C₂×C₄×C₁₃₇	—	2	1	no	—
157	C₂×C₄×C₁₃₇	—	2	1	no	—
158	C₂×C₄×C₁₃₇	—	4	2	no	—
159	C₂×C₄×C₁₃₇	—	4	2	no	—
160	C₂×C₄×C₁₃₇	—	4	2	no	—
161	C₂×C₄×C₁₃₇	—	4	2	no	—
162	C₂×C₄×C₁₃₇	—	4	2	no	—
163	C₂×C₄×C₁₃₇	—	4	2	no	—
164	C₂×C₄×C₁₃₇	—	4	2	no	—
165	C₂×C₄×C₁₃₇	—	4	2	no	—
166	C₂×C₄×C₁₃₇	—	8	8	no	—
167	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₁₃₇	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₁₃₇	#124	1	2	yes	—
170	C₂×C₄×C₁₃₇	#124	1	2	yes	—
171	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₁₃₇	#137	1	2	yes	—
175	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
177	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
178	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
179	C₈×C₁₃₇	—	2	4	no	—
180	C₈×C₁₃₇	—	2	4	no	—
181	C₂²×C₁₃₇	—	4	4	no	—
182	C₂²×C₁₃₇	—	4	4	no	—
183	C₂²×C₁₃₇	—	4	4	no	—
184	C₂²×C₁₃₇	—	8	2	no	—
185	C₂²×C₁₃₇	—	8	2	no	—
186	C₂²×C₁₃₇	—	8	2	no	—
187	C₂²×C₁₃₇	—	8	2	no	—
188	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
190	C₂²×C₁₃₇	#532	1	1	yes	characteristic
191	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂²×C₁₃₇	C₂×#137	1	1	yes	characteristic
195	C₄×C₁₃₇	—	2	2	no	—
196	C₄×C₁₃₇	—	2	2	no	—
197	C₄×C₁₃₇	—	4	1	no	—
198	C₄×C₁₃₇	—	4	1	no	—
199	C₄×C₁₃₇	—	4	4	no	—
200	C₄×C₁₃₇	—	8	2	no	—
201	C₄×C₁₃₇	—	8	2	no	—
202	C₄×C₁₃₇	#440	1	4	yes	—
203	C₄×C₁₃₇	#442	1	4	yes	—
204	C₄×C₁₃₇	C₄×#41	1	2	yes	—
205	C₄×C₁₃₇	#529	1	2	yes	—
206	C₂×C₁₃₇	—	8	4	no	—
207	C₂×C₁₃₇	C₄×#124	1	1	yes	characteristic
208	C₂×C₁₃₇	C₂×#440	1	1	yes	characteristic
209	C₂×C₁₃₇	C₂×#442	1	1	yes	characteristic
210	C₂×C₁₃₇	#3138	1	1	yes	characteristic
211	C₂×C₁₃₇	#3141	1	1	yes	characteristic
212	C₂×C₁₃₇	#3182	1	1	yes	characteristic
213	C₂×C₁₃₇	#3184	1	1	yes	characteristic
214	#21646	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₁₃₇	#21646	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×#486	C₂×C₁₃₇	1	1	yes	characteristic
217	C₄×C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
218	C₄×#127	C₂×C₁₃₇	1	1	yes	characteristic
219	#1990	C₂×C₁₃₇	1	1	yes	characteristic, complemented
220	#1957	C₂×C₁₃₇	1	1	yes	characteristic
221	#1936	C₂×C₁₃₇	1	4	yes	complemented
222	#1834	C₂×C₁₃₇	1	1	yes	characteristic, complemented
223	#1762	C₂×C₁₃₇	1	4	yes	complemented
224	C₄×#97	C₂×C₁₃₇	1	1	yes	characteristic, complemented
225	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
226	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
227	C₂×C₄×Q₈	—	2	1	no	—
228	C₂×C₄×D₄	—	2	1	no	—
229	C₂×C₄×D₄	—	2	1	no	—
230	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
231	#486	C₄×C₁₃₇	1	8	yes	complemented
232	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
233	C₂×#102	C₂²×C₁₃₇	1	4	yes	—
234	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
235	C₂×#97	C₂²×C₁₃₇	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
238	#391	—	2	1	no	—
239	#390	—	2	1	no	—
240	#390	—	2	1	no	—
241	#390	C₂²×C₁₃₇	1	2	yes	—
242	#389	—	2	1	no	—
243	#389	C₂²×C₁₃₇	1	2	yes	—
244	#384	C₂²×C₁₃₇	1	1	yes	characteristic
245	#384	C₂²×C₁₃₇	1	1	yes	characteristic
246	C₄×C₄⋊C₄	—	2	1	no	—
247	C₄×C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
248	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
249	#340	C₂²×C₁₃₇	1	4	yes	—
250	#340	C₂²×C₁₃₇	1	4	yes	—
251	C₂²×Q₈	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
255	C₂³×C₄	—	2	2	no	—
256	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
257	C₂²×C₈	C₂³×C₁₃₇	1	2	yes	—
258	C₄⋊Q₈	—	2	2	no	—
259	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
260	#127	—	2	2	no	—
261	#127	C₂×C₄×C₁₃₇	1	4	yes	—
262	C₄×Q₈	—	4	4	no	—
263	C₄×D₄	—	2	8	no	—
264	C₄×D₄	—	4	4	no	—
265	C₄×D₄	—	4	4	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
