Subgroups — C₁₃₇×#19188 #46868574

Order 35072.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3226×C₁₃₇	C₂	1	2	yes	maximal
2	C₂×#494×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1689×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1669×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1564×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1546×C₁₃₇	C₂	1	2	yes	maximal, complemented
8	C₂²×Q₁₆×C₁₃₇	—	2	1	no	—
9	C₂²×Q₁₆×C₁₃₇	—	2	1	no	—
10	C₂×#131×C₁₃₇	—	2	1	no	—
11	#532×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
13	#500×C₁₃₇	—	2	2	no	—
14	#494×C₁₃₇	—	2	1	no	—
15	#494×C₁₃₇	—	2	1	no	—
16	#494×C₁₃₇	C₂²	1	2	yes	—
17	#494×C₁₃₇	C₂²	1	2	yes	—
18	#486×C₁₃₇	—	2	2	no	—
19	#480×C₁₃₇	—	2	4	no	—
20	#474×C₁₃₇	C₂²	1	8	yes	—
21	#436×C₁₃₇	—	2	1	no	—
22	#436×C₁₃₇	C₂²	1	2	yes	—
23	C₈×D₄×C₁₃₇	—	2	1	no	—
24	C₈×D₄×C₁₃₇	C₂²	1	2	yes	—
25	#431×C₁₃₇	C₂²	1	2	yes	—
26	#423×C₁₃₇	C₂²	1	1	yes	characteristic
27	#408×C₁₃₇	—	2	1	no	—
28	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
29	#335×C₁₃₇	C₂²	1	2	yes	—
30	#334×C₁₃₇	C₂²	1	2	yes	—
31	#332×C₁₃₇	C₂²	1	8	yes	—
32	C₂×#41×C₁₃₇	—	2	1	no	—
33	C₂²×Q₈×C₁₃₇	—	2	2	no	—
34	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
35	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
36	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
37	C₂×Q₁₆×C₁₃₇	—	2	2	no	—
38	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
39	C₂×Q₁₆×C₁₃₇	—	4	4	no	—
40	#131×C₁₃₇	—	4	4	no	—
41	C₂×M₁₆×C₁₃₇	—	2	1	no	—
42	C₂×M₁₆×C₁₃₇	—	2	2	no	—
43	C₂²×C₈×C₁₃₇	—	2	2	no	—
44	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
47	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
48	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
49	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₁₃₇	C₂³	1	4	yes	—
51	#127×C₁₃₇	D₄	1	1	yes	characteristic
52	#124×C₁₃₇	—	2	1	no	—
53	#121×C₁₃₇	—	2	1	no	—
54	#121×C₁₃₇	D₄	1	2	yes	—
55	C₄×D₄×C₁₃₇	—	2	1	no	—
56	C₄×D₄×C₁₃₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
59	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
60	#107×C₁₃₇	—	2	8	no	—
61	C₄⋊C₈×C₁₃₇	—	2	1	no	—
62	C₄⋊C₈×C₁₃₇	C₂³	1	2	yes	—
63	C₄⋊C₈×C₁₃₇	C₂³	1	4	yes	—
64	#102×C₁₃₇	—	2	8	no	—
65	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
66	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
67	C₂²⋊C₈×C₁₃₇	—	2	2	no	—
68	C₄×C₈×C₁₃₇	—	2	2	no	—
69	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃₇	D₄	1	1	yes	characteristic
71	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
73	#41×C₁₃₇	—	4	2	no	—
74	#41×C₁₃₇	—	4	2	no	—
75	C₂×Q₈×C₁₃₇	—	2	1	no	—
76	C₂×Q₈×C₁₃₇	—	2	4	no	—
77	C₂×Q₈×C₁₃₇	—	4	2	no	—
78	C₂×Q₈×C₁₃₇	—	4	4	no	—
79	C₂×Q₈×C₁₃₇	—	4	4	no	—
80	C₂×D₄×C₁₃₇	—	2	2	no	—
81	C₂×D₄×C₁₃₇	—	2	1	no	—
82	C₂×D₄×C₁₃₇	—	2	1	no	—
83	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃₇	—	2	2	no	—
85	C₂²×C₄×C₁₃₇	—	4	2	no	—
86	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃₇	#41	1	1	yes	characteristic
89	Q₁₆×C₁₃₇	—	4	4	no	—
90	Q₁₆×C₁₃₇	—	4	4	no	—
