Subgroups — C₁₃₇×#28733 #46878119

Order 35072.

666 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#3142×C₁₃₇	C₂	1	2	yes	maximal
3	#2931×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#2078×C₁₃₇	C₂	1	4	yes	maximal, complemented
5	#1998×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#1815×C₁₃₇	C₂	1	4	yes	maximal, complemented
7	C₄×#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₇	C₂²	1	2	yes	—
9	#532×C₁₃₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₁₃₇	C₂²	1	2	yes	—
11	#499×C₁₃₇	—	2	2	no	—
12	#499×C₁₃₇	—	2	2	no	—
13	#491×C₁₃₇	—	2	2	no	—
14	#490×C₁₃₇	—	2	2	no	—
15	#488×C₁₃₇	—	2	4	no	—
16	#436×C₁₃₇	—	2	2	no	—
17	C₈×D₄×C₁₃₇	—	2	2	no	—
18	C₂×#108×C₁₃₇	C₂²	1	2	yes	—
19	C₂×#107×C₁₃₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₃₇	C₂²	1	2	yes	—
21	#423×C₁₃₇	C₂²	1	1	yes	characteristic
22	#417×C₁₃₇	C₂²	1	8	yes	—
23	#408×C₁₃₇	C₂²	1	2	yes	—
24	C₄×M₁₆×C₁₃₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
26	#343×C₁₃₇	C₂²	1	8	yes	—
27	#341×C₁₃₇	C₂²	1	2	yes	—
28	C₂×#41×C₁₃₇	C₂³	1	2	yes	—
29	#131×C₁₃₇	—	2	8	no	—
30	C₂×M₁₆×C₁₃₇	—	2	2	no	—
31	C₂×M₁₆×C₁₃₇	C₂³	1	4	yes	—
32	C₂×M₁₆×C₁₃₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃₇	—	2	2	no	—
34	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊Q₈×C₁₃₇	D₄	1	1	yes	characteristic
37	#127×C₁₃₇	D₄	1	1	yes	characteristic
38	#125×C₁₃₇	—	2	2	no	—
39	#125×C₁₃₇	—	2	2	no	—
40	#125×C₁₃₇	—	2	4	no	—
41	#124×C₁₃₇	D₄	1	2	yes	—
42	#121×C₁₃₇	—	2	2	no	—
43	C₄×D₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₃₇	C₂³	1	4	yes	—
46	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
47	#108×C₁₃₇	—	2	4	no	—
48	#107×C₁₃₇	—	2	4	no	—
49	#106×C₁₃₇	—	2	2	no	—
50	#106×C₁₃₇	—	2	2	no	—
51	C₄⋊C₈×C₁₃₇	—	2	2	no	—
52	#103×C₁₃₇	—	2	8	no	—
53	#102×C₁₃₇	—	2	8	no	—
54	C₂²⋊C₈×C₁₃₇	—	2	4	no	—
55	#97×C₁₃₇	—	2	2	no	—
56	C₄×C₈×C₁₃₇	—	2	2	no	—
57	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
58	C₄×C₈×C₁₃₇	D₄	1	2	yes	—
59	#41×C₁₃₇	—	2	4	no	—
60	#41×C₁₃₇	—	2	4	no	—
61	C₂×Q₈×C₁₃₇	C₂×D₄	1	2	yes	—
62	C₂×D₄×C₁₃₇	—	2	2	no	—
63	C₂×D₄×C₁₃₇	—	2	2	no	—
64	C₂×D₄×C₁₃₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃₇	—	2	2	no	—
66	C₂²×C₄×C₁₃₇	—	2	4	no	—
67	C₂²×C₄×C₁₃₇	#41	1	2	yes	—
68	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₁₃₇	—	2	4	no	—
70	M₁₆×C₁₃₇	—	2	4	no	—
71	M₁₆×C₁₃₇	—	2	4	no	—
72	M₁₆×C₁₃₇	—	4	4	no	—
73	C₂×C₈×C₁₃₇	—	2	2	no	—
74	C₂×C₈×C₁₃₇	—	2	4	no	—
75	C₂×C₈×C₁₃₇	—	2	2	no	—
76	C₂×C₈×C₁₃₇	—	4	4	no	—
77	C₂×C₈×C₁₃₇	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₇	#41	1	4	yes	—