274	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
280	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
281	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
282	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
286	#102	—	2	8	no	—
287	#102	C₂×C₄×C₁₃₇	1	16	yes	complemented
288	#97	—	2	4	no	—
289	#97	—	2	4	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	C₂×C₄×C₁₃₇	1	4	yes	complemented
292	#95	—	2	2	no	—
293	#95	—	2	2	no	—
294	C₂⁴	—	2	2	no	—
295	C₂×Q₈	—	2	4	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	C₂²×C₄×C₁₃₇	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	—	4	4	no	—
314	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
320	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	C₂²×C₄×C₁₃₇	1	4	yes	—
326	C₂×C₈	#41×C₁₃₇	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	4	2	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₂²⋊C₄	—	4	4	no	—
339	C₂²⋊C₄	—	4	4	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	4	no	—
346	C₄²	C₂²×C₄×C₁₃₇	1	2	yes	—
347	C₄²	#41×C₁₃₇	1	2	yes	—
348	C₄²	#41×C₁₃₇	1	4	yes	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
357	Q₈	—	4	8	no	—
358	D₄	—	2	8	no	—
359	D₄	—	4	8	no	—
360	D₄	—	4	8	no	—
361	D₄	—	4	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	8	no	—
382	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₁₃₇	1	2	yes	—
384	C₂×C₄	#124×C₁₃₇	1	2	yes	—
385	C₂×C₄	#124×C₁₃₇	1	2	yes	—
386	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
387	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
388	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
389	C₂×C₄	#137×C₁₃₇	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
392	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
393	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
394	C₈	—	2	4	no	—
395	C₈	—	2	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
404	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
405	C₂²	#532×C₁₃₇	1	1	yes	characteristic
406	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
407	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
408	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
409	C₂²	C₂×#137×C₁₃₇	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	4	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	#440×C₁₃₇	1	4	yes	—
418	C₄	#442×C₁₃₇	1	4	yes	—
419	C₄	C₄×#41×C₁₃₇	1	2	yes	—
420	C₄	#529×C₁₃₇	1	2	yes	—
421	C₂	—	8	4	no	—
422	C₂	C₄×#124×C₁₃₇	1	1	yes	characteristic
423	C₂	C₂×#440×C₁₃₇	1	1	yes	characteristic
424	C₂	C₂×#442×C₁₃₇	1	1	yes	characteristic
425	C₂	#3138×C₁₃₇	1	1	yes	characteristic
426	C₂	#3141×C₁₃₇	1	1	yes	characteristic
427	C₂	#3182×C₁₃₇	1	1	yes	characteristic
428	C₂	#3184×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊Q₈×C₁₃₇	C₂	semidirect
4. #1990×C₁₃₇	C₂	semidirect
6. #1936×C₁₃₇	C₂	semidirect
7. #1834×C₁₃₇	C₂	semidirect
8. #1762×C₁₃₇	C₂	semidirect
9. C₄×#97×C₁₃₇	C₂	semidirect
16. #486×C₁₃₇	C₄	semidirect
44. C₄⋊Q₈×C₁₃₇	C₂×C₄	semidirect
72. #102×C₁₃₇	C₂×C₄	semidirect
76. C₄×C₈×C₁₃₇	C₂×C₄	semidirect
214. #21646	C₁₃₇	direct (N × H)
215. C₁₃₇	#21646	direct (N × H)
217. C₄×C₄⋊Q₈	C₂×C₁₃₇	semidirect
219. #1990	C₂×C₁₃₇	semidirect
221. #1936	C₂×C₁₃₇	semidirect
222. #1834	C₂×C₁₃₇	semidirect
223. #1762	C₂×C₁₃₇	semidirect
224. C₄×#97	C₂×C₁₃₇	semidirect
231. #486	C₄×C₁₃₇	semidirect
259. C₄⋊Q₈	C₂×C₄×C₁₃₇	semidirect
287. #102	C₂×C₄×C₁₃₇	semidirect
291. C₄×C₈	C₂×C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₁₃₇	2
C₄×C₄⋊Q₈×C₁₃₇	2
C₄×#127×C₁₃₇	2
#1990×C₁₃₇	2
#1957×C₁₃₇	2
#1936×C₁₃₇	2
#1834×C₁₃₇	2
#1762×C₁₃₇	2
C₄×#97×C₁₃₇	2
#21646	137

Subgroups — C137×#21646 #46871032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21646 #46871032