Subgroups — C₂×C₈×Dic₇₄ #18003517

Order 4736.

440 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₇₄	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924465	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₃₇	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₄	1	1	yes	characteristic
8	C₂²×#64436	C₂²	1	2	yes	—
9	C₄×Dic₇₄	C₄	1	4	yes	—
10	C₄×Dic₇₄	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
12	C₂×#10596936	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
14	C₄⋊#64436	C₂²	1	4	yes	—
15	#11924465	C₂²	1	8	yes	—
16	C₈×Dic₃₇	C₂²	1	8	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₃₇	C₂²	1	4	yes	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₂×Dic₇₄	C₈	1	8	yes	direct product factor
27	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₃₇	—	2	2	no	—
29	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
30	C₂×C₈×C₃₇	C₂³	1	2	yes	—
31	C₂×C₈×C₃₇	C₂³	1	2	yes	—
32	C₄²×C₃₇	C₂×C₄	1	2	yes	—
33	C₄²×C₃₇	C₂³	1	2	yes	—
34	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
35	#10596936	C₂×C₄	1	8	yes	—
36	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
37	C₄×Dic₃₇	C₂³	1	4	yes	—
38	C₂×#64436	—	2	4	no	—
39	C₂×#64436	C₂³	1	4	yes	—
40	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₃₇	#41	1	2	yes	—
48	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	C₂×C₈	1	8	yes	—
51	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
52	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
53	Dic₇₄	C₂×C₈	1	16	yes	direct product factor
54	C₈×C₃₇	—	2	2	no	—
55	C₈×C₃₇	C₂×Q₈	1	4	yes	complemented
56	#64436	—	2	4	no	—
57	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
58	C₂²×C₃₇	C₂²×C₈	1	2	yes	—
59	C₂²×C₃₇	#131	1	2	yes	—
60	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	C₄×Q₈	1	2	yes	—
63	C₄×C₃₇	C₂²×C₈	1	4	yes	—
64	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
66	Dic₃₇	—	2	4	no	—
67	Dic₃₇	C₂²×C₈	1	8	yes	—
68	C₂×C₈×Q₈	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	C₈×Q₈	1	4	yes	complemented
70	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
72	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
73	C₂×C₄×Q₈	—	37	1	no	—
74	C₈×Q₈	—	37	8	no	—
75	C₂×C₄⋊C₈	—	37	2	no	—
76	C₂×C₄⋊C₈	—	37	1	no	—
77	C₂×C₄×C₈	—	37	2	no	—
78	C₂×C₄×C₈	D₃₇	1	1	yes	characteristic
79	C₃₇	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	37	1	no	—
81	C₂²×C₈	—	37	2	no	—
82	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
84	C₄×Q₈	—	37	4	no	—
85	C₄×Q₈	—	37	4	no	—
86	C₂×C₄⋊C₄	—	37	2	no	—
87	C₂×C₄⋊C₄	—	37	1	no	—
88	C₂×C₄²	—	37	2	no	—
89	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
90	C₄⋊C₈	—	37	8	no	—
91	C₄⋊C₈	—	37	4	no	—
92	C₄×C₈	—	37	8	no	—
93	C₄×C₈	C₂×D₃₇	1	4	yes	—
94	C₂×Q₈	—	37	8	no	—
95	C₂×Q₈	—	37	4	no	—
96	C₂²×C₄	—	37	2	no	—
97	C₂²×C₄	—	37	2	no	—
98	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
99	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	37	4	no	—
103	C₂×C₈	—	74	4	no	—
104	C₂×C₈	Dic₇₄	1	4	yes	direct product factor
105	C₂×C₈	C₂²×D₃₇	1	2	yes	—
106	C₂×C₈	C₂²×D₃₇	1	2	yes	—
107	C₄⋊C₄	—	37	8	no	—
108	C₄⋊C₄	—	37	4	no	—
109	C₄²	—	37	4	no	—
110	C₄²	—	37	4	no	—
111	C₄²	C₄×D₃₇	1	2	yes	—
112	C₄²	C₂²×D₃₇	1	2	yes	—
113	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
114	Q₈	—	37	16	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	37	8	no	—
117	C₂×C₄	—	37	4	no	—
118	C₂×C₄	—	37	4	no	—
119	C₂×C₄	—	74	4	no	—
120	C₂×C₄	C₈×D₃₇	1	4	yes	—
121	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
125	C₂×C₄	#10596962	1	2	yes	—
126	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	74	4	no	—
129	C₈	C₂×Dic₇₄	1	4	yes	direct product factor
130	C₂²	C₄×Dic₇₄	1	2	yes	—
131	C₂²	C₂×C₈×D₃₇	1	2	yes	—
132	C₂²	#11924542	1	2	yes	—
133	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	37	8	no	—
136	C₄	—	74	4	no	—
137	C₄	C₄×Dic₇₄	1	2	yes	—
138	C₄	C₂×C₈×D₃₇	1	4	yes	—
139	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
140	C₄	C₂×#10596962	1	1	yes	characteristic
141	C₂	C₈×Dic₇₄	1	4	yes	direct product factor
142	C₂	C₂×C₄×Dic₇₄	1	1	yes	characteristic
143	C₂	C₂²×C₈×D₃₇	1	1	yes	characteristic
144	C₂	C₂×#11924542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₃₇
commutator	C₂×C₃₇	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×C₄×C₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₃₇
hypocenter	C₃₇	C₂×C₈×Q₈
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₇₄	C₂	direct (N × H)
26. C₂×Dic₇₄	C₈	direct (N × H)
29. C₂×C₈×C₃₇	Q₈	semidirect
53. Dic₇₄	C₂×C₈	direct (N × H)
55. C₈×C₃₇	C₂×Q₈	semidirect
69. C₂×C₃₇	C₈×Q₈	semidirect
79. C₃₇	C₂×C₈×Q₈	semidirect
104. C₂×C₈	Dic₇₄	direct (N × H)
129. C₈	C₂×Dic₇₄	direct (N × H)
141. C₂	C₈×Dic₇₄	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×C₄×Dic₇₄	2
C₈×Dic₇₄	2
C₂×C₄⋊#64436	2
C₂×#11924465	2
C₂×C₈×Dic₃₇	2
C₂×C₄×C₈×C₃₇	2
C₂×C₈×Q₈	37

Subgroups — C2×C8×Dic74 #18003517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Dic₇₄ #18003517