81	C₄⋊C₄×C₁₃₇	—	2	2	no	—
82	C₄⋊C₄×C₁₃₇	—	2	4	no	—
83	C₄⋊C₄×C₁₃₇	—	2	4	no	—
84	C₄⋊C₄×C₁₃₇	—	4	4	no	—
85	C₄⋊C₄×C₁₃₇	—	4	4	no	—
86	C₂²⋊C₄×C₁₃₇	—	2	4	no	—
87	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₁₃₇	—	2	2	no	—
91	C₂³×C₁₃₇	C₂×#41	1	1	yes	characteristic
92	Q₈×C₁₃₇	—	2	4	no	—
93	D₄×C₁₃₇	—	2	4	no	—
94	D₄×C₁₃₇	—	4	4	no	—
95	D₄×C₁₃₇	—	4	4	no	—
96	C₂×C₄×C₁₃₇	—	2	2	no	—
97	C₂×C₄×C₁₃₇	—	4	2	no	—
98	C₂×C₄×C₁₃₇	—	4	2	no	—
99	C₂×C₄×C₁₃₇	—	4	2	no	—
100	C₂×C₄×C₁₃₇	—	4	4	no	—
101	C₂×C₄×C₁₃₇	—	8	4	no	—
102	C₂×C₄×C₁₃₇	#121	1	2	yes	—
103	C₂×C₄×C₁₃₇	#121	1	2	yes	—
104	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₁₃₇	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₁₃₇	—	2	2	no	—
109	C₈×C₁₃₇	—	4	2	no	—
110	C₈×C₁₃₇	—	4	4	no	—
111	C₈×C₁₃₇	#124	1	2	yes	—
112	C₈×C₁₃₇	#124	1	2	yes	—
113	C₂²×C₁₃₇	—	2	2	no	—
114	C₂²×C₁₃₇	—	4	2	no	—
115	C₂²×C₁₃₇	—	4	4	no	—
116	C₂²×C₁₃₇	C₂×#121	1	1	yes	characteristic
117	C₂²×C₁₃₇	#556	1	2	yes	—
118	C₄×C₁₃₇	—	2	2	no	—
119	C₄×C₁₃₇	—	2	2	no	—
120	C₄×C₁₃₇	—	4	2	no	—
121	C₄×C₁₃₇	—	8	4	no	—
122	C₄×C₁₃₇	C₂×#124	1	1	yes	characteristic
123	C₄×C₁₃₇	#536	1	1	yes	characteristic
124	C₄×C₁₃₇	#547	1	2	yes	—
125	C₂×C₁₃₇	—	2	2	no	—
126	C₂×C₁₃₇	—	8	2	no	—
127	C₂×C₁₃₇	#2547	1	1	yes	characteristic
128	C₂×C₁₃₇	#3067	1	2	yes	—
129	#28733	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₁₃₇	#28733	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×#499	C₂×C₁₃₇	1	1	yes	characteristic, complemented
132	#3142	C₂×C₁₃₇	1	2	yes	—
133	#2931	C₂×C₁₃₇	1	1	yes	characteristic
134	#2078	C₂×C₁₃₇	1	4	yes	complemented
135	#1998	C₂×C₁₃₇	1	2	yes	complemented
136	#1815	C₂×C₁₃₇	1	4	yes	complemented
137	C₄×#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
138	C₂×#131	C₂²×C₁₃₇	1	2	yes	—
139	#532	C₂²×C₁₃₇	1	1	yes	characteristic
140	C₂×#125	C₂²×C₁₃₇	1	2	yes	—
141	#499	—	2	2	no	—
142	#499	—	2	2	no	—
143	#491	—	2	2	no	—
144	#490	—	2	2	no	—
145	#488	—	2	4	no	—
146	#436	—	2	2	no	—
147	C₈×D₄	—	2	2	no	—
148	C₂×#108	C₂²×C₁₃₇	1	2	yes	—
149	C₂×#107	C₂²×C₁₃₇	1	2	yes	—
150	C₂×#106	C₂²×C₁₃₇	1	2	yes	—
151	#423	C₂²×C₁₃₇	1	1	yes	characteristic
152	#417	C₂²×C₁₃₇	1	8	yes	—
153	#408	C₂²×C₁₃₇	1	2	yes	—
154	C₄×M₁₆	C₂²×C₁₃₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
156	#343	C₂²×C₁₃₇	1	8	yes	—
157	#341	C₂²×C₁₃₇	1	2	yes	—
158	C₂×#41	C₂³×C₁₃₇	1	2	yes	—
159	#131	—	2	8	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	C₂³×C₁₃₇	1	2	yes	—
162	C₂×M₁₆	C₂³×C₁₃₇	1	4	yes	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
166	C₄⋊Q₈	D₄×C₁₃₇	1	1	yes	characteristic
