Subgroups — C₂×C₄×C₂²⋊C₄×C₄₇ #20239878

Order 6016.

1360 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₄×C₄₇	C₂	1	2	yes	maximal
3	C₂²×C₄²×C₄₇	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊C₄×C₄₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₄₇	C₂	1	2	yes	maximal, complemented
6	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₄₇	—	2	8	no	—
8	C₂³×C₄×C₄₇	C₄	1	8	yes	complemented
9	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₄₇	C₂²	1	4	yes	—
11	C₂³×C₄×C₄₇	C₂²	1	4	yes	—
12	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₄₇	C₄	1	16	yes	direct product factor
14	C₂×C₂²⋊C₄×C₄₇	C₂²	1	8	yes	—
15	C₂×C₄²×C₄₇	—	2	8	no	—
16	C₂×C₄²×C₄₇	C₂²	1	8	yes	complemented
17	#95×C₄₇	C₂²	1	8	yes	complemented
18	C₂⁴×C₄₇	—	2	4	no	—
19	C₂⁴×C₄₇	—	2	4	no	—
20	C₂⁴×C₄₇	C₂×C₄	1	8	yes	—
21	C₂⁴×C₄₇	C₂×C₄	1	4	yes	—
22	C₂⁴×C₄₇	C₂³	1	2	yes	—
23	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	2	8	no	—
25	C₂²×C₄×C₄₇	—	2	4	no	—
26	C₂²×C₄×C₄₇	—	2	8	no	—
27	C₂²×C₄×C₄₇	—	2	32	no	—
28	C₂²×C₄×C₄₇	—	2	16	no	—
29	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₄₇	C₂×C₄	1	16	yes	complemented
32	C₂²×C₄×C₄₇	C₂×C₄	1	16	yes	complemented
33	C₂²×C₄×C₄₇	D₄	1	8	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	8	yes	—
36	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂²⋊C₄×C₄₇	C₂×C₄	1	32	yes	direct product factor
38	C₄²×C₄₇	—	2	16	no	—
39	C₂³×C₄₇	—	2	8	no	—
40	C₂³×C₄₇	—	2	8	no	—
41	C₂³×C₄₇	—	2	32	no	—
42	C₂³×C₄₇	—	2	4	no	—
43	C₂³×C₄₇	—	2	4	no	—
44	C₂³×C₄₇	C₄²	1	16	yes	complemented
45	C₂³×C₄₇	C₂²×C₄	1	8	yes	—
46	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂³×C₄₇	C₂×D₄	1	4	yes	—
50	C₂³×C₄₇	#41	1	4	yes	—
51	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	8	no	—
53	C₂×C₄×C₄₇	—	2	32	no	—
54	C₂×C₄×C₄₇	—	2	8	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	8	no	—
57	C₂×C₄×C₄₇	—	2	32	no	—
58	C₂×C₄×C₄₇	C₂²⋊C₄	1	16	yes	direct product factor
59	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₄₇	C₂²×C₄	1	16	yes	complemented
62	C₂×C₄×C₄₇	C₂×D₄	1	8	yes	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	—	2	16	no	—
66	C₂²×C₄₇	—	2	32	no	—
67	C₂²×C₄₇	C₂×C₄²	1	4	yes	—
68	C₂²×C₄₇	C₂×C₄²	1	8	yes	complemented
69	C₂²×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄₇	#117	1	4	yes	—
71	C₂²×C₄₇	C₄×D₄	1	16	yes	—
72	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
73	C₂²×C₄₇	C₂³×C₄	1	2	yes	—
