Subgroups — C₁₃₇×#17972 #46867358

Order 35072.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#2985×C₁₃₇	C₂	1	2	yes	maximal
3	#1593×C₁₃₇	C₂	1	2	yes	maximal, complemented
4	#1591×C₁₃₇	C₂	1	2	yes	maximal, complemented
5	#1579×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	C₈×Q₁₆×C₁₃₇	C₂	1	2	yes	maximal, complemented
7	#1568×C₁₃₇	C₂	1	1	yes	maximal, characteristic
8	#1497×C₁₃₇	C₂	1	2	yes	maximal
9	#1450×C₁₃₇	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₃₇	C₂²	1	2	yes	—
11	#500×C₁₃₇	C₄	1	1	yes	characteristic
12	#494×C₁₃₇	C₄	1	1	yes	characteristic
13	#487×C₁₃₇	C₄	1	2	yes	—
14	#467×C₁₃₇	C₄	1	2	yes	—
15	#467×C₁₃₇	C₄	1	2	yes	—
16	#446×C₁₃₇	—	2	4	no	—
17	#446×C₁₃₇	C₂²	1	4	yes	—
18	C₈×Q₈×C₁₃₇	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
21	#433×C₁₃₇	—	2	4	no	—
22	#432×C₁₃₇	C₂²	1	2	yes	—
23	#431×C₁₃₇	C₂²	1	1	yes	characteristic
24	#431×C₁₃₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#335×C₁₃₇	C₂²	1	2	yes	—
27	#334×C₁₃₇	C₂²	1	2	yes	—
28	Q₈⋊C₈×C₁₃₇	C₂²	1	4	yes	—
29	Q₈⋊C₈×C₁₃₇	C₂²	1	4	yes	—
30	#322×C₁₃₇	C₂²	1	1	yes	characteristic
31	#322×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
33	C₈²×C₁₃₇	C₂²	1	1	yes	characteristic
34	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
39	C₄⋊Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
40	#125×C₁₃₇	—	2	2	no	—
41	C₂²⋊Q₈×C₁₃₇	—	2	2	no	—
42	C₂²⋊Q₈×C₁₃₇	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₁₃₇	—	2	2	no	—
44	C₄×Q₈×C₁₃₇	C₂³	1	4	yes	—
45	#117×C₁₃₇	—	2	2	no	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#107×C₁₃₇	C₂×C₄	1	2	yes	—
48	#107×C₁₃₇	C₂×C₄	1	2	yes	—
49	C₄⋊C₈×C₁₃₇	—	2	4	no	—
50	C₄⋊C₈×C₁₃₇	—	2	4	no	—
51	C₄⋊C₈×C₁₃₇	—	2	1	no	—
52	C₄⋊C₈×C₁₃₇	D₄	1	2	yes	—
53	C₄⋊C₈×C₁₃₇	C₂³	1	4	yes	—
54	#103×C₁₃₇	C₂×C₄	1	4	yes	—
55	#103×C₁₃₇	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
57	C₂²⋊C₈×C₁₃₇	—	2	1	no	—
58	C₂²⋊C₈×C₁₃₇	D₄	1	2	yes	—
59	#97×C₁₃₇	—	2	4	no	—
60	#97×C₁₃₇	—	2	2	no	—
61	C₄×C₈×C₁₃₇	—	2	4	no	—
62	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
70	C₂×Q₈×C₁₃₇	—	2	2	no	—
71	C₂×Q₈×C₁₃₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
75	Q₁₆×C₁₃₇	—	2	4	no	—
76	Q₁₆×C₁₃₇	—	2	4	no	—
77	C₂×C₈×C₁₃₇	—	2	2	no	—
78	C₂×C₈×C₁₃₇	—	2	2	no	—
79	C₂×C₈×C₁₃₇	—	2	4	no	—
80	C₂×C₈×C₁₃₇	—	2	1	no	—
81	C₂×C₈×C₁₃₇	—	2	1	no	—
82	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₁₃₇	—	2	2	no	—
89	C₄⋊C₄×C₁₃₇	—	2	2	no	—
90	C₄⋊C₄×C₁₃₇	—	2	2	no	—
91	C₄⋊C₄×C₁₃₇	—	2	4	no	—
92	C₄⋊C₄×C₁₃₇	—	2	4	no	—
93	C₄⋊C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
95	C₄²×C₁₃₇	—	2	4	no	—
