Subgroups — C₁₃₇×#18073 #46867459

Order 35072.

714 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3169×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2935×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#1594×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1586×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1580×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×#334×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1480×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	#1451×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₁₃₇	—	2	2	no	—
10	#532×C₁₃₇	C₂²	1	2	yes	—
11	#500×C₁₃₇	C₄	1	1	yes	characteristic
12	#493×C₁₃₇	C₄	1	1	yes	characteristic
13	#486×C₁₃₇	C₄	1	2	yes	—
14	#466×C₁₃₇	C₄	1	2	yes	—
15	#466×C₁₃₇	C₄	1	2	yes	—
16	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
17	C₄×D₈×C₁₃₇	C₂²	1	2	yes	—
18	#436×C₁₃₇	C₂²	1	4	yes	—
19	#435×C₁₃₇	—	2	4	no	—
20	C₈×D₄×C₁₃₇	C₂²	1	4	yes	—
21	#431×C₁₃₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
23	#408×C₁₃₇	—	2	2	no	—
24	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#334×C₁₃₇	C₂²	1	1	yes	characteristic
27	#334×C₁₃₇	C₂²	1	2	yes	—
28	#334×C₁₃₇	C₂²	1	1	yes	characteristic
29	D₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
30	D₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
31	#322×C₁₃₇	C₂²	1	2	yes	—
32	#322×C₁₃₇	C₂²	1	2	yes	—
33	C₂×#41×C₁₃₇	—	2	2	no	—
34	C₂×D₈×C₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₁₃₇	C₂×C₄	1	2	yes	—
36	#131×C₁₃₇	—	4	8	no	—
37	C₂×M₁₆×C₁₃₇	—	2	4	no	—
38	C₂×M₁₆×C₁₃₇	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃₇	—	2	2	no	—
40	C₂²×C₈×C₁₃₇	—	2	4	no	—
41	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
44	#127×C₁₃₇	C₂×C₄	1	2	yes	—
45	#124×C₁₃₇	—	2	2	no	—
46	#121×C₁₃₇	—	2	2	no	—
47	#121×C₁₃₇	C₂×C₄	1	4	yes	—
48	C₄×D₄×C₁₃₇	—	2	2	no	—
49	C₄×D₄×C₁₃₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
51	#107×C₁₃₇	C₂×C₄	1	2	yes	—
52	#107×C₁₃₇	C₂×C₄	1	2	yes	—
53	C₄⋊C₈×C₁₃₇	—	2	2	no	—
54	C₄⋊C₈×C₁₃₇	—	2	1	no	—
55	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
56	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
57	#102×C₁₃₇	C₂×C₄	1	4	yes	—
58	#102×C₁₃₇	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
60	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
61	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
62	#97×C₁₃₇	D₄	1	2	yes	—
63	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
66	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃₇	C₂³	1	2	yes	—
69	#41×C₁₃₇	—	4	4	no	—
70	#41×C₁₃₇	—	4	4	no	—
71	C₂×Q₈×C₁₃₇	—	2	2	no	—
72	C₂×D₄×C₁₃₇	—	2	4	no	—
73	C₂×D₄×C₁₃₇	—	2	2	no	—
74	C₂×D₄×C₁₃₇	—	2	2	no	—
75	C₂×D₄×C₁₃₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₃₇	—	2	4	no	—
77	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
80	D₈×C₁₃₇	—	2	4	no	—
81	D₈×C₁₃₇	—	4	2	no	—
82	M₁₆×C₁₃₇	—	2	4	no	—
83	M₁₆×C₁₃₇	—	4	8	no	—
84	C₂×C₈×C₁₃₇	—	2	2	no	—
85	C₂×C₈×C₁₃₇	—	2	2	no	—
86	C₂×C₈×C₁₃₇	—	2	2	no	—
87	C₂×C₈×C₁₃₇	—	2	2	no	—
88	C₂×C₈×C₁₃₇	—	4	1	no	—
89	C₂×C₈×C₁₃₇	—	4	8	no	—
90	C₂×C₈×C₁₃₇	—	4	4	no	—
91	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₇	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₁₃₇	—	2	2	no	—
