Subgroups — C₃₇×#7470 #22628723

Order 9472.

178 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₇×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic
3	C₃₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₃₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
6	C₃₇×#405	C₄	1	1	yes	characteristic
7	C₃₇×M₆₄	—	2	1	no	—
8	C₃₇×M₆₄	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₃₇	—	2	1	no	—
10	C₂×C₃₂×C₃₇	—	2	1	no	—
11	C₂×C₃₂×C₃₇	—	2	1	no	—
12	C₂×C₃₂×C₃₇	C₄	1	2	yes	complemented
13	C₃₇×#346	—	2	1	no	—
14	C₄×C₁₆×C₃₇	—	2	1	no	—
15	C₂×M₁₆×C₃₇	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
17	#117×C₃₇	C₈	1	1	yes	characteristic
18	M₃₂×C₃₇	—	2	1	no	—
19	M₃₂×C₃₇	C₈	1	2	yes	—
20	C₂×C₁₆×C₃₇	—	2	1	no	—
21	C₂×C₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₃₇	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₃₇	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₃₇	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₃₇	Q₈	1	1	yes	characteristic
27	#97×C₃₇	—	2	1	no	—
28	C₄×C₈×C₃₇	—	2	1	no	—
29	C₃₂×C₃₇	—	2	2	no	—
30	C₃₂×C₃₇	—	2	2	no	—
31	C₂²×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
32	M₁₆×C₃₇	—	2	1	no	—
33	M₁₆×C₃₇	C₁₆	1	2	yes	—
34	C₂×C₈×C₃₇	—	2	1	no	—
35	C₂×C₈×C₃₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₃₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₃₇	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₃₇	M₁₆	1	1	yes	characteristic
41	C₄⋊C₄×C₃₇	C₁₆	1	2	yes	—
42	C₂²⋊C₄×C₃₇	—	2	1	no	—
43	C₄²×C₃₇	—	2	1	no	—
44	C₁₆×C₃₇	—	2	2	no	—
45	C₁₆×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₁₆×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₆×C₃₇	C₄⋊C₄	1	1	yes	characteristic
48	C₁₆×C₃₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₃₇	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₃₇	—	2	2	no	—
51	C₂×C₄×C₃₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₃₇	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₃₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₃₇	M₃₂	1	1	yes	characteristic
57	C₈×C₃₇	—	2	2	no	—
58	C₈×C₃₇	#95	1	1	yes	characteristic
59	C₈×C₃₇	#97	1	1	yes	characteristic
60	C₈×C₃₇	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₃₇	C₄⋊C₈	1	1	yes	characteristic
62	#7470	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₃₇	—	2	2	no	—
64	C₂²×C₃₇	#346	1	1	yes	characteristic
65	C₂²×C₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
66	C₂²×C₃₇	C₄⋊C₁₆	1	1	yes	characteristic
67	C₄×C₃₇	—	4	2	no	—
68	C₄×C₃₇	#336	1	1	yes	characteristic
69	C₄×C₃₇	C₄×C₁₆	1	1	yes	characteristic
70	C₄×C₃₇	C₂²⋊C₁₆	1	1	yes	characteristic
71	C₄×C₃₇	C₄⋊C₁₆	1	1	yes	characteristic
72	C₂×M₆₄	C₂×C₃₇	1	1	yes	characteristic
73	C₂²×C₃₂	C₂×C₃₇	1	1	yes	characteristic
74	#2108	C₂×C₃₇	1	1	yes	characteristic
75	C₂×C₃₇	—	2	1	no	—
76	C₂×C₃₇	—	2	1	no	—
77	C₂×C₃₇	#1374	1	1	yes	characteristic
78	C₂×C₃₇	#1401	1	2	yes	—
79	C₂×M₃₂	C₄×C₃₇	1	1	yes	characteristic
80	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
81	#405	C₄×C₃₇	1	1	yes	characteristic
