Subgroups — C₁₃₇×#13138 #46862524

Order 35072.

354 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#365×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#361×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#361×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#1972×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1778×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#1383×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#1377×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×#108×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×#106×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#102×C₁₃₇	—	2	1	no	—
15	C₄×M₁₆×C₁₃₇	—	2	1	no	—
16	C₂×#97×C₁₃₇	C₂²	1	1	yes	characteristic
17	#390×C₁₃₇	—	2	1	no	—
18	#365×C₁₃₇	—	2	2	no	—
19	#361×C₁₃₇	—	2	2	no	—
20	#361×C₁₃₇	C₄	1	4	yes	complemented
21	#341×C₁₃₇	—	2	1	no	—
22	C₂²×D₄×C₁₃₇	—	2	1	no	—
23	C₂×D₈×C₁₃₇	—	2	1	no	—
24	C₂×D₈×C₁₃₇	—	2	1	no	—
25	C₂×D₈×C₁₃₇	—	2	1	no	—
26	C₂×D₈×C₁₃₇	—	2	1	no	—
27	C₂×D₈×C₁₃₇	—	2	1	no	—
28	C₂×D₈×C₁₃₇	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₁₃₇	—	2	1	no	—
30	C₂×M₁₆×C₁₃₇	—	2	1	no	—
31	C₂²×C₈×C₁₃₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₁₃₇	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₁₃₇	—	4	2	no	—
35	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
36	M₃₂×C₁₃₇	—	2	1	no	—
37	M₃₂×C₁₃₇	—	2	1	no	—
38	M₃₂×C₁₃₇	—	2	1	no	—
39	M₃₂×C₁₃₇	C₂×C₄	1	2	yes	—
40	C₂×C₁₆×C₁₃₇	—	2	1	no	—
41	C₂×C₁₆×C₁₃₇	—	2	1	no	—
42	#108×C₁₃₇	—	2	1	no	—
43	#108×C₁₃₇	—	2	1	no	—
44	#108×C₁₃₇	C₂×C₄	1	2	yes	—
45	#108×C₁₃₇	D₄	1	2	yes	—
46	#106×C₁₃₇	—	2	1	no	—
47	#106×C₁₃₇	—	2	1	no	—
48	#102×C₁₃₇	—	4	2	no	—
49	#97×C₁₃₇	—	2	1	no	—
50	#97×C₁₃₇	—	2	1	no	—
51	#97×C₁₃₇	—	4	1	no	—
52	C₄×C₈×C₁₃₇	—	4	1	no	—
53	C₂⁴×C₁₃₇	—	4	1	no	—
54	C₂×D₄×C₁₃₇	—	2	2	no	—
55	C₂×D₄×C₁₃₇	—	4	1	no	—
56	C₂×D₄×C₁₃₇	—	4	1	no	—
57	C₂×D₄×C₁₃₇	—	4	1	no	—
58	C₂×D₄×C₁₃₇	—	4	1	no	—
59	C₂×D₄×C₁₃₇	—	4	1	no	—
60	C₂²×C₄×C₁₃₇	—	4	1	no	—
61	C₂²×C₄×C₁₃₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
63	D₈×C₁₃₇	—	2	2	no	—
64	D₈×C₁₃₇	—	2	2	no	—
65	D₈×C₁₃₇	—	4	1	no	—
66	D₈×C₁₃₇	—	4	1	no	—
67	M₁₆×C₁₃₇	—	2	2	no	—
68	M₁₆×C₁₃₇	—	2	2	no	—
69	M₁₆×C₁₃₇	—	4	1	no	—
70	M₁₆×C₁₃₇	—	4	1	no	—
71	C₂×C₈×C₁₃₇	—	2	1	no	—
72	C₂×C₈×C₁₃₇	—	2	1	no	—
73	C₂×C₈×C₁₃₇	—	2	1	no	—
74	C₂×C₈×C₁₃₇	—	4	1	no	—
75	C₂×C₈×C₁₃₇	—	4	1	no	—
76	C₂×C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
82	C₄⋊C₄×C₁₃₇	—	4	1	no	—
83	C₄⋊C₄×C₁₃₇	—	4	1	no	—
84	C₂²⋊C₄×C₁₃₇	—	8	4	no	—
85	C₄²×C₁₃₇	—	2	1	no	—
86	C₄²×C₁₃₇	—	2	1	no	—
87	C₁₆×C₁₃₇	—	2	2	no	—
88	C₁₆×C₁₃₇	—	4	1	no	—
89	C₂³×C₁₃₇	—	4	2	no	—
90	C₂³×C₁₃₇	—	8	2	no	—
91	C₂³×C₁₃₇	—	8	2	no	—
92	C₂³×C₁₃₇	—	8	1	no	—
93	C₂³×C₁₃₇	—	8	1	no	—
94	C₂³×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
95	D₄×C₁₃₇	—	4	2	no	—
96	D₄×C₁₃₇	—	4	2	no	—
97	D₄×C₁₃₇	—	8	1	no	—
98	D₄×C₁₃₇	—	8	1	no	—
99	C₂×C₄×C₁₃₇	—	2	2	no	—
100	C₂×C₄×C₁₃₇	—	4	2	no	—
