Subgroups — C₁₃₇×#21681 #46871067

Order 35072.

1230 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₃₇	C₂	1	6	yes	maximal, complemented
2	C₄×#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1954×C₁₃₇	C₂	1	6	yes	maximal, complemented
4	#1842×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×M₁₆×C₁₃₇	C₂²	1	4	yes	—
7	C₂×#127×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃₇	C₂²	1	6	yes	—
9	#436×C₁₃₇	—	2	24	no	—
10	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	6	yes	—
11	C₂×C₂²⋊C₈×C₁₃₇	C₂²	1	12	yes	—
12	C₂×C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
14	#390×C₁₃₇	C₄	1	6	yes	—
15	#384×C₁₃₇	C₄	1	1	yes	characteristic
16	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₃₇	C₂×C₄	1	6	yes	—
18	C₂³×C₄×C₁₃₇	C₂³	1	4	yes	—
19	C₂×M₁₆×C₁₃₇	—	2	24	no	—
20	C₂²×C₈×C₁₃₇	C₂³	1	4	yes	—
21	C₂²×C₈×C₁₃₇	C₂³	1	3	yes	—
22	C₂²×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
23	#127×C₁₃₇	—	2	4	no	—
24	C₄×D₄×C₁₃₇	—	2	12	no	—
25	C₄×D₄×C₁₃₇	—	2	12	no	—
26	C₂×C₄⋊C₄×C₁₃₇	C₂×C₄	1	6	yes	—
27	C₂×C₂²⋊C₄×C₁₃₇	C₂×C₄	1	12	yes	—
28	C₂×C₄²×C₁₃₇	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃₇	C₂³	1	3	yes	—
31	C₄⋊C₈×C₁₃₇	—	2	12	no	—
32	C₂²⋊C₈×C₁₃₇	—	2	24	no	—
33	C₄×C₈×C₁₃₇	—	2	6	no	—
34	C₄×C₈×C₁₃₇	D₄	1	12	yes	complemented
35	C₂⁴×C₁₃₇	C₂²×C₄	1	4	yes	—
36	C₂×D₄×C₁₃₇	—	2	12	no	—
37	C₂×D₄×C₁₃₇	—	2	12	no	—
38	C₂×D₄×C₁₃₇	—	2	12	no	—
39	C₂²×C₄×C₁₃₇	—	2	12	no	—
40	C₂²×C₄×C₁₃₇	—	2	12	no	—
41	C₂²×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₁₃₇	C₂²×C₄	1	3	yes	—
44	C₂²×C₄×C₁₃₇	C₂⁴	1	1	yes	characteristic
45	M₁₆×C₁₃₇	—	4	16	no	—
46	C₂×C₈×C₁₃₇	—	2	6	no	—
47	C₂×C₈×C₁₃₇	—	2	12	no	—
48	C₂×C₈×C₁₃₇	C₂×D₄	1	6	yes	—
49	C₂×C₈×C₁₃₇	C₂×D₄	1	6	yes	—
50	C₄⋊C₄×C₁₃₇	—	2	12	no	—
51	C₂²⋊C₄×C₁₃₇	—	2	24	no	—
52	C₄²×C₁₃₇	—	2	6	no	—
53	C₄²×C₁₃₇	M₁₆	1	4	yes	complemented
54	C₄²×C₁₃₇	C₂×D₄	1	6	yes	—
55	C₄²×C₁₃₇	#41	1	6	yes	—
56	C₂³×C₁₃₇	—	2	12	no	—
57	C₂³×C₁₃₇	—	2	4	no	—
58	C₂³×C₁₃₇	—	2	12	no	—
59	C₂³×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
60	D₄×C₁₃₇	—	4	24	no	—
61	C₂×C₄×C₁₃₇	—	2	6	no	—
62	C₂×C₄×C₁₃₇	—	2	12	no	—
63	C₂×C₄×C₁₃₇	—	4	4	no	—
64	C₂×C₄×C₁₃₇	—	4	12	no	—
65	C₂×C₄×C₁₃₇	C₄×D₄	1	6	yes	—
66	C₂×C₄×C₁₃₇	C₄×D₄	1	6	yes	—
67	C₂×C₄×C₁₃₇	C₂×M₁₆	1	6	yes	—
68	C₂×C₄×C₁₃₇	#131	1	6	yes	—