97	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
99	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
100	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃₇	#41	1	1	yes	characteristic
103	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
104	C₂³×C₁₃₇	—	2	4	no	—
105	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₁₃₇	—	4	4	no	—
107	D₄×C₁₃₇	—	2	4	no	—
108	D₄×C₁₃₇	—	2	4	no	—
109	D₄×C₁₃₇	—	4	8	no	—
110	D₄×C₁₃₇	—	4	4	no	—
111	D₄×C₁₃₇	—	8	2	no	—
112	C₂×C₄×C₁₃₇	—	2	1	no	—
113	C₂×C₄×C₁₃₇	—	2	1	no	—
114	C₂×C₄×C₁₃₇	—	2	4	no	—
115	C₂×C₄×C₁₃₇	—	4	1	no	—
116	C₂×C₄×C₁₃₇	—	4	4	no	—
117	C₂×C₄×C₁₃₇	—	4	4	no	—
118	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₃₇	—	2	1	no	—
126	C₈×C₁₃₇	—	2	1	no	—
127	C₈×C₁₃₇	—	2	1	no	—
128	C₈×C₁₃₇	—	2	4	no	—
129	C₈×C₁₃₇	—	4	2	no	—
130	C₈×C₁₃₇	—	4	2	no	—
131	C₈×C₁₃₇	—	4	2	no	—
132	C₈×C₁₃₇	#131	1	2	yes	complemented
133	C₂²×C₁₃₇	—	2	1	no	—
134	C₂²×C₁₃₇	—	2	1	no	—
135	C₂²×C₁₃₇	—	4	4	no	—
136	C₂²×C₁₃₇	—	4	4	no	—
137	C₂²×C₁₃₇	—	4	4	no	—
138	C₂²×C₁₃₇	#442	1	2	yes	—
139	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₃₇	—	2	1	no	—
141	C₄×C₁₃₇	—	2	1	no	—
142	C₄×C₁₃₇	—	2	1	no	—
143	C₄×C₁₃₇	—	4	4	no	—
144	C₄×C₁₃₇	#442	1	2	yes	—
145	C₄×C₁₃₇	C₂×#131	1	1	yes	characteristic
146	C₄×C₁₃₇	#568	1	1	yes	characteristic
147	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₁₃₇	—	2	2	no	—
149	C₂×C₁₃₇	—	8	4	no	—
150	C₂×C₁₃₇	#2935	1	1	yes	characteristic
151	C₂×C₁₃₇	C₂×#442	1	1	yes	characteristic
152	C₂×C₁₃₇	#2957	1	1	yes	characteristic
153	#18073	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₇	#18073	1	1	yes	characteristic, direct product factor, Sylow
155	#3169	C₂×C₁₃₇	1	1	yes	characteristic
156	#2935	C₂×C₁₃₇	1	2	yes	complemented
157	#1594	C₂×C₁₃₇	1	2	yes	complemented
158	#1586	C₂×C₁₃₇	1	4	yes	complemented
159	#1580	C₂×C₁₃₇	1	2	yes	complemented
160	C₂×#334	C₂×C₁₃₇	1	1	yes	characteristic, complemented
161	#1480	C₂×C₁₃₇	1	2	yes	complemented
162	#1451	C₂×C₁₃₇	1	1	yes	characteristic, complemented
163	C₂×#131	—	2	2	no	—
164	#532	C₂²×C₁₃₇	1	2	yes	—
165	#500	C₄×C₁₃₇	1	1	yes	characteristic
166	#493	C₄×C₁₃₇	1	1	yes	characteristic
167	#486	C₄×C₁₃₇	1	2	yes	—
168	#466	C₄×C₁₃₇	1	2	yes	—
169	#466	C₄×C₁₃₇	1	2	yes	—
170	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
171	C₄×D₈	C₂²×C₁₃₇	1	2	yes	—
172	#436	C₂²×C₁₃₇	1	4	yes	—
173	#435	—	2	4	no	—
174	C₈×D₄	C₂²×C₁₃₇	1	4	yes	—
175	#431	C₂²×C₁₃₇	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
177	#408	—	2	2	no	—
178	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
180	#334	C₂²×C₁₃₇	1	1	yes	characteristic
181	#334	C₂²×C₁₃₇	1	2	yes	—
182	#334	C₂²×C₁₃₇	1	1	yes	characteristic
183	D₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
184	D₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
185	#322	C₂²×C₁₃₇	1	2	yes	—
186	#322	C₂²×C₁₃₇	1	2	yes	—
187	C₂×#41	—	2	2	no	—
188	C₂×D₈	C₂×C₄×C₁₃₇	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₁₃₇	1	2	yes	—
190	#131	—	4	8	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	D₄×C₁₃₇	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
196	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
198	#127	C₂×C₄×C₁₃₇	1	2	yes	—
199	#124	—	2	2	no	—