82	M₆₄	—	2	1	no	—
83	M₆₄	C₄×C₃₇	1	2	yes	complemented
84	C₂×C₃₂	—	2	1	no	—
85	C₂×C₃₂	—	2	1	no	—
86	C₂×C₃₂	—	2	1	no	—
87	C₂×C₃₂	C₄×C₃₇	1	2	yes	complemented
88	#346	—	2	1	no	—
89	C₄×C₁₆	—	2	1	no	—
90	C₃₇	#7470	1	1	yes	characteristic, direct product factor, Sylow
91	C₂×M₁₆	C₈×C₃₇	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
93	#117	C₈×C₃₇	1	1	yes	characteristic
94	M₃₂	—	2	1	no	—
95	M₃₂	C₈×C₃₇	1	2	yes	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
98	C₂×C₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
99	C₂×C₁₆	D₄×C₃₇	1	1	yes	characteristic
100	C₂×C₁₆	D₄×C₃₇	1	1	yes	characteristic
101	C₂×C₁₆	D₄×C₃₇	1	1	yes	characteristic
102	C₂×C₁₆	Q₈×C₃₇	1	1	yes	characteristic
103	#97	—	2	1	no	—
104	C₄×C₈	—	2	1	no	—
105	C₃₂	—	2	2	no	—
106	C₃₂	—	2	2	no	—
107	C₂²×C₄	C₂×C₈×C₃₇	1	1	yes	characteristic
108	M₁₆	—	2	1	no	—
109	M₁₆	C₁₆×C₃₇	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	C₄²×C₃₇	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
113	C₂×C₈	C₄⋊C₄×C₃₇	1	1	yes	characteristic
114	C₂×C₈	C₂×C₈×C₃₇	1	1	yes	characteristic
115	C₂×C₈	M₁₆×C₃₇	1	1	yes	characteristic
116	C₂×C₈	M₁₆×C₃₇	1	1	yes	characteristic
117	C₄⋊C₄	C₁₆×C₃₇	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
122	C₁₆	C₂²⋊C₄×C₃₇	1	1	yes	characteristic
123	C₁₆	C₄⋊C₄×C₃₇	1	1	yes	characteristic
124	C₁₆	C₄⋊C₄×C₃₇	1	1	yes	characteristic
125	C₂³	M₃₂×C₃₇	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₄×C₈×C₃₇	1	1	yes	characteristic
128	C₂×C₄	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
129	C₂×C₄	C₄⋊C₈×C₃₇	1	1	yes	characteristic
130	C₂×C₄	C₂×C₁₆×C₃₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₁₆×C₃₇	1	1	yes	characteristic
132	C₂×C₄	M₃₂×C₃₇	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	#95×C₃₇	1	1	yes	characteristic
135	C₈	#97×C₃₇	1	1	yes	characteristic
136	C₈	C₂²⋊C₈×C₃₇	1	1	yes	characteristic
137	C₈	C₄⋊C₈×C₃₇	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₃₇×#346	1	1	yes	characteristic
140	C₂²	C₃₇×C₂²⋊C₁₆	1	1	yes	characteristic
141	C₂²	C₃₇×C₄⋊C₁₆	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	C₃₇×#336	1	1	yes	characteristic
144	C₄	C₄×C₁₆×C₃₇	1	1	yes	characteristic
145	C₄	C₃₇×C₂²⋊C₁₆	1	1	yes	characteristic
146	C₄	C₃₇×C₄⋊C₁₆	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	C₃₇×#1374	1	1	yes	characteristic
150	C₂	C₃₇×#1401	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₃₇	D₄
commutator	C₄	C₄×C₁₆×C₃₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₃₇
socle	C₂²×C₃₇	C₂²⋊C₁₆
prime radical	C₂³×C₃₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₇ ◁ C₂²×C₁₆×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₇×M₆₄	C₄	semidirect
12. C₂×C₃₂×C₃₇	C₄	semidirect
62. #7470	C₃₇	direct (N × H)
83. M₆₄	C₄×C₃₇	semidirect
87. C₂×C₃₂	C₄×C₃₇	semidirect
90. C₃₇	#7470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₇×M₆₄	2
C₂²×C₃₂×C₃₇	2
C₃₇×#2108	2
#7470	37

Subgroups — C37×#7470 #22628723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#7470 #22628723