69	C₂×C₄×C₁₃₇	C₂²×D₄	1	3	yes	—
70	C₂×C₄×C₁₃₇	C₂×#41	1	3	yes	—
71	C₈×C₁₃₇	—	2	6	no	—
72	C₈×C₁₃₇	—	4	4	no	—
73	C₈×C₁₃₇	#127	1	4	yes	complemented
74	C₂²×C₁₃₇	—	4	4	no	—
75	C₂²×C₁₃₇	—	4	12	no	—
76	C₂²×C₁₃₇	—	4	12	no	—
77	C₂²×C₁₃₇	C₂×C₄×D₄	1	3	yes	—
78	C₂²×C₁₃₇	C₂²×M₁₆	1	1	yes	characteristic
79	C₂²×C₁₃₇	C₂×#131	1	3	yes	—
80	C₄×C₁₃₇	—	2	6	no	—
81	C₄×C₁₃₇	—	4	4	no	—
82	C₄×C₁₃₇	#436	1	12	yes	complemented
83	C₄×C₁₃₇	C₂×#127	1	1	yes	characteristic
84	C₄×C₁₃₇	#532	1	3	yes	—
85	C₂×C₁₃₇	—	8	4	no	—
86	C₂×C₁₃₇	C₄×#127	1	1	yes	characteristic
87	C₂×C₁₃₇	C₂×#436	1	3	yes	—
88	C₂×C₁₃₇	#2931	1	3	yes	—
89	#21681	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₁₃₇	#21681	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×#436	C₂×C₁₃₇	1	6	yes	complemented
92	C₄×#127	C₂×C₁₃₇	1	1	yes	characteristic
93	#1954	C₂×C₁₃₇	1	6	yes	complemented
94	#1842	C₂×C₁₃₇	1	1	yes	characteristic, complemented
95	C₄²×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
96	C₂²×M₁₆	C₂²×C₁₃₇	1	4	yes	—
97	C₂×#127	C₄×C₁₃₇	1	1	yes	characteristic
98	C₂×C₄×D₄	C₂²×C₁₃₇	1	6	yes	—
99	#436	—	2	24	no	—
100	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	6	yes	—
101	C₂×C₂²⋊C₈	C₂²×C₁₃₇	1	12	yes	—
102	C₂×C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
103	C₂×C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
104	#390	C₄×C₁₃₇	1	6	yes	—
105	#384	C₄×C₁₃₇	1	1	yes	characteristic
106	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
107	C₂²×D₄	C₂×C₄×C₁₃₇	1	6	yes	—
108	C₂³×C₄	C₂³×C₁₃₇	1	4	yes	—
109	C₂×M₁₆	—	2	24	no	—
110	C₂²×C₈	C₂³×C₁₃₇	1	4	yes	—
111	C₂²×C₈	C₂³×C₁₃₇	1	3	yes	—
112	C₂²×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
113	#127	—	2	4	no	—
114	C₄×D₄	—	2	12	no	—
115	C₄×D₄	—	2	12	no	—
116	C₂×C₄⋊C₄	C₂×C₄×C₁₃₇	1	6	yes	—
117	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₇	1	12	yes	—
118	C₂×C₄²	C₂×C₄×C₁₃₇	1	3	yes	—
119	C₂×C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₁₃₇	1	3	yes	—
121	C₄⋊C₈	—	2	12	no	—
122	C₂²⋊C₈	—	2	24	no	—
123	C₄×C₈	—	2	6	no	—
124	C₄×C₈	D₄×C₁₃₇	1	12	yes	complemented
125	C₂⁴	C₂²×C₄×C₁₃₇	1	4	yes	—
126	C₂×D₄	—	2	12	no	—
127	C₂×D₄	—	2	12	no	—
128	C₂×D₄	—	2	12	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	C₂²×C₄×C₁₃₇	1	4	yes	—
