Subgroups — C₁₃₇×#14665 #46864051

Order 35072.

1530 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#384×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#383×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#1441×C₁₃₇	C₂	1	6	yes	maximal, complemented
5	#1439×C₁₃₇	C₂	1	6	yes	maximal
6	C₂×#126×C₁₃₇	—	2	4	no	—
7	C₂×#126×C₁₃₇	—	2	4	no	—
8	C₂×#126×C₁₃₇	C₄	1	8	yes	complemented
9	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	3	yes	—
10	C₂²×C₄⋊C₄×C₁₃₇	C₂²	1	6	yes	—
11	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₁₃₇	C₂²	1	4	yes	—
13	C₂²×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	#384×C₁₃₇	—	2	4	no	—
15	#383×C₁₃₇	—	2	4	no	—
16	C₂×#95×C₁₃₇	C₂²	1	6	yes	—
17	C₂×#95×C₁₃₇	C₂²	1	12	yes	—
18	C₂⁵×C₁₃₇	C₂³	1	1	yes	characteristic
19	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
20	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
21	C₂³×C₄×C₁₃₇	C₂³	1	3	yes	—
22	C₂³×C₄×C₁₃₇	C₂³	1	3	yes	—
23	#126×C₁₃₇	—	2	32	no	—
24	C₂×C₄⋊C₄×C₁₃₇	—	2	6	no	—
25	C₂×C₄⋊C₄×C₁₃₇	—	2	6	no	—
26	C₂×C₄⋊C₄×C₁₃₇	—	2	12	no	—
27	C₂×C₄⋊C₄×C₁₃₇	—	4	12	no	—
28	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	12	yes	—
29	C₂×C₂²⋊C₄×C₁₃₇	—	2	6	no	—
30	C₂×C₂²⋊C₄×C₁₃₇	—	2	6	no	—
31	C₂×C₂²⋊C₄×C₁₃₇	—	4	12	no	—
32	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	12	yes	—
33	C₂×C₄²×C₁₃₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
35	C₂×C₄²×C₁₃₇	D₄	1	4	yes	—
36	#95×C₁₃₇	—	2	12	no	—
37	#95×C₁₃₇	—	4	12	no	—
38	C₂⁴×C₁₃₇	—	2	1	no	—
39	C₂⁴×C₁₃₇	—	2	1	no	—
40	C₂⁴×C₁₃₇	—	2	6	no	—
41	C₂⁴×C₁₃₇	—	2	6	no	—
42	C₂⁴×C₁₃₇	C₂²×C₄	1	2	yes	—
43	C₂⁴×C₁₃₇	C₂⁴	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₇	—	2	6	no	—
45	C₂²×C₄×C₁₃₇	—	2	6	no	—
46	C₂²×C₄×C₁₃₇	—	2	3	no	—
47	C₂²×C₄×C₁₃₇	—	2	3	no	—
48	C₂²×C₄×C₁₃₇	—	2	6	no	—
49	C₂²×C₄×C₁₃₇	—	2	6	no	—
50	C₂²×C₄×C₁₃₇	—	2	4	no	—
51	C₂²×C₄×C₁₃₇	—	2	4	no	—
52	C₂²×C₄×C₁₃₇	—	2	12	no	—
53	C₂²×C₄×C₁₃₇	—	2	12	no	—
54	C₂²×C₄×C₁₃₇	—	2	12	no	—
55	C₂²×C₄×C₁₃₇	—	2	12	no	—
56	C₂²×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
58	C₂²×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
59	C₂²×C₄×C₁₃₇	C₂×D₄	1	6	yes	—
60	C₄⋊C₄×C₁₃₇	—	2	24	no	—
61	C₄⋊C₄×C₁₃₇	—	4	24	no	—
62	C₂²⋊C₄×C₁₃₇	—	2	24	no	—
63	C₂²⋊C₄×C₁₃₇	—	4	16	no	—
64	C₄²×C₁₃₇	C₂²⋊C₄	1	16	yes	complemented
65	C₂³×C₁₃₇	—	2	2	no	—
66	C₂³×C₁₃₇	—	2	12	no	—
67	C₂³×C₁₃₇	—	4	3	no	—
68	C₂³×C₁₃₇	—	4	3	no	—
69	C₂³×C₁₃₇	—	4	6	no	—
70	C₂³×C₁₃₇	—	4	4	no	—
71	C₂³×C₁₃₇	—	4	12	no	—
72	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
73	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
74	C₂³×C₁₃₇	C₂²×D₄	1	1	yes	characteristic
75	C₂³×C₁₃₇	H(5,𝔽₂)	1	6	yes	—
76	C₂³×C₁₃₇	2₋¹⁺⁴	1	6	yes	—
