Subgroups — C₁₃₇×#28635 #46878021

Order 35072.

806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3151×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#3142×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#490×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#2932×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#2051×C₁₃₇	C₂	1	4	yes	maximal, complemented
6	#1998×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1965×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#1907×C₁₃₇	C₂	1	4	yes	maximal, complemented
9	#1778×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₁₃₇	C₂²	1	4	yes	—
11	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
12	#532×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#125×C₁₃₇	C₂²	1	1	yes	characteristic
15	#496×C₁₃₇	—	2	2	no	—
16	#493×C₁₃₇	—	2	1	no	—
17	#492×C₁₃₇	—	2	1	no	—
18	#491×C₁₃₇	—	2	1	no	—
19	#490×C₁₃₇	—	2	1	no	—
20	#490×C₁₃₇	—	2	4	no	—
21	#488×C₁₃₇	—	2	2	no	—
22	#435×C₁₃₇	—	2	4	no	—
23	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
24	#423×C₁₃₇	C₂²	1	1	yes	characteristic
25	#417×C₁₃₇	C₂²	1	4	yes	—
26	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
27	C₂×#102×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#100×C₁₃₇	C₂²	1	4	yes	—
29	#408×C₁₃₇	C₂²	1	2	yes	—
30	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
31	C₄×M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
33	#343×C₁₃₇	C₂²	1	4	yes	—
34	#341×C₁₃₇	C₂²	1	2	yes	—
35	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₁₃₇	—	2	2	no	—
37	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
38	AGL(1,ℤ/8)×C₁₃₇	—	2	8	no	—
39	AGL(1,ℤ/8)×C₁₃₇	—	4	4	no	—
40	C₂×SD₁₆×C₁₃₇	—	2	4	no	—
41	C₂×D₈×C₁₃₇	—	2	4	no	—
42	#131×C₁₃₇	—	4	4	no	—
43	C₂×M₁₆×C₁₃₇	—	2	2	no	—
44	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
45	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃₇	—	2	2	no	—
48	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
51	#127×C₁₃₇	—	2	2	no	—
52	#127×C₁₃₇	—	2	1	no	—
53	#127×C₁₃₇	—	2	1	no	—
54	#127×C₁₃₇	D₄	1	1	yes	characteristic
55	#125×C₁₃₇	—	2	1	no	—
56	#125×C₁₃₇	—	2	2	no	—
57	#125×C₁₃₇	—	2	1	no	—
58	#124×C₁₃₇	D₄	1	2	yes	—
59	#121×C₁₃₇	—	2	2	no	—
60	C₄×D₄×C₁₃₇	—	2	2	no	—
61	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
64	#108×C₁₃₇	—	2	4	no	—
65	C₄⋊C₈×C₁₃₇	—	2	2	no	—
66	#103×C₁₃₇	—	2	4	no	—
67	#102×C₁₃₇	—	2	4	no	—
68	#102×C₁₃₇	—	2	4	no	—
69	#102×C₁₃₇	—	2	4	no	—
70	#100×C₁₃₇	—	4	4	no	—
71	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
72	#97×C₁₃₇	—	2	1	no	—
73	#97×C₁₃₇	—	2	1	no	—
74	#97×C₁₃₇	D₄	1	4	yes	—
75	C₄×C₈×C₁₃₇	—	2	1	no	—
76	C₄×C₈×C₁₃₇	—	2	1	no	—
77	C₂⁴×C₁₃₇	—	2	2	no	—
78	#41×C₁₃₇	—	2	4	no	—
79	#41×C₁₃₇	—	2	4	no	—
80	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
81	C₂×D₄×C₁₃₇	—	2	2	no	—
82	C₂×D₄×C₁₃₇	—	2	2	no	—
83	C₂×D₄×C₁₃₇	—	2	2	no	—
84	C₂×D₄×C₁₃₇	—	2	2	no	—
85	C₂×D₄×C₁₃₇	—	2	4	no	—
86	C₂×D₄×C₁₃₇	—	2	4	no	—
87	C₂×D₄×C₁₃₇	—	4	2	no	—
88	C₂×D₄×C₁₃₇	—	4	2	no	—
89	C₂×D₄×C₁₃₇	—	8	4	no	—
90	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃₇	—	2	2	no	—
92	C₂²×C₄×C₁₃₇	—	2	2	no	—
93	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
94	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
95	SD₁₆×C₁₃₇	—	4	8	no	—
96	D₈×C₁₃₇	—	4	8	no	—
97	M₁₆×C₁₃₇	—	2	2	no	—
98	M₁₆×C₁₃₇	—	2	2	no	—
99	M₁₆×C₁₃₇	—	2	2	no	—
100	M₁₆×C₁₃₇	—	2	4	no	—
101	M₁₆×C₁₃₇	—	4	4	no	—
102	M₁₆×C₁₃₇	#41	1	4	yes	—
103	C₂×C₈×C₁₃₇	—	2	2	no	—
