Subgroups — C₈×C₄⋊C₄×C₃₇ #17990587

Order 4736.

296 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₈×C₃₇	C₂²
commutator	C₂	C₂×C₄×C₈×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₄⋊C₈×C₃₇	C₄	semidirect
14. C₄×C₈×C₃₇	C₄	semidirect
29. C₄⋊C₄×C₃₇	C₈	direct (N × H)
50. C₈×C₃₇	C₄⋊C₄	direct (N × H)
58. C₄×C₃₇	C₄×C₈	semidirect
63. C₈×C₄⋊C₄	C₃₇	direct (N × H)
74. C₃₇	C₈×C₄⋊C₄	direct (N × H)
81. C₄⋊C₈	C₄×C₃₇	semidirect
83. C₄×C₈	C₄×C₃₇	semidirect
98. C₄⋊C₄	C₈×C₃₇	direct (N × H)
119. C₈	C₄⋊C₄×C₃₇	direct (N × H)
127. C₄	C₄×C₈×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×C₃₇	2
C₂×C₄×C₈×C₃₇	2
C₄×C₄⋊C₄×C₃₇	2
C₃₇×#336	2
C₈×C₄⋊C₄	37

Subgroups — C8×C4⋊C4×C37 #17990587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊C₄×C₃₇ #17990587