74	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
75	C₂²×C₄₇	C₂×#41	1	2	yes	—
76	C₄×C₄₇	—	2	4	no	—
77	C₄×C₄₇	—	2	16	no	—
78	C₄×C₄₇	C₂×C₂²⋊C₄	1	8	yes	direct product factor
79	C₂×C₄×C₂²⋊C₄	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×C₄₇	—	2	8	no	—
81	C₂×C₄₇	C₄×C₂²⋊C₄	1	8	yes	direct product factor
82	C₂×C₄₇	C₂²×C₄²	1	1	yes	characteristic
83	C₂×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
85	C₂×C₄₇	C₂×C₄×D₄	1	4	yes	—
86	C₂⁴×C₄	C₂×C₄₇	1	1	yes	characteristic
87	C₂²×C₂²⋊C₄	C₂×C₄₇	1	2	yes	—
88	C₂²×C₄²	C₂×C₄₇	1	2	yes	complemented
89	C₄×C₂²⋊C₄	C₂×C₄₇	1	8	yes	direct product factor
90	C₂×#95	C₂×C₄₇	1	2	yes	complemented
91	C₄₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
92	C₂⁵	C₂²×C₄₇	1	1	yes	characteristic
93	C₂³×C₄	—	2	8	no	—
94	C₂³×C₄	C₄×C₄₇	1	8	yes	complemented
95	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
96	C₂³×C₄	C₂²×C₄₇	1	4	yes	—
97	C₂³×C₄	C₂²×C₄₇	1	4	yes	—
98	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
99	C₂×C₂²⋊C₄	C₄×C₄₇	1	16	yes	direct product factor
100	C₂×C₂²⋊C₄	C₂²×C₄₇	1	8	yes	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₂²×C₄₇	1	8	yes	complemented
103	#95	C₂²×C₄₇	1	8	yes	complemented
104	C₂⁴	—	2	4	no	—
105	C₂⁴	—	2	4	no	—
106	C₂⁴	C₂×C₄×C₄₇	1	8	yes	—
107	C₂⁴	C₂×C₄×C₄₇	1	4	yes	—
108	C₂⁴	C₂³×C₄₇	1	2	yes	—
109	C₂⁴	C₂³×C₄₇	1	1	yes	characteristic
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	8	no	—
113	C₂²×C₄	—	2	32	no	—
114	C₂²×C₄	—	2	16	no	—
115	C₂²×C₄	C₂×C₄×C₄₇	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₄₇	1	4	yes	—
117	C₂²×C₄	C₂×C₄×C₄₇	1	16	yes	complemented
118	C₂²×C₄	C₂×C₄×C₄₇	1	16	yes	complemented
119	C₂²×C₄	D₄×C₄₇	1	8	yes	—
120	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
121	C₂²×C₄	C₂³×C₄₇	1	8	yes	—
122	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
123	C₂²⋊C₄	C₂×C₄×C₄₇	1	32	yes	direct product factor
124	C₄²	—	2	16	no	—
125	C₂³	—	2	8	no	—
126	C₂³	—	2	8	no	—
127	C₂³	—	2	32	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	C₄²×C₄₇	1	16	yes	complemented
131	C₂³	C₂²×C₄×C₄₇	1	8	yes	—
132	C₂³	C₂²×C₄×C₄₇	1	4	yes	—
133	C₂³	C₂²×C₄×C₄₇	1	2	yes	—
134	C₂³	C₂²×C₄×C₄₇	1	4	yes	—
135	C₂³	C₂×D₄×C₄₇	1	4	yes	—
136	C₂³	#41×C₄₇	1	4	yes	—
137	C₂³	C₂⁴×C₄₇	1	1	yes	characteristic
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	32	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	32	no	—
144	C₂×C₄	C₂²⋊C₄×C₄₇	1	16	yes	direct product factor
145	C₂×C₄	C₂²×C₄×C₄₇	1	4	yes	—