77	C₂×C₄×C₁₃₇	—	2	12	no	—
78	C₂×C₄×C₁₃₇	—	2	6	no	—
79	C₂×C₄×C₁₃₇	—	2	6	no	—
80	C₂×C₄×C₁₃₇	—	2	12	no	—
81	C₂×C₄×C₁₃₇	—	2	6	no	—
82	C₂×C₄×C₁₃₇	—	2	12	no	—
83	C₂×C₄×C₁₃₇	—	4	3	no	—
84	C₂×C₄×C₁₃₇	—	4	3	no	—
85	C₂×C₄×C₁₃₇	—	4	6	no	—
86	C₂×C₄×C₁₃₇	—	4	12	no	—
87	C₂×C₄×C₁₃₇	—	4	12	no	—
88	C₂×C₄×C₁₃₇	—	4	4	no	—
89	C₂×C₄×C₁₃₇	—	4	4	no	—
90	C₂×C₄×C₁₃₇	—	4	12	no	—
91	C₂×C₄×C₁₃₇	—	4	12	no	—
92	C₂×C₄×C₁₃₇	C₂×C₂²⋊C₄	1	12	yes	—
93	C₂²×C₁₃₇	—	4	6	no	—
94	C₂²×C₁₃₇	—	4	8	no	—
95	C₂²×C₁₃₇	—	8	1	no	—
96	C₂²×C₁₃₇	—	8	3	no	—
97	C₂²×C₁₃₇	—	8	4	no	—
98	C₂²×C₁₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₁₃₇	#520	1	6	yes	—
100	C₂²×C₁₃₇	#535	1	3	yes	—
101	C₂²×C₁₃₇	#536	1	3	yes	—
102	C₂²×C₁₃₇	#537	1	6	yes	—
103	C₂²×C₁₃₇	#560	1	4	yes	—
104	C₂²×C₁₃₇	#563	1	12	yes	—
105	C₄×C₁₃₇	—	2	12	no	—
106	C₄×C₁₃₇	—	4	6	no	—
107	C₄×C₁₃₇	—	8	4	no	—
108	C₄×C₁₃₇	—	8	4	no	—
109	C₂×C₁₃₇	—	8	2	no	—
110	C₂×C₁₃₇	#2311	1	3	yes	—
111	C₂×C₁₃₇	#2383	1	4	yes	—
112	C₂×C₁₃₇	#2818	1	1	yes	characteristic
113	C₂×C₁₃₇	#2821	1	3	yes	—
114	C₂×C₁₃₇	#2823	1	4	yes	—
115	#14665	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₁₃₇	#14665	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²×#126	C₂×C₁₃₇	1	1	yes	characteristic
118	C₂×#384	C₂×C₁₃₇	1	1	yes	characteristic, complemented
119	C₂×#383	C₂×C₁₃₇	1	1	yes	characteristic, complemented
120	#1441	C₂×C₁₃₇	1	6	yes	complemented
121	#1439	C₂×C₁₃₇	1	6	yes	—
122	C₂×#126	—	2	4	no	—
123	C₂×#126	—	2	4	no	—
124	C₂×#126	C₄×C₁₃₇	1	8	yes	complemented
125	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	3	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₃₇	1	6	yes	—
127	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	3	yes	—
128	C₂²×C₂²⋊C₄	C₂²×C₁₃₇	1	4	yes	—
129	C₂²×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
130	#384	—	2	4	no	—
131	#383	—	2	4	no	—
132	C₂×#95	C₂²×C₁₃₇	1	6	yes	—
133	C₂×#95	C₂²×C₁₃₇	1	12	yes	—
134	C₂⁵	C₂³×C₁₃₇	1	1	yes	characteristic
135	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
136	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
137	C₂³×C₄	C₂³×C₁₃₇	1	3	yes	—
138	C₂³×C₄	C₂³×C₁₃₇	1	3	yes	—
139	#126	—	2	32	no	—
140	C₂×C₄⋊C₄	—	2	6	no	—
141	C₂×C₄⋊C₄	—	2	6	no	—
142	C₂×C₄⋊C₄	—	2	12	no	—
143	C₂×C₄⋊C₄	—	4	12	no	—
144	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	12	yes	—
145	C₂×C₂²⋊C₄	—	2	6	no	—
146	C₂×C₂²⋊C₄	—	2	6	no	—
147	C₂×C₂²⋊C₄	—	4	12	no	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	12	yes	—
149	C₂×C₄²	C₂×C₄×C₁₃₇	1	4	yes	—
150	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
151	C₂×C₄²	D₄×C₁₃₇	1	4	yes	—
152	#95	—	2	12	no	—
153	#95	—	4	12	no	—
154	C₂⁴	—	2	1	no	—
155	C₂⁴	—	2	1	no	—