167	#127	D₄×C₁₃₇	1	1	yes	characteristic
168	#125	—	2	2	no	—
169	#125	—	2	2	no	—
170	#125	—	2	4	no	—
171	#124	D₄×C₁₃₇	1	2	yes	—
172	#121	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	C₂³×C₁₃₇	1	4	yes	—
176	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
177	#108	—	2	4	no	—
178	#107	—	2	4	no	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	#103	—	2	8	no	—
183	#102	—	2	8	no	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
188	C₄×C₈	D₄×C₁₃₇	1	2	yes	—
189	#41	—	2	4	no	—
190	#41	—	2	4	no	—
191	C₂×Q₈	C₂×D₄×C₁₃₇	1	2	yes	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	C₂×D₄×C₁₃₇	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	#41×C₁₃₇	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	C₂×D₄×C₁₃₇	1	2	yes	—
208	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
210	C₂×C₈	#41×C₁₃₇	1	4	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	4	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
219	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
220	C₂³	—	2	2	no	—
221	C₂³	C₂×#41×C₁₃₇	1	1	yes	characteristic
222	Q₈	—	2	4	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	4	no	—
232	C₂×C₄	#121×C₁₃₇	1	2	yes	—
233	C₂×C₄	#121×C₁₃₇	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
236	C₂×C₄	C₂×#41×C₁₃₇	1	2	yes	—
237	C₂×C₄	2₋¹⁺⁴×C₁₃₇	1	2	yes	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	4	no	—
241	C₈	#124×C₁₃₇	1	2	yes	—
242	C₈	#124×C₁₃₇	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	4	no	—
246	C₂²	C₂×#121×C₁₃₇	1	1	yes	characteristic
247	C₂²	#556×C₁₃₇	1	2	yes	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	C₂×#124×C₁₃₇	1	1	yes	characteristic
253	C₄	#536×C₁₃₇	1	1	yes	characteristic
254	C₄	#547×C₁₃₇	1	2	yes	—
255	C₂	—	2	2	no	—
256	C₂	—	8	2	no	—
257	C₂	#2547×C₁₃₇	1	1	yes	characteristic
258	C₂	#3067×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
prime radical	#532×C₁₃₇	C₂²
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
1. C₂×#499×C₁₃₇	C₂	semidirect
4. #2078×C₁₃₇	C₂	semidirect
5. #1998×C₁₃₇	C₂	semidirect
6. #1815×C₁₃₇	C₂	semidirect
7. C₄×#108×C₁₃₇	C₂	semidirect
129. #28733	C₁₃₇	direct (N × H)
130. C₁₃₇	#28733	direct (N × H)
131. C₂×#499	C₂×C₁₃₇	semidirect
134. #2078	C₂×C₁₃₇	semidirect
135. #1998	C₂×C₁₃₇	semidirect
136. #1815	C₂×C₁₃₇	semidirect
137. C₄×#108	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×#499×C₁₃₇	2
#3142×C₁₃₇	2
#2931×C₁₃₇	2
#2078×C₁₃₇	2
#1998×C₁₃₇	2
#1815×C₁₃₇	2
C₄×#108×C₁₃₇	2
#28733	137

Subgroups — C137×#28733 #46878119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#28733 #46878119