Subgroups — C₄×#15688845 #26105627

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688845	C₂	1	4	yes	maximal
4	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688845	C₄	1	16	yes	direct product factor
10	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
12	C₄³×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×C₄₇⋊C₈	—	2	8	no	—
16	C₄×C₄₇⋊C₈	C₈	1	16	yes	complemented
17	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
18	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₄₇	—	2	8	no	—
25	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
26	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	—	2	8	no	—
31	C₂×C₈×C₄₇	C₄²	1	8	yes	—
32	C₄²×C₄₇	C₂×C₈	1	8	yes	—
33	C₄²×C₄₇	M₁₆	1	8	yes	—
34	C₄²×C₄₇	M₁₆	1	8	yes	—
35	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
36	C₂×C₄₇⋊C₈	—	2	8	no	—
37	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
38	C₂×C₄₇⋊C₈	C₄²	1	8	yes	—
39	C₂×C₄₇⋊C₈	C₂×C₈	1	16	yes	—
40	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₄×C₈	1	8	yes	—
42	C₂×C₄×C₄₇	#97	1	8	yes	—
43	C₂×C₄×C₄₇	#97	1	8	yes	—
44	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
48	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
49	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
50	C₈×C₄₇	—	2	8	no	—
51	C₄₇⋊C₈	—	2	8	no	—
52	C₄₇⋊C₈	C₄×C₈	1	16	yes	complemented
53	C₄×#322	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	C₄³	1	1	yes	characteristic
55	C₂²×C₄₇	C₂×C₄×C₈	1	2	yes	—
56	C₂²×C₄₇	C₂×#97	1	2	yes	—
57	C₂²×C₄₇	C₂×#97	1	2	yes	—
58	C₄×C₄₇	#322	1	16	yes	complemented
59	C₄×C₄₇	C₂×C₄×C₈	1	4	yes	—
60	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
61	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
62	C₄²×C₈	—	47	1	no	—
63	C₄²×C₈	—	47	1	no	—
64	C₄²×C₈	D₄₇	1	1	yes	characteristic
65	C₂×#322	—	47	4	no	—
66	C₂×C₄₇	C₂×#322	1	4	yes	—
67	C₂×C₄₇	C₄²×C₈	1	1	yes	characteristic
68	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
69	C₂×C₄₇	C₄×#97	1	1	yes	characteristic
70	C₂×C₄×C₈	—	47	4	no	—
71	C₂×C₄×C₈	—	47	2	no	—
72	C₂×C₄×C₈	—	47	2	no	—
73	C₂×C₄×C₈	—	47	4	no	—
74	C₂×C₄×C₈	Dic₄₇	1	4	yes	—
75	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
76	C₄³	C₂×D₄₇	1	1	yes	characteristic
77	#322	—	47	16	no	—
78	C₄₇	C₄×#322	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	47	4	no	—
80	C₂²×C₈	—	47	4	no	—
81	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
82	C₂×C₄²	C₄×D₄₇	1	2	yes	—
83	C₂×C₄²	C₄×D₄₇	1	2	yes	—
84	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
85	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
86	C₄×C₈	—	2	8	no	—
87	C₄×C₈	—	47	16	no	—
88	C₄×C₈	—	47	8	no	—
89	C₄×C₈	—	47	8	no	—
90	C₄×C₈	—	94	8	no	—
91	C₄×C₈	C₄×D₄₇	1	8	yes	—
92	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
93	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	47	8	no	—
98	C₂×C₈	—	47	16	no	—
99	C₂×C₈	—	47	8	no	—
100	C₂×C₈	—	94	8	no	—
101	C₂×C₈	C₄×Dic₄₇	1	8	yes	—
102	C₄²	C₈×D₄₇	1	8	yes	—
103	C₄²	#10625324	1	8	yes	—
104	C₄²	#10625329	1	8	yes	—
105	C₄²	C₂×C₄×D₄₇	1	4	yes	—
106	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
107	C₂×C₄	C₈×Dic₄₇	1	8	yes	—
108	C₂×C₄	#13093934	1	8	yes	—
109	C₂×C₄	#13093935	1	8	yes	—
110	C₂×C₄	C₄²×D₄₇	1	2	yes	—
111	C₂×C₄	C₄²×D₄₇	1	2	yes	—
112	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
113	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
114	C₂×C₄	C₂×#10625324	1	4	yes	—
115	C₂×C₄	C₂×#10625329	1	4	yes	—
116	C₈	—	2	8	no	—
117	C₈	—	47	16	no	—
118	C₈	—	94	8	no	—
119	C₂²	C₄²×Dic₄₇	1	1	yes	characteristic
120	C₂²	C₂×C₈×Dic₄₇	1	2	yes	—
121	C₂²	C₂×#13093934	1	2	yes	—
122	C₂²	C₂×#13093935	1	2	yes	—
123	C₄	#15688845	1	16	yes	direct product factor
124	C₄	C₄×C₈×D₄₇	1	4	yes	—
125	C₄	C₄×#10625324	1	4	yes	—
126	C₄	C₄×#10625329	1	4	yes	—
127	C₂	C₂×#15688845	1	4	yes	—
128	C₂	C₄×C₈×Dic₄₇	1	1	yes	characteristic
129	C₂	C₄×#13093934	1	1	yes	characteristic
130	C₂	C₄×#13093935	1	1	yes	characteristic

	subgroup	G/H
center	C₄³	C₂×D₄₇
commutator	C₂×C₄₇	C₄²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	C₄×#322
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
9. #15688845	C₄	direct (N × H)
16. C₄×C₄₇⋊C₈	C₈	semidirect
52. C₄₇⋊C₈	C₄×C₈	semidirect
58. C₄×C₄₇	#322	semidirect
78. C₄₇	C₄×#322	semidirect
123. C₄	#15688845	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₄₇⋊C₈	2
C₂×#15688845	2
C₄²×C₈×C₄₇	2
C₄×#322	47

Subgroups — C4×#15688845 #26105627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688845 #26105627