Subgroups — C₁₃₇×#28718 #46878104

Order 35072.

946 conjugacy classes of subgroups.

The subgroup lattice has 292 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	2	yes	maximal
2	#2862×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2073×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1990×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1804×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
9	C₂×#127×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃₇	—	2	2	no	—
11	C₂×C₄×D₄×C₁₃₇	C₂²	1	2	yes	—
12	#486×C₁₃₇	—	2	8	no	—
13	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	2	yes	—
14	C₂×#102×C₁₃₇	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
16	#391×C₁₃₇	—	2	2	no	—
17	#391×C₁₃₇	—	2	2	no	—
18	#390×C₁₃₇	—	2	2	no	—
19	#390×C₁₃₇	C₂²	1	2	yes	—
20	#384×C₁₃₇	C₂²	1	1	yes	characteristic
21	#382×C₁₃₇	C₂²	1	2	yes	—
22	#382×C₁₃₇	C₂²	1	2	yes	—
23	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
24	#340×C₁₃₇	C₂²	1	8	yes	—
25	#336×C₁₃₇	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₃₇	—	2	2	no	—
27	C₂²×D₄×C₁₃₇	—	2	2	no	—
28	C₂²×D₄×C₁₃₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃₇	—	2	2	no	—
30	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
31	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
32	C₄⋊Q₈×C₁₃₇	—	2	4	no	—
33	#127×C₁₃₇	—	2	2	no	—
34	#127×C₁₃₇	D₄	1	4	yes	—
35	C₄×D₄×C₁₃₇	—	2	8	no	—
36	C₄×D₄×C₁₃₇	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	2	yes	—
41	C₂×C₂²⋊C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₃₇	—	2	4	no	—
43	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₁₃₇	—	2	4	no	—
49	#102×C₁₃₇	—	2	16	no	—
50	C₄×C₈×C₁₃₇	—	2	4	no	—
51	#95×C₁₃₇	—	2	4	no	—
52	#95×C₁₃₇	—	2	2	no	—
53	#95×C₁₃₇	—	2	2	no	—
54	C₂⁴×C₁₃₇	—	2	2	no	—
55	C₂×Q₈×C₁₃₇	—	4	4	no	—
56	C₂×Q₈×C₁₃₇	—	4	4	no	—
57	C₂×Q₈×C₁₃₇	—	4	4	no	—
58	C₂×D₄×C₁₃₇	—	2	8	no	—
59	C₂×D₄×C₁₃₇	—	2	4	no	—
60	C₂×D₄×C₁₃₇	—	2	2	no	—
61	C₂×D₄×C₁₃₇	—	2	4	no	—
62	C₂×D₄×C₁₃₇	—	4	4	no	—
63	C₂×D₄×C₁₃₇	—	4	4	no	—
64	C₂×D₄×C₁₃₇	C₂×D₄	1	4	yes	—
65	C₂²×C₄×C₁₃₇	—	2	4	no	—
66	C₂²×C₄×C₁₃₇	—	2	2	no	—
67	C₂²×C₄×C₁₃₇	—	4	8	no	—
68	C₂²×C₄×C₁₃₇	—	4	4	no	—
69	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	4	no	—
76	C₂×C₈×C₁₃₇	—	2	4	no	—
77	C₄⋊C₄×C₁₃₇	—	2	4	no	—
78	C₄⋊C₄×C₁₃₇	—	2	4	no	—
79	C₄⋊C₄×C₁₃₇	—	2	4	no	—
80	C₄⋊C₄×C₁₃₇	—	4	4	no	—
81	C₄⋊C₄×C₁₃₇	—	4	4	no	—
82	C₂²⋊C₄×C₁₃₇	—	4	8	no	—
83	C₂²⋊C₄×C₁₃₇	—	4	4	no	—
84	C₄²×C₁₃₇	—	2	4	no	—
85	C₄²×C₁₃₇	—	2	4	no	—
86	C₄²×C₁₃₇	—	2	2	no	—
87	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₇	#41	1	2	yes	—
89	C₄²×C₁₃₇	#41	1	4	yes	—
90	C₂³×C₁₃₇	—	2	4	no	—
91	C₂³×C₁₃₇	—	4	2	no	—
92	C₂³×C₁₃₇	—	4	2	no	—
93	C₂³×C₁₃₇	—	4	4	no	—
94	C₂³×C₁₃₇	—	4	4	no	—
95	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₃₇	—	8	8	no	—
97	D₄×C₁₃₇	—	2	8	no	—
98	D₄×C₁₃₇	—	4	4	no	—
99	D₄×C₁₃₇	—	4	16	no	—
100	C₂×C₄×C₁₃₇	—	2	2	no	—