91	Q₁₆×C₁₃₇	—	4	4	no	—
92	Q₁₆×C₁₃₇	—	4	4	no	—
93	M₁₆×C₁₃₇	—	4	2	no	—
94	M₁₆×C₁₃₇	—	4	4	no	—
95	C₂×C₈×C₁₃₇	—	2	2	no	—
96	C₂×C₈×C₁₃₇	—	2	4	no	—
97	C₂×C₈×C₁₃₇	—	2	2	no	—
98	C₂×C₈×C₁₃₇	—	4	4	no	—
99	C₂×C₈×C₁₃₇	D₈	1	4	yes	—
100	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₃₇	—	2	1	no	—
105	C₄⋊C₄×C₁₃₇	—	2	4	no	—
106	C₄⋊C₄×C₁₃₇	—	2	4	no	—
107	C₄⋊C₄×C₁₃₇	—	4	2	no	—
108	C₄⋊C₄×C₁₃₇	—	4	2	no	—
109	C₄⋊C₄×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
111	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
112	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃₇	#41	1	1	yes	characteristic
115	C₄²×C₁₃₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₁₃₇	—	2	2	no	—
117	C₂³×C₁₃₇	#121	1	1	yes	characteristic
118	Q₈×C₁₃₇	—	4	2	no	—
119	Q₈×C₁₃₇	—	4	4	no	—
120	Q₈×C₁₃₇	—	4	4	no	—
121	Q₈×C₁₃₇	—	8	4	no	—
122	D₄×C₁₃₇	—	2	4	no	—
123	D₄×C₁₃₇	—	4	4	no	—
124	D₄×C₁₃₇	—	4	2	no	—
125	D₄×C₁₃₇	—	4	2	no	—
126	C₂×C₄×C₁₃₇	—	2	2	no	—
127	C₂×C₄×C₁₃₇	—	2	2	no	—
128	C₂×C₄×C₁₃₇	—	2	2	no	—
129	C₂×C₄×C₁₃₇	—	4	2	no	—
130	C₂×C₄×C₁₃₇	—	4	2	no	—
131	C₂×C₄×C₁₃₇	—	4	4	no	—
132	C₂×C₄×C₁₃₇	—	8	4	no	—
133	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
141	C₈×C₁₃₇	—	2	1	no	—
142	C₈×C₁₃₇	—	2	1	no	—
143	C₈×C₁₃₇	—	4	2	no	—
144	C₈×C₁₃₇	—	4	4	no	—
145	C₈×C₁₃₇	C₂×D₈	1	2	yes	—
146	C₈×C₁₃₇	C₂×D₈	1	2	yes	—
147	C₂²×C₁₃₇	—	2	1	no	—
148	C₂²×C₁₃₇	—	2	1	no	—
149	C₂²×C₁₃₇	—	4	2	no	—
150	C₂²×C₁₃₇	—	4	2	no	—
151	C₂²×C₁₃₇	—	4	2	no	—
152	C₂²×C₁₃₇	#459	1	2	yes	—
153	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
154	C₄×C₁₃₇	—	2	1	no	—
155	C₄×C₁₃₇	—	2	1	no	—
156	C₄×C₁₃₇	—	4	2	no	—
157	C₄×C₁₃₇	—	8	4	no	—
158	C₄×C₁₃₇	#459	1	2	yes	—
159	C₄×C₁₃₇	#473	1	2	yes	—
160	C₄×C₁₃₇	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₁₃₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₄×C₁₃₇	#578	1	2	yes	—
163	C₂×C₁₃₇	—	2	2	no	—
164	C₂×C₁₃₇	—	8	2	no	—
165	C₂×C₁₃₇	C₂×#459	1	1	yes	characteristic
166	C₂×C₁₃₇	#3059	1	1	yes	characteristic
167	C₂×C₁₃₇	C₂×#473	1	1	yes	characteristic
168	#19188	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₁₃₇	#19188	1	1	yes	characteristic, direct product factor, Sylow
170	#3226	C₂×C₁₃₇	1	2	yes	—
171	C₂×#494	C₂×C₁₃₇	1	1	yes	characteristic, complemented
172	#2931	C₂×C₁₃₇	1	1	yes	characteristic
173	#1689	C₂×C₁₃₇	1	4	yes	complemented
174	#1669	C₂×C₁₃₇	1	4	yes	complemented
175	#1564	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	#1546	C₂×C₁₃₇	1	2	yes	complemented
177	C₂²×Q₁₆	—	2	1	no	—
178	C₂²×Q₁₆	—	2	1	no	—
179	C₂×#131	—	2	1	no	—
180	#532	C₂²×C₁₃₇	1	1	yes	characteristic
181	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
182	#500	—	2	2	no	—
183	#494	—	2	1	no	—
184	#494	—	2	1	no	—
185	#494	C₂²×C₁₃₇	1	2	yes	—
186	#494	C₂²×C₁₃₇	1	2	yes	—
187	#486	—	2	2	no	—
188	#480	—	2	4	no	—
189	#474	C₂²×C₁₃₇	1	8	yes	—
190	#436	—	2	1	no	—
191	#436	C₂²×C₁₃₇	1	2	yes	—
192	C₈×D₄	—	2	1	no	—
193	C₈×D₄	C₂²×C₁₃₇	1	2	yes	—
194	#431	C₂²×C₁₃₇	1	2	yes	—
195	#423	C₂²×C₁₃₇	1	1	yes	characteristic
196	#408	—	2	1	no	—