96	C₄²×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₃₇	#41	1	1	yes	characteristic
99	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
100	C₂³×C₁₃₇	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₁₃₇	—	2	4	no	—
102	Q₈×C₁₃₇	—	2	4	no	—
103	Q₈×C₁₃₇	—	8	2	no	—
104	C₂×C₄×C₁₃₇	—	2	1	no	—
105	C₂×C₄×C₁₃₇	—	2	1	no	—
106	C₂×C₄×C₁₃₇	—	2	2	no	—
107	C₂×C₄×C₁₃₇	—	2	4	no	—
108	C₂×C₄×C₁₃₇	—	2	4	no	—
109	C₂×C₄×C₁₃₇	—	4	1	no	—
110	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
116	C₈×C₁₃₇	—	2	1	no	—
117	C₈×C₁₃₇	—	2	1	no	—
118	C₈×C₁₃₇	—	2	2	no	—
119	C₈×C₁₃₇	—	2	2	no	—
120	C₈×C₁₃₇	—	4	1	no	—
121	C₈×C₁₃₇	—	4	1	no	—
122	C₈×C₁₃₇	—	4	4	no	—
123	C₈×C₁₃₇	#131	1	2	yes	—
124	C₈×C₁₃₇	#131	1	2	yes	—
125	C₂²×C₁₃₇	—	2	1	no	—
126	C₂²×C₁₃₇	—	2	1	no	—
127	C₂²×C₁₃₇	—	2	1	no	—
128	C₂²×C₁₃₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₁₃₇	—	2	1	no	—
130	C₄×C₁₃₇	—	2	1	no	—
131	C₄×C₁₃₇	—	2	1	no	—
132	C₄×C₁₃₇	—	2	1	no	—
133	C₄×C₁₃₇	—	4	4	no	—
134	C₄×C₁₃₇	—	4	4	no	—
135	C₄×C₁₃₇	C₂×#131	1	1	yes	characteristic
136	C₄×C₁₃₇	C₂×#131	1	1	yes	characteristic
137	C₄×C₁₃₇	#578	1	2	yes	—
138	C₂×C₁₃₇	—	4	1	no	—
139	C₂×C₁₃₇	#2931	1	1	yes	characteristic
140	C₂×C₁₃₇	#2950	1	1	yes	characteristic
141	C₂×C₁₃₇	#2957	1	1	yes	characteristic
142	#17972	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₁₃₇	#17972	1	1	yes	characteristic, direct product factor, Sylow
144	#3170	C₂×C₁₃₇	1	1	yes	characteristic
145	#2985	C₂×C₁₃₇	1	2	yes	—
146	#1593	C₂×C₁₃₇	1	2	yes	complemented
147	#1591	C₂×C₁₃₇	1	2	yes	complemented
148	#1579	C₂×C₁₃₇	1	2	yes	complemented
149	C₈×Q₁₆	C₂×C₁₃₇	1	2	yes	complemented
150	#1568	C₂×C₁₃₇	1	1	yes	characteristic
151	#1497	C₂×C₁₃₇	1	2	yes	—
152	#1450	C₂×C₁₃₇	1	1	yes	characteristic
153	#533	C₂²×C₁₃₇	1	2	yes	—
154	#500	C₄×C₁₃₇	1	1	yes	characteristic
155	#494	C₄×C₁₃₇	1	1	yes	characteristic
156	#487	C₄×C₁₃₇	1	2	yes	—
157	#467	C₄×C₁₃₇	1	2	yes	—
158	#467	C₄×C₁₃₇	1	2	yes	—
159	#446	—	2	4	no	—
160	#446	C₂²×C₁₃₇	1	4	yes	—
161	C₈×Q₈	C₂²×C₁₃₇	1	4	yes	—
162	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
163	C₄×Q₁₆	C₂²×C₁₃₇	1	2	yes	—
164	#433	—	2	4	no	—
165	#432	C₂²×C₁₃₇	1	2	yes	—
166	#431	C₂²×C₁₃₇	1	1	yes	characteristic
167	#431	C₂²×C₁₃₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
169	#335	C₂²×C₁₃₇	1	2	yes	—
170	#334	C₂²×C₁₃₇	1	2	yes	—
171	Q₈⋊C₈	C₂²×C₁₃₇	1	4	yes	—
172	Q₈⋊C₈	C₂²×C₁₃₇	1	4	yes	—
173	#322	C₂²×C₁₃₇	1	1	yes	characteristic
174	#322	C₂²×C₁₃₇	1	1	yes	characteristic
175	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
176	C₈²	C₂²×C₁₃₇	1	1	yes	characteristic
177	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
178	C₂×Q₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