200	#121	—	2	2	no	—
201	#121	C₂×C₄×C₁₃₇	1	4	yes	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂³×C₁₃₇	1	4	yes	—
204	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
205	#107	C₂×C₄×C₁₃₇	1	2	yes	—
206	#107	C₂×C₄×C₁₃₇	1	2	yes	—
207	C₄⋊C₈	—	2	2	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
210	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
211	#102	C₂×C₄×C₁₃₇	1	4	yes	—
212	#102	C₂×C₄×C₁₃₇	1	4	yes	—
213	C₂²⋊C₈	—	2	1	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	#97	D₄×C₁₃₇	1	2	yes	—
217	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
218	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
220	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₁₃₇	1	2	yes	—
223	#41	—	4	4	no	—
224	#41	—	4	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₁₃₇	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
233	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
234	D₈	—	2	4	no	—
235	D₈	—	4	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	4	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	8	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₁₃₇	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
256	C₄²	#41×C₁₃₇	1	1	yes	characteristic
257	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
258	C₂³	—	2	4	no	—
259	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	4	8	no	—
264	D₄	—	4	4	no	—
265	D₄	—	8	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
274	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
275	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
276	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	2	no	—
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	#131×C₁₃₇	1	2	yes	complemented
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	#442×C₁₃₇	1	2	yes	—
293	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	4	no	—
298	C₄	#442×C₁₃₇	1	2	yes	—
299	C₄	C₂×#131×C₁₃₇	1	1	yes	characteristic
300	C₄	#568×C₁₃₇	1	1	yes	characteristic
301	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
302	C₂	—	2	2	no	—
303	C₂	—	8	4	no	—
304	C₂	#2935×C₁₃₇	1	1	yes	characteristic
305	C₂	C₂×#442×C₁₃₇	1	1	yes	characteristic
306	C₂	#2957×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
socle	C₂²×C₁₃₇	C₂×C₄×D₄
prime radical	#3169×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ C₄×M₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2935×C₁₃₇	C₂	semidirect
3. #1594×C₁₃₇	C₂	semidirect
4. #1586×C₁₃₇	C₂	semidirect
5. #1580×C₁₃₇	C₂	semidirect
6. C₂×#334×C₁₃₇	C₂	semidirect
7. #1480×C₁₃₇	C₂	semidirect
8. #1451×C₁₃₇	C₂	semidirect
132. C₈×C₁₃₇	#131	semidirect
153. #18073	C₁₃₇	direct (N × H)
154. C₁₃₇	#18073	direct (N × H)
156. #2935	C₂×C₁₃₇	semidirect
157. #1594	C₂×C₁₃₇	semidirect
158. #1586	C₂×C₁₃₇	semidirect
159. #1580	C₂×C₁₃₇	semidirect
160. C₂×#334	C₂×C₁₃₇	semidirect
161. #1480	C₂×C₁₃₇	semidirect
162. #1451	C₂×C₁₃₇	semidirect
286. C₈	#131×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3169×C₁₃₇	2
#2935×C₁₃₇	2
#1594×C₁₃₇	2
#1586×C₁₃₇	2
#1580×C₁₃₇	2
C₂×#334×C₁₃₇	2
#1480×C₁₃₇	2
#1451×C₁₃₇	2
#18073	137

Subgroups — C137×#18073 #46867459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#18073 #46867459