101	C₂×C₄×C₁₃₇	—	8	2	no	—
102	C₂×C₄×C₁₃₇	—	8	1	no	—
103	C₂×C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	#123	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	#135	1	1	yes	characteristic
107	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₁₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₈×C₁₃₇	—	4	2	no	—
110	C₈×C₁₃₇	—	4	2	no	—
111	C₈×C₁₃₇	—	4	1	no	—
112	C₈×C₁₃₇	#117	1	1	yes	characteristic
113	C₈×C₁₃₇	C₄×D₄	1	1	yes	characteristic
114	C₈×C₁₃₇	#124	1	1	yes	characteristic
115	C₈×C₁₃₇	#126	1	1	yes	characteristic
116	C₂²×C₁₃₇	—	2	2	no	—
117	C₂²×C₁₃₇	—	8	4	no	—
118	C₂²×C₁₃₇	—	8	2	no	—
119	C₂²×C₁₃₇	—	16	1	no	—
120	C₂²×C₁₃₇	—	16	1	no	—
121	C₂²×C₁₃₇	—	16	2	no	—
122	C₂²×C₁₃₇	C₄×SD₁₆	1	1	yes	characteristic
123	C₂²×C₁₃₇	#460	1	1	yes	characteristic
124	C₂²×C₁₃₇	#481	1	1	yes	characteristic
125	C₄×C₁₃₇	—	4	2	no	—
126	C₄×C₁₃₇	—	16	1	no	—
127	C₄×C₁₃₇	#388	1	1	yes	characteristic
128	C₄×C₁₃₇	#442	1	1	yes	characteristic
129	C₄×C₁₃₇	#463	1	1	yes	characteristic
130	C₄×C₁₃₇	#482	1	1	yes	characteristic
131	C₂×C₁₃₇	—	2	1	no	—
132	C₂×C₁₃₇	—	2	1	no	—
133	C₂×C₁₃₇	—	16	2	no	—
134	C₂×C₁₃₇	#1926	1	1	yes	characteristic
135	C₂×C₁₃₇	#2238	1	2	yes	—
136	#13138	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₁₃₇	#13138	1	1	yes	characteristic, direct product factor, Sylow
138	C₂×#365	C₂×C₁₃₇	1	1	yes	characteristic, complemented
139	C₂×#361	C₂×C₁₃₇	1	1	yes	characteristic
140	C₂×#361	C₂×C₁₃₇	1	1	yes	characteristic
141	#1972	C₂×C₁₃₇	1	1	yes	characteristic
142	#1778	C₂×C₁₃₇	1	1	yes	characteristic, complemented
143	#1383	C₂×C₁₃₇	1	1	yes	characteristic, complemented
144	#1377	C₂×C₁₃₇	1	1	yes	characteristic, complemented
145	C₂²×D₈	C₂²×C₁₃₇	1	1	yes	characteristic
146	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₁₃₇	1	1	yes	characteristic
148	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
149	C₂×#108	C₂²×C₁₃₇	1	1	yes	characteristic
150	C₂×#106	C₂²×C₁₃₇	1	1	yes	characteristic
151	C₂×#102	—	2	1	no	—
152	C₄×M₁₆	—	2	1	no	—
153	C₂×#97	C₂²×C₁₃₇	1	1	yes	characteristic
154	#390	—	2	1	no	—
155	#365	—	2	2	no	—
156	#361	—	2	2	no	—
157	#361	C₄×C₁₃₇	1	4	yes	complemented
158	#341	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	1	no	—
163	C₂×D₈	—	2	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂×C₄×C₁₃₇	1	2	yes	—
166	C₂×M₁₆	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂²×C₈	D₄×C₁₃₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₂²⋊C₄	—	4	2	no	—
172	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
173	M₃₂	—	2	1	no	—
174	M₃₂	—	2	1	no	—
175	M₃₂	—	2	1	no	—
176	M₃₂	C₂×C₄×C₁₃₇	1	2	yes	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	—	2	1	no	—
181	#108	C₂×C₄×C₁₃₇	1	2	yes	—
182	#108	D₄×C₁₃₇	1	2	yes	—
183	#106	—	2	1	no	—
184	#106	—	2	1	no	—
185	#102	—	4	2	no	—
186	#97	—	2	1	no	—
187	#97	—	2	1	no	—
188	#97	—	4	1	no	—
189	C₄×C₈	—	4	1	no	—