104	C₂×C₈×C₁₃₇	—	2	2	no	—
105	C₂×C₈×C₁₃₇	—	2	2	no	—
106	C₂×C₈×C₁₃₇	—	2	2	no	—
107	C₂×C₈×C₁₃₇	—	2	2	no	—
108	C₂×C₈×C₁₃₇	—	4	4	no	—
109	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₁₃₇	—	2	2	no	—
112	C₄⋊C₄×C₁₃₇	—	2	2	no	—
113	C₄⋊C₄×C₁₃₇	—	2	2	no	—
114	C₄⋊C₄×C₁₃₇	—	4	2	no	—
115	C₄⋊C₄×C₁₃₇	—	4	2	no	—
116	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
117	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
120	C₂³×C₁₃₇	—	2	2	no	—
121	C₂³×C₁₃₇	—	4	2	no	—
122	C₂³×C₁₃₇	—	4	4	no	—
123	C₂³×C₁₃₇	—	8	4	no	—
124	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
125	Q₈×C₁₃₇	—	2	4	no	—
126	D₄×C₁₃₇	—	2	4	no	—
127	D₄×C₁₃₇	—	4	4	no	—
128	D₄×C₁₃₇	—	4	2	no	—
129	D₄×C₁₃₇	—	4	2	no	—
130	D₄×C₁₃₇	—	4	4	no	—
131	D₄×C₁₃₇	—	8	2	no	—
132	D₄×C₁₃₇	—	8	4	no	—
133	D₄×C₁₃₇	—	8	4	no	—
134	C₂×C₄×C₁₃₇	—	2	2	no	—
135	C₂×C₄×C₁₃₇	—	4	2	no	—
136	C₂×C₄×C₁₃₇	—	4	2	no	—
137	C₂×C₄×C₁₃₇	—	4	2	no	—
138	C₂×C₄×C₁₃₇	—	4	2	no	—
139	C₂×C₄×C₁₃₇	—	8	2	no	—
140	C₂×C₄×C₁₃₇	#121	1	2	yes	—
141	C₂×C₄×C₁₃₇	#121	1	2	yes	—
142	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
145	C₂×C₄×C₁₃₇	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₁₃₇	—	2	2	no	—
147	C₈×C₁₃₇	—	2	2	no	—
148	C₈×C₁₃₇	—	2	4	no	—
149	C₈×C₁₃₇	—	4	2	no	—
150	C₈×C₁₃₇	—	4	2	no	—
151	C₂²×C₁₃₇	—	2	2	no	—
152	C₂²×C₁₃₇	—	4	2	no	—
153	C₂²×C₁₃₇	—	4	4	no	—
154	C₂²×C₁₃₇	—	8	2	no	—
155	C₂²×C₁₃₇	—	8	4	no	—
156	C₂²×C₁₃₇	—	16	4	no	—
157	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
158	C₂²×C₁₃₇	#559	1	2	yes	—
159	C₄×C₁₃₇	—	2	2	no	—
160	C₄×C₁₃₇	—	2	2	no	—
161	C₄×C₁₃₇	—	4	2	no	—
162	C₄×C₁₃₇	—	8	2	no	—
163	C₄×C₁₃₇	#532	1	1	yes	characteristic
164	C₄×C₁₃₇	#535	1	1	yes	characteristic
165	C₄×C₁₃₇	#546	1	2	yes	—
166	C₂×C₁₃₇	—	2	2	no	—
167	C₂×C₁₃₇	—	8	2	no	—
168	C₂×C₁₃₇	—	16	2	no	—
169	C₂×C₁₃₇	#2544	1	1	yes	characteristic
170	C₂×C₁₃₇	#3068	1	2	yes	—
171	#28635	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₁₃₇	#28635	1	1	yes	characteristic, direct product factor, Sylow
173	#3151	C₂×C₁₃₇	1	1	yes	characteristic
174	#3142	C₂×C₁₃₇	1	1	yes	characteristic, complemented
175	C₂×#490	C₂×C₁₃₇	1	1	yes	characteristic, complemented
176	#2932	C₂×C₁₃₇	1	1	yes	characteristic, complemented
177	#2051	C₂×C₁₃₇	1	4	yes	complemented
178	#1998	C₂×C₁₃₇	1	1	yes	characteristic, complemented
179	#1965	C₂×C₁₃₇	1	1	yes	characteristic, complemented
180	#1907	C₂×C₁₃₇	1	4	yes	complemented
181	#1778	C₂×C₁₃₇	1	1	yes	characteristic, complemented
182	C₂×AGL(1,ℤ/8)	C₂²×C₁₃₇	1	4	yes	—
183	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
184	#532	C₂²×C₁₃₇	1	1	yes	characteristic
185	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
186	C₂×#125	C₂²×C₁₃₇	1	1	yes	characteristic
187	#496	—	2	2	no	—
188	#493	—	2	1	no	—
189	#492	—	2	1	no	—
190	#491	—	2	1	no	—
191	#490	—	2	1	no	—
192	#490	—	2	4	no	—
193	#488	—	2	2	no	—
194	#435	—	2	4	no	—
195	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
196	#423	C₂²×C₁₃₇	1	1	yes	characteristic
197	#417	C₂²×C₁₃₇	1	4	yes	—
198	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
199	C₂×#102	C₂²×C₁₃₇	1	2	yes	—
200	C₂×#100	C₂²×C₁₃₇	1	4	yes	—
201	#408	C₂²×C₁₃₇	1	2	yes	—
202	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
203	C₄×M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
204	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
205	#343	C₂²×C₁₃₇	1	4	yes	—
206	#341	C₂²×C₁₃₇	1	2	yes	—
207	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