101	C₂×C₄×C₁₃₇	—	2	2	no	—
102	C₂×C₄×C₁₃₇	—	2	2	no	—
103	C₂×C₄×C₁₃₇	—	2	2	no	—
104	C₂×C₄×C₁₃₇	—	2	2	no	—
105	C₂×C₄×C₁₃₇	—	4	4	no	—
106	C₂×C₄×C₁₃₇	—	4	2	no	—
107	C₂×C₄×C₁₃₇	—	4	4	no	—
108	C₂×C₄×C₁₃₇	—	4	4	no	—
109	C₂×C₄×C₁₃₇	—	4	4	no	—
110	C₂×C₄×C₁₃₇	—	8	8	no	—
111	C₂×C₄×C₁₃₇	#121	1	4	yes	—
112	C₂×C₄×C₁₃₇	#124	1	2	yes	—
113	C₂×C₄×C₁₃₇	#124	1	2	yes	—
114	C₂×C₄×C₁₃₇	#135	1	4	yes	—
115	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	4	yes	—
116	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃₇	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
120	C₈×C₁₃₇	—	4	4	no	—
121	C₂²×C₁₃₇	—	4	8	no	—
122	C₂²×C₁₃₇	—	4	4	no	—
123	C₂²×C₁₃₇	—	8	2	no	—
124	C₂²×C₁₃₇	—	8	2	no	—
125	C₂²×C₁₃₇	—	8	4	no	—
126	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
127	C₂²×C₁₃₇	C₂×#124	1	1	yes	characteristic
128	C₂²×C₁₃₇	#536	1	1	yes	characteristic
129	C₂²×C₁₃₇	C₂×#135	1	2	yes	—
130	C₂²×C₁₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₁₃₇	—	2	2	no	—
132	C₄×C₁₃₇	—	2	2	no	—
133	C₄×C₁₃₇	—	4	2	no	—
134	C₄×C₁₃₇	—	8	2	no	—
135	C₄×C₁₃₇	—	8	4	no	—
136	C₄×C₁₃₇	#463	1	8	yes	—
137	C₄×C₁₃₇	#547	1	2	yes	—
138	C₄×C₁₃₇	#556	1	2	yes	—
139	C₂×C₁₃₇	—	8	4	no	—
140	C₂×C₁₃₇	#2547	1	1	yes	characteristic
141	C₂×C₁₃₇	C₂×#463	1	2	yes	—
142	C₂×C₁₃₇	#3131	1	1	yes	characteristic
143	C₂×C₁₃₇	#3138	1	1	yes	characteristic
144	C₂×C₁₃₇	#3228	1	2	yes	—
145	#28718	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₇	#28718	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#486	C₂×C₁₃₇	1	2	yes	—
148	#2862	C₂×C₁₃₇	1	1	yes	characteristic, complemented
149	C₄×#127	C₂×C₁₃₇	1	1	yes	characteristic
150	#2073	C₂×C₁₃₇	1	4	yes	complemented
151	#1990	C₂×C₁₃₇	1	2	yes	complemented
152	#1804	C₂×C₁₃₇	1	4	yes	complemented
153	C₄×C₄⋊C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
154	C₂×C₄⋊Q₈	C₂²×C₁₃₇	1	2	yes	—
155	C₂×#127	C₂²×C₁₃₇	1	1	yes	characteristic
156	C₂×C₄×D₄	—	2	2	no	—
157	C₂×C₄×D₄	C₂²×C₁₃₇	1	2	yes	—
158	#486	—	2	8	no	—
159	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	2	yes	—
160	C₂×#102	C₂²×C₁₃₇	1	8	yes	—
161	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
162	#391	—	2	2	no	—
163	#391	—	2	2	no	—
164	#390	—	2	2	no	—
165	#390	C₂²×C₁₃₇	1	2	yes	—
166	#384	C₂²×C₁₃₇	1	1	yes	characteristic
167	#382	C₂²×C₁₃₇	1	2	yes	—
168	#382	C₂²×C₁₃₇	1	2	yes	—
169	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
170	#340	C₂²×C₁₃₇	1	8	yes	—
171	#336	C₂²×C₁₃₇	1	2	yes	—
172	C₂²×Q₈	—	2	2	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	C₂³×C₁₃₇	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
177	C₄⋊Q₈	—	2	4	no	—
178	C₄⋊Q₈	—	2	4	no	—
179	#127	—	2	2	no	—
180	#127	D₄×C₁₃₇	1	4	yes	—
181	C₄×D₄	—	2	8	no	—
182	C₄×D₄	—	4	8	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	2	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	4	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