190	C₂⁴	—	4	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	SD₁₆×C₁₃₇	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
200	D₈	—	2	2	no	—
201	D₈	—	2	2	no	—
202	D₈	—	4	1	no	—
203	D₈	—	4	1	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
214	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
215	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
216	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
217	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
218	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
219	C₄⋊C₄	—	4	1	no	—
220	C₄⋊C₄	—	4	1	no	—
221	C₂²⋊C₄	—	8	4	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₁₆	—	2	2	no	—
225	C₁₆	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	1	no	—
230	C₂³	—	8	1	no	—
231	C₂³	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic
232	D₄	—	4	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	8	1	no	—
235	D₄	—	8	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	C₄×D₄×C₁₃₇	1	1	yes	characteristic
241	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
242	C₂×C₄	#123×C₁₃₇	1	1	yes	characteristic
243	C₂×C₄	#135×C₁₃₇	1	1	yes	characteristic
244	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
245	C₂×C₄	AGL(1,ℤ/8)×C₁₃₇	1	1	yes	characteristic
246	C₈	—	4	2	no	—
247	C₈	—	4	2	no	—
248	C₈	—	4	1	no	—
249	C₈	#117×C₁₃₇	1	1	yes	characteristic
250	C₈	C₄×D₄×C₁₃₇	1	1	yes	characteristic
251	C₈	#124×C₁₃₇	1	1	yes	characteristic
252	C₈	#126×C₁₃₇	1	1	yes	characteristic
253	C₂²	—	2	2	no	—
254	C₂²	—	8	4	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	16	1	no	—
257	C₂²	—	16	1	no	—
258	C₂²	—	16	2	no	—
259	C₂²	C₄×SD₁₆×C₁₃₇	1	1	yes	characteristic
260	C₂²	#460×C₁₃₇	1	1	yes	characteristic
261	C₂²	#481×C₁₃₇	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	16	1	no	—
264	C₄	#388×C₁₃₇	1	1	yes	characteristic
265	C₄	#442×C₁₃₇	1	1	yes	characteristic
266	C₄	#463×C₁₃₇	1	1	yes	characteristic
267	C₄	#482×C₁₃₇	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	16	2	no	—
271	C₂	#1926×C₁₃₇	1	1	yes	characteristic
272	C₂	#2238×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#481
commutator	C₂×C₈	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
prime radical	C₂²×D₈×C₁₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×#97×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#365×C₁₃₇	C₂	semidirect
5. #1778×C₁₃₇	C₂	semidirect
6. #1383×C₁₃₇	C₂	semidirect
7. #1377×C₁₃₇	C₂	semidirect
20. #361×C₁₃₇	C₄	semidirect
136. #13138	C₁₃₇	direct (N × H)
137. C₁₃₇	#13138	direct (N × H)
138. C₂×#365	C₂×C₁₃₇	semidirect
142. #1778	C₂×C₁₃₇	semidirect
143. #1383	C₂×C₁₃₇	semidirect
144. #1377	C₂×C₁₃₇	semidirect
157. #361	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#365×C₁₃₇	2
C₂×#361×C₁₃₇	2
#1972×C₁₃₇	2
#1778×C₁₃₇	2
#1383×C₁₃₇	2
#1377×C₁₃₇	2
#13138	137

Subgroups — C137×#13138 #46862524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#13138 #46862524