210	AGL(1,ℤ/8)	—	2	8	no	—
211	AGL(1,ℤ/8)	—	4	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	4	no	—
214	#131	—	4	4	no	—
215	C₂×M₁₆	—	2	2	no	—
216	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
217	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
218	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
219	C₂²×C₈	—	2	2	no	—
220	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
222	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
223	#127	—	2	2	no	—
224	#127	—	2	1	no	—
225	#127	—	2	1	no	—
226	#127	D₄×C₁₃₇	1	1	yes	characteristic
227	#125	—	2	1	no	—
228	#125	—	2	2	no	—
229	#125	—	2	1	no	—
230	#124	D₄×C₁₃₇	1	2	yes	—
231	#121	—	2	2	no	—
232	C₄×D₄	—	2	2	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
235	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
236	#108	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	#100	—	4	4	no	—
243	C₂²⋊C₈	—	2	4	no	—
244	#97	—	2	1	no	—
245	#97	—	2	1	no	—
246	#97	D₄×C₁₃₇	1	4	yes	—
247	C₄×C₈	—	2	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₂⁴	—	2	2	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	4	no	—
258	C₂×D₄	—	2	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	8	4	no	—
262	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
267	SD₁₆	—	4	8	no	—
268	D₈	—	4	8	no	—
269	M₁₆	—	2	2	no	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	4	no	—
274	M₁₆	#41×C₁₃₇	1	4	yes	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	4	no	—
281	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
282	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	2	4	no	—
289	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
292	C₂³	—	2	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	8	4	no	—
296	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	#121×C₁₃₇	1	2	yes	—
313	C₂×C₄	#121×C₁₃₇	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
316	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
317	C₂×C₄	H(5,𝔽₂)×C₁₃₇	1	2	yes	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	2	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	4	no	—
328	C₂²	—	16	4	no	—
329	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
330	C₂²	#559×C₁₃₇	1	2	yes	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	#532×C₁₃₇	1	1	yes	characteristic
336	C₄	#535×C₁₃₇	1	1	yes	characteristic
337	C₄	#546×C₁₃₇	1	2	yes	—
338	C₂	—	2	2	no	—
339	C₂	—	8	2	no	—
340	C₂	—	16	2	no	—
341	C₂	#2544×C₁₃₇	1	1	yes	characteristic
342	C₂	#3068×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#3151×C₁₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. #3142×C₁₃₇	C₂	semidirect
3. C₂×#490×C₁₃₇	C₂	semidirect
4. #2932×C₁₃₇	C₂	semidirect
5. #2051×C₁₃₇	C₂	semidirect
6. #1998×C₁₃₇	C₂	semidirect
7. #1965×C₁₃₇	C₂	semidirect
8. #1907×C₁₃₇	C₂	semidirect
9. #1778×C₁₃₇	C₂	semidirect
171. #28635	C₁₃₇	direct (N × H)
172. C₁₃₇	#28635	direct (N × H)
174. #3142	C₂×C₁₃₇	semidirect
175. C₂×#490	C₂×C₁₃₇	semidirect
176. #2932	C₂×C₁₃₇	semidirect
177. #2051	C₂×C₁₃₇	semidirect
178. #1998	C₂×C₁₃₇	semidirect
179. #1965	C₂×C₁₃₇	semidirect
180. #1907	C₂×C₁₃₇	semidirect
181. #1778	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#3151×C₁₃₇	2
#3142×C₁₃₇	2
C₂×#490×C₁₃₇	2
#2932×C₁₃₇	2
#2051×C₁₃₇	2
#1998×C₁₃₇	2
#1965×C₁₃₇	2
#1907×C₁₃₇	2
#1778×C₁₃₇	2
#28635	137

Subgroups — C137×#28635 #46878021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28635 #46878021