190	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
191	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
192	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	#102	—	2	16	no	—
196	C₄×C₈	—	2	4	no	—
197	#95	—	2	4	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₂⁴	—	2	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×D₄	—	2	8	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂×D₄	C₂×D₄×C₁₃₇	1	4	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	8	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	—	4	8	no	—
229	C₂²⋊C₄	—	4	4	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	2	no	—
233	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
234	C₄²	#41×C₁₃₇	1	2	yes	—
235	C₄²	#41×C₁₃₇	1	4	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	—	4	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	4	no	—
241	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
242	Q₈	—	8	8	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	16	no	—
245	D₄	—	4	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	#121×C₁₃₇	1	4	yes	—
258	C₂×C₄	#124×C₁₃₇	1	2	yes	—
259	C₂×C₄	#124×C₁₃₇	1	2	yes	—
260	C₂×C₄	#135×C₁₃₇	1	4	yes	—
261	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	4	yes	—
262	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
263	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
264	C₂×C₄	C₂×#41×C₁₃₇	1	1	yes	characteristic
265	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
266	C₈	—	4	4	no	—
267	C₂²	—	4	4	no	—
268	C₂²	—	4	8	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	4	no	—
272	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
273	C₂²	C₂×#124×C₁₃₇	1	1	yes	characteristic
274	C₂²	#536×C₁₃₇	1	1	yes	characteristic
275	C₂²	C₂×#135×C₁₃₇	1	2	yes	—
276	C₂²	C₂×AGL(1,ℤ/8)×C₁₃₇	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	4	no	—
282	C₄	#463×C₁₃₇	1	8	yes	—
283	C₄	#547×C₁₃₇	1	2	yes	—
284	C₄	#556×C₁₃₇	1	2	yes	—
285	C₂	—	8	4	no	—
286	C₂	#2547×C₁₃₇	1	1	yes	characteristic
287	C₂	C₂×#463×C₁₃₇	1	2	yes	—
288	C₂	#3131×C₁₃₇	1	1	yes	characteristic
289	C₂	#3138×C₁₃₇	1	1	yes	characteristic
290	C₂	#3228×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	C₂×#127×C₁₃₇	C₂²
LCS	G ▷ C₂²×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. #2862×C₁₃₇	C₂	semidirect
4. #2073×C₁₃₇	C₂	semidirect
5. #1990×C₁₃₇	C₂	semidirect
6. #1804×C₁₃₇	C₂	semidirect
7. C₄×C₄⋊C₈×C₁₃₇	C₂	semidirect
145. #28718	C₁₃₇	direct (N × H)
146. C₁₃₇	#28718	direct (N × H)
148. #2862	C₂×C₁₃₇	semidirect
150. #2073	C₂×C₁₃₇	semidirect
151. #1990	C₂×C₁₃₇	semidirect
152. #1804	C₂×C₁₃₇	semidirect
153. C₄×C₄⋊C₈	C₂×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
#2862×C₁₃₇	2
C₄×#127×C₁₃₇	2
#2073×C₁₃₇	2
#1990×C₁₃₇	2
#1804×C₁₃₇	2
C₄×C₄⋊C₈×C₁₃₇	2
#28718	137

Subgroups — C137×#28718 #46878104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28718 #46878104