197	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
198	#335	C₂²×C₁₃₇	1	2	yes	—
199	#334	C₂²×C₁₃₇	1	2	yes	—
200	#332	C₂²×C₁₃₇	1	8	yes	—
201	C₂×#41	—	2	1	no	—
202	C₂²×Q₈	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	4	4	no	—
208	C₂×Q₁₆	—	4	4	no	—
209	#131	—	4	4	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
214	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
219	C₄⋊Q₈	C₂³×C₁₃₇	1	4	yes	—
220	#127	D₄×C₁₃₇	1	1	yes	characteristic
221	#124	—	2	1	no	—
222	#121	—	2	1	no	—
223	#121	D₄×C₁₃₇	1	2	yes	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	C₂³×C₁₃₇	1	2	yes	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
229	#107	—	2	8	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	C₂³×C₁₃₇	1	2	yes	—
232	C₄⋊C₈	C₂³×C₁₃₇	1	4	yes	—
233	#102	—	2	8	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₄×C₈	—	2	2	no	—
238	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
239	C₄×C₈	D₄×C₁₃₇	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
242	#41	—	4	2	no	—
243	#41	—	4	2	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
257	C₂²×C₄	#41×C₁₃₇	1	1	yes	characteristic
258	Q₁₆	—	4	4	no	—
259	Q₁₆	—	4	4	no	—
260	Q₁₆	—	4	4	no	—
261	Q₁₆	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	4	no	—
268	C₂×C₈	D₈×C₁₃₇	1	4	yes	—
269	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₁₃₇	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
283	C₄²	#41×C₁₃₇	1	1	yes	characteristic
284	C₄²	C₂⁴×C₁₃₇	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	C₂³	#121×C₁₃₇	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	4	no	—
289	Q₈	—	4	4	no	—
290	Q₈	—	8	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
303	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
304	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
306	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×D₈×C₁₃₇	1	2	yes	—
315	C₈	C₂×D₈×C₁₃₇	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#459×C₁₃₇	1	2	yes	—
322	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	4	no	—
327	C₄	#459×C₁₃₇	1	2	yes	—
328	C₄	#473×C₁₃₇	1	2	yes	—
329	C₄	C₂²×D₈×C₁₃₇	1	1	yes	characteristic
330	C₄	C₂×AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
331	C₄	#578×C₁₃₇	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×#459×C₁₃₇	1	1	yes	characteristic
335	C₂	#3059×C₁₃₇	1	1	yes	characteristic
336	C₂	C₂×#473×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#494×C₁₃₇	C₂	semidirect
4. #1689×C₁₃₇	C₂	semidirect
5. #1669×C₁₃₇	C₂	semidirect
6. #1564×C₁₃₇	C₂	semidirect
7. #1546×C₁₃₇	C₂	semidirect
168. #19188	C₁₃₇	direct (N × H)
169. C₁₃₇	#19188	direct (N × H)
171. C₂×#494	C₂×C₁₃₇	semidirect
173. #1689	C₂×C₁₃₇	semidirect
174. #1669	C₂×C₁₃₇	semidirect
175. #1564	C₂×C₁₃₇	semidirect
176. #1546	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3226×C₁₃₇	2
C₂×#494×C₁₃₇	2
#2931×C₁₃₇	2
#1689×C₁₃₇	2
#1669×C₁₃₇	2
#1564×C₁₃₇	2
#1546×C₁₃₇	2
#19188	137

Subgroups — C137×#19188 #46868574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#19188 #46868574