179	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
180	C₂²×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
181	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
182	C₄⋊Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	C₂×C₄×C₁₃₇	1	4	yes	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	C₂³×C₁₃₇	1	4	yes	—
188	#117	—	2	2	no	—
189	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
190	#107	C₂×C₄×C₁₃₇	1	2	yes	—
191	#107	C₂×C₄×C₁₃₇	1	2	yes	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	D₄×C₁₃₇	1	2	yes	—
196	C₄⋊C₈	C₂³×C₁₃₇	1	4	yes	—
197	#103	C₂×C₄×C₁₃₇	1	4	yes	—
198	#103	C₂×C₄×C₁₃₇	1	4	yes	—
199	C₂²⋊C₈	—	2	2	no	—
200	C₂²⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	D₄×C₁₃₇	1	2	yes	—
202	#97	—	2	4	no	—
203	#97	—	2	2	no	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
206	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
210	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
211	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₂²×C₄×C₁₃₇	1	4	yes	—
215	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
230	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
241	C₄²	#41×C₁₃₇	1	1	yes	characteristic
242	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
243	C₂³	C₄×D₄×C₁₃₇	1	1	yes	characteristic
244	Q₈	—	2	4	no	—
245	Q₈	—	2	4	no	—
246	Q₈	—	8	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	4	4	no	—
266	C₈	#131×C₁₃₇	1	2	yes	—
267	C₈	#131×C₁₃₇	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₄×D₄×C₁₃₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	4	4	no	—
278	C₄	C₂×#131×C₁₃₇	1	1	yes	characteristic
279	C₄	C₂×#131×C₁₃₇	1	1	yes	characteristic
280	C₄	#578×C₁₃₇	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#2931×C₁₃₇	1	1	yes	characteristic
283	C₂	#2950×C₁₃₇	1	1	yes	characteristic
284	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	C₂³×C₁₃₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ #431×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1593×C₁₃₇	C₂	semidirect
4. #1591×C₁₃₇	C₂	semidirect
5. #1579×C₁₃₇	C₂	semidirect
6. C₈×Q₁₆×C₁₃₇	C₂	semidirect
142. #17972	C₁₃₇	direct (N × H)
143. C₁₃₇	#17972	direct (N × H)
146. #1593	C₂×C₁₃₇	semidirect
147. #1591	C₂×C₁₃₇	semidirect
148. #1579	C₂×C₁₃₇	semidirect
149. C₈×Q₁₆	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3170×C₁₃₇	2
#2985×C₁₃₇	2
#1593×C₁₃₇	2
#1591×C₁₃₇	2
#1579×C₁₃₇	2
C₈×Q₁₆×C₁₃₇	2
#1568×C₁₃₇	2
#1497×C₁₃₇	2
#1450×C₁₃₇	2
#17972	137

Subgroups — C137×#17972 #46867358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#17972 #46867358