132	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
133	C₂²×C₄	C₂²×C₄×C₁₃₇	1	3	yes	—
134	C₂²×C₄	C₂⁴×C₁₃₇	1	1	yes	characteristic
135	M₁₆	—	4	16	no	—
136	C₂×C₈	—	2	6	no	—
137	C₂×C₈	—	2	12	no	—
138	C₂×C₈	C₂×D₄×C₁₃₇	1	6	yes	—
139	C₂×C₈	C₂×D₄×C₁₃₇	1	6	yes	—
140	C₄⋊C₄	—	2	12	no	—
141	C₂²⋊C₄	—	2	24	no	—
142	C₄²	—	2	6	no	—
143	C₄²	M₁₆×C₁₃₇	1	4	yes	complemented
144	C₄²	C₂×D₄×C₁₃₇	1	6	yes	—
145	C₄²	#41×C₁₃₇	1	6	yes	—
146	C₂³	—	2	12	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	12	no	—
149	C₂³	C₂³×C₄×C₁₃₇	1	1	yes	characteristic
150	D₄	—	4	24	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	12	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	C₄×D₄×C₁₃₇	1	6	yes	—
156	C₂×C₄	C₄×D₄×C₁₃₇	1	6	yes	—
157	C₂×C₄	C₂×M₁₆×C₁₃₇	1	6	yes	—
158	C₂×C₄	#131×C₁₃₇	1	6	yes	—
159	C₂×C₄	C₂²×D₄×C₁₃₇	1	3	yes	—
160	C₂×C₄	C₂×#41×C₁₃₇	1	3	yes	—
161	C₈	—	2	6	no	—
162	C₈	—	4	4	no	—
163	C₈	#127×C₁₃₇	1	4	yes	complemented
164	C₂²	—	4	4	no	—
165	C₂²	—	4	12	no	—
166	C₂²	—	4	12	no	—
167	C₂²	C₂×C₄×D₄×C₁₃₇	1	3	yes	—
168	C₂²	C₂²×M₁₆×C₁₃₇	1	1	yes	characteristic
169	C₂²	C₂×#131×C₁₃₇	1	3	yes	—
170	C₄	—	2	6	no	—
171	C₄	—	4	4	no	—
172	C₄	#436×C₁₃₇	1	12	yes	complemented
173	C₄	C₂×#127×C₁₃₇	1	1	yes	characteristic
174	C₄	#532×C₁₃₇	1	3	yes	—
175	C₂	—	8	4	no	—
176	C₂	C₄×#127×C₁₃₇	1	1	yes	characteristic
177	C₂	C₂×#436×C₁₃₇	1	3	yes	—
178	C₂	#2931×C₁₃₇	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₇
socle	C₂³×C₁₃₇	C₂³×C₄
prime radical	C₂×#127×C₁₃₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#436×C₁₃₇	C₂	semidirect
3. #1954×C₁₃₇	C₂	semidirect
4. #1842×C₁₃₇	C₂	semidirect
5. C₄²×C₈×C₁₃₇	C₂	semidirect
34. C₄×C₈×C₁₃₇	D₄	semidirect
53. C₄²×C₁₃₇	M₁₆	semidirect
73. C₈×C₁₃₇	#127	semidirect
82. C₄×C₁₃₇	#436	semidirect
89. #21681	C₁₃₇	direct (N × H)
90. C₁₃₇	#21681	direct (N × H)
91. C₂×#436	C₂×C₁₃₇	semidirect
93. #1954	C₂×C₁₃₇	semidirect
94. #1842	C₂×C₁₃₇	semidirect
95. C₄²×C₈	C₂×C₁₃₇	semidirect
124. C₄×C₈	D₄×C₁₃₇	semidirect
143. C₄²	M₁₆×C₁₃₇	semidirect
163. C₈	#127×C₁₃₇	semidirect
172. C₄	#436×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₃₇	2
C₄×#127×C₁₃₇	2
#1954×C₁₃₇	2
#1842×C₁₃₇	2
C₄²×C₈×C₁₃₇	2
#21681	137

Subgroups — C137×#21681 #46871067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#21681 #46871067