146	C₂×C₄	C₂²×C₄×C₄₇	1	4	yes	—
147	C₂×C₄	C₂²×C₄×C₄₇	1	16	yes	complemented
148	C₂×C₄	C₂×D₄×C₄₇	1	8	yes	—
149	C₂²	—	2	4	no	—
150	C₂²	—	2	4	no	—
151	C₂²	—	2	16	no	—
152	C₂²	—	2	32	no	—
153	C₂²	C₂×C₄²×C₄₇	1	4	yes	—
154	C₂²	C₂×C₄²×C₄₇	1	8	yes	complemented
155	C₂²	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
156	C₂²	#117×C₄₇	1	4	yes	—
157	C₂²	C₄×D₄×C₄₇	1	16	yes	—
158	C₂²	C₂³×C₄×C₄₇	1	1	yes	characteristic
159	C₂²	C₂³×C₄×C₄₇	1	2	yes	—
160	C₂²	C₂²×D₄×C₄₇	1	2	yes	—
161	C₂²	C₂×#41×C₄₇	1	2	yes	—
162	C₄	—	2	4	no	—
163	C₄	—	2	16	no	—
164	C₄	C₂×C₂²⋊C₄×C₄₇	1	8	yes	direct product factor
165	C₂	—	2	8	no	—
166	C₂	C₄×C₂²⋊C₄×C₄₇	1	8	yes	direct product factor
167	C₂	C₂²×C₄²×C₄₇	1	1	yes	characteristic
168	C₂	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
169	C₂	C₂×#117×C₄₇	1	1	yes	characteristic
170	C₂	C₂×C₄×D₄×C₄₇	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₄₇	C₂²
commutator	C₂	C₂²×C₄²×C₄₇
Frattini, cosocle	C₂³	C₂⁴×C₄₇
socle	C₂⁴×C₄₇	C₂³
prime radical	C₂⁵×C₄₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄²×C₄₇	C₂	semidirect
4. C₄×C₂²⋊C₄×C₄₇	C₂	direct (N × H)
5. C₂×#95×C₄₇	C₂	semidirect
8. C₂³×C₄×C₄₇	C₄	semidirect
13. C₂×C₂²⋊C₄×C₄₇	C₄	direct (N × H)
16. C₂×C₄²×C₄₇	C₂²	semidirect
17. #95×C₄₇	C₂²	semidirect
31. C₂²×C₄×C₄₇	C₂×C₄	semidirect
32. C₂²×C₄×C₄₇	C₂×C₄	semidirect
37. C₂²⋊C₄×C₄₇	C₂×C₄	direct (N × H)
44. C₂³×C₄₇	C₄²	semidirect
58. C₂×C₄×C₄₇	C₂²⋊C₄	direct (N × H)
61. C₂×C₄×C₄₇	C₂²×C₄	semidirect
68. C₂²×C₄₇	C₂×C₄²	semidirect
78. C₄×C₄₇	C₂×C₂²⋊C₄	direct (N × H)
79. C₂×C₄×C₂²⋊C₄	C₄₇	direct (N × H)
81. C₂×C₄₇	C₄×C₂²⋊C₄	direct (N × H)
88. C₂²×C₄²	C₂×C₄₇	semidirect
89. C₄×C₂²⋊C₄	C₂×C₄₇	direct (N × H)
90. C₂×#95	C₂×C₄₇	semidirect
91. C₄₇	C₂×C₄×C₂²⋊C₄	direct (N × H)
94. C₂³×C₄	C₄×C₄₇	semidirect
99. C₂×C₂²⋊C₄	C₄×C₄₇	direct (N × H)
102. C₂×C₄²	C₂²×C₄₇	semidirect
103. #95	C₂²×C₄₇	semidirect
117. C₂²×C₄	C₂×C₄×C₄₇	semidirect
118. C₂²×C₄	C₂×C₄×C₄₇	semidirect
123. C₂²⋊C₄	C₂×C₄×C₄₇	direct (N × H)
130. C₂³	C₄²×C₄₇	semidirect
144. C₂×C₄	C₂²⋊C₄×C₄₇	direct (N × H)
147. C₂×C₄	C₂²×C₄×C₄₇	semidirect
154. C₂²	C₂×C₄²×C₄₇	semidirect
164. C₄	C₂×C₂²⋊C₄×C₄₇	direct (N × H)
166. C₂	C₄×C₂²⋊C₄×C₄₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₄×C₄₇	2
C₂²×C₂²⋊C₄×C₄₇	2
C₂²×C₄²×C₄₇	2
C₄×C₂²⋊C₄×C₄₇	2
C₂×#95×C₄₇	2
C₂×C₄×C₂²⋊C₄	47

Subgroups — C2×C4×C22⋊C4×C47 #20239878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂²⋊C₄×C₄₇ #20239878