156	C₂⁴	—	2	6	no	—
157	C₂⁴	—	2	6	no	—
158	C₂⁴	C₂²×C₄×C₁₃₇	1	2	yes	—
159	C₂⁴	C₂⁴×C₁₃₇	1	1	yes	characteristic
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	2	6	no	—
162	C₂²×C₄	—	2	3	no	—
163	C₂²×C₄	—	2	3	no	—
164	C₂²×C₄	—	2	6	no	—
165	C₂²×C₄	—	2	6	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	2	12	no	—
170	C₂²×C₄	—	2	12	no	—
171	C₂²×C₄	—	2	12	no	—
172	C₂²×C₄	C₂²×C₄×C₁₃₇	1	6	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃₇	1	6	yes	—
174	C₂²×C₄	C₂×D₄×C₁₃₇	1	6	yes	—
175	C₂²×C₄	C₂×D₄×C₁₃₇	1	6	yes	—
176	C₄⋊C₄	—	2	24	no	—
177	C₄⋊C₄	—	4	24	no	—
178	C₂²⋊C₄	—	2	24	no	—
179	C₂²⋊C₄	—	4	16	no	—
180	C₄²	C₂²⋊C₄×C₁₃₇	1	16	yes	complemented
181	C₂³	—	2	2	no	—
182	C₂³	—	2	12	no	—
183	C₂³	—	4	3	no	—
184	C₂³	—	4	3	no	—
185	C₂³	—	4	6	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	4	12	no	—
188	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
189	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
190	C₂³	C₂²×D₄×C₁₃₇	1	1	yes	characteristic
191	C₂³	H(5,𝔽₂)×C₁₃₇	1	6	yes	—
192	C₂³	2₋¹⁺⁴×C₁₃₇	1	6	yes	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	12	no	—
197	C₂×C₄	—	2	6	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	—	4	3	no	—
201	C₂×C₄	—	4	6	no	—
202	C₂×C₄	—	4	12	no	—
203	C₂×C₄	—	4	12	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	12	no	—
207	C₂×C₄	—	4	12	no	—
208	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₇	1	12	yes	—
209	C₂²	—	4	6	no	—
210	C₂²	—	4	8	no	—
211	C₂²	—	8	1	no	—
212	C₂²	—	8	3	no	—
213	C₂²	—	8	4	no	—
214	C₂²	C₂²×C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
215	C₂²	#520×C₁₃₇	1	6	yes	—
216	C₂²	#535×C₁₃₇	1	3	yes	—
217	C₂²	#536×C₁₃₇	1	3	yes	—
218	C₂²	#537×C₁₃₇	1	6	yes	—
219	C₂²	#560×C₁₃₇	1	4	yes	—
220	C₂²	#563×C₁₃₇	1	12	yes	—
221	C₄	—	2	12	no	—
222	C₄	—	4	6	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	4	no	—
225	C₂	—	8	2	no	—
226	C₂	#2311×C₁₃₇	1	3	yes	—
227	C₂	#2383×C₁₃₇	1	4	yes	—
228	C₂	#2818×C₁₃₇	1	1	yes	characteristic
229	C₂	#2821×C₁₃₇	1	3	yes	—
230	C₂	#2823×C₁₃₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#384×C₁₃₇	C₂	semidirect
3. C₂×#383×C₁₃₇	C₂	semidirect
4. #1441×C₁₃₇	C₂	semidirect
8. C₂×#126×C₁₃₇	C₄	semidirect
64. C₄²×C₁₃₇	C₂²⋊C₄	semidirect
115. #14665	C₁₃₇	direct (N × H)
116. C₁₃₇	#14665	direct (N × H)
118. C₂×#384	C₂×C₁₃₇	semidirect
119. C₂×#383	C₂×C₁₃₇	semidirect
120. #1441	C₂×C₁₃₇	semidirect
124. C₂×#126	C₄×C₁₃₇	semidirect
180. C₄²	C₂²⋊C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#126×C₁₃₇	2
C₂×#384×C₁₃₇	2
C₂×#383×C₁₃₇	2
#1441×C₁₃₇	2
#1439×C₁₃₇	2
#14665	137

Subgroups — C137×#14665 #46864051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#14665 #46864051