Subgroups — C₂²×C₂²⋊Q₈×C₄₇ #20241043

Order 6016.

1840 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Q₈×C₄₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄⋊C₄×C₄₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂²⋊C₄×C₄₇	C₂	1	2	yes	maximal
7	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×C₄₇	—	2	8	no	—
9	C₂²×Q₈×C₄₇	C₂²	1	12	yes	complemented
10	C₂³×C₄×C₄₇	—	2	8	no	—
11	C₂³×C₄×C₄₇	C₂²	1	12	yes	—
12	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
13	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₄₇	C₂²	1	64	yes	direct product factor
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	12	yes	complemented
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	24	yes	complemented
19	C₂×C₂²⋊C₄×C₄₇	C₂²	1	24	yes	—
20	C₂⁴×C₄₇	—	2	4	no	—
21	C₂⁴×C₄₇	—	2	4	no	—
22	C₂⁴×C₄₇	Q₈	1	8	yes	complemented
23	C₂⁴×C₄₇	C₂³	1	6	yes	—
24	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	—	2	48	no	—
26	C₂×Q₈×C₄₇	C₂³	1	16	yes	complemented
27	C₂²×C₄×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	—	2	8	no	—
29	C₂²×C₄×C₄₇	—	2	8	no	—
30	C₂²×C₄×C₄₇	—	2	48	no	—
31	C₂²×C₄×C₄₇	D₄	1	8	yes	—
32	C₂²×C₄×C₄₇	C₂³	1	12	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	12	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	6	yes	—
35	C₂²×C₄×C₄₇	C₂³	1	16	yes	—
36	C₂²×C₄×C₄₇	C₂³	1	6	yes	—
37	C₄⋊C₄×C₄₇	C₂³	1	16	yes	complemented
38	C₄⋊C₄×C₄₇	C₂³	1	32	yes	complemented
39	C₂²⋊C₄×C₄₇	C₂³	1	32	yes	—
40	C₂³×C₄₇	—	2	32	no	—
41	C₂³×C₄₇	—	2	12	no	—
42	C₂³×C₄₇	—	2	12	no	—
43	C₂³×C₄₇	C₂×D₄	1	4	yes	—
44	C₂³×C₄₇	C₂×Q₈	1	4	yes	—
45	C₂³×C₄₇	C₂×Q₈	1	24	yes	complemented
46	C₂³×C₄₇	#41	1	4	yes	—
47	C₂³×C₄₇	C₂⁴	1	4	yes	—
48	C₂³×C₄₇	C₂⁴	1	3	yes	—
49	Q₈×C₄₇	—	2	32	no	—
50	C₂×C₄×C₄₇	—	2	12	no	—
51	C₂×C₄×C₄₇	—	2	24	no	—
52	C₂×C₄×C₄₇	—	2	24	no	—
53	C₂×C₄×C₄₇	—	2	32	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	24	yes	—
55	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
56	C₂×C₄×C₄₇	C₂⁴	1	8	yes	—
57	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
58	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
59	C₂²×C₄₇	—	2	48	no	—
60	C₂²×C₄₇	—	2	4	no	—
61	C₂²×C₄₇	—	2	4	no	—
62	C₂²×C₄₇	C₂²⋊Q₈	1	16	yes	direct product factor
63	C₂²×C₄₇	C₂²×D₄	1	6	yes	—
64	C₂²×C₄₇	C₂²×Q₈	1	6	yes	—
65	C₂²×C₄₇	C₂²×Q₈	1	8	yes	complemented
66	C₂²×C₄₇	C₂×#41	1	6	yes	—
67	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	2	8	no	—
70	C₄×C₄₇	—	2	8	no	—
71	C₄×C₄₇	C₂²×D₄	1	8	yes	—
72	C₂²×C₂²⋊Q₈	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂×C₄₇	—	2	8	no	—
74	C₂×C₄₇	C₂×C₂²⋊Q₈	1	12	yes	direct product factor
75	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
76	C₂×C₄₇	C₂³×Q₈	1	1	yes	characteristic
77	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
78	C₂³×Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
79	C₂⁴×C₄	C₂×C₄₇	1	1	yes	characteristic
80	C₂×C₂²⋊Q₈	C₂×C₄₇	1	24	yes	direct product factor
81	C₂²×C₄⋊C₄	C₂×C₄₇	1	1	yes	characteristic, complemented
82	C₂²×C₄⋊C₄	C₂×C₄₇	1	2	yes	complemented
83	C₂²×C₂²⋊C₄	C₂×C₄₇	1	2	yes	—
84	C₄₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
85	C₂⁵	C₂²×C₄₇	1	1	yes	characteristic
86	C₂²×Q₈	—	2	8	no	—
87	C₂²×Q₈	C₂²×C₄₇	1	12	yes	complemented
88	C₂³×C₄	—	2	8	no	—
89	C₂³×C₄	C₂²×C₄₇	1	2	yes	—
90	C₂³×C₄	C₂²×C₄₇	1	2	yes	—
91	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
92	C₂³×C₄	C₂²×C₄₇	1	12	yes	—
93	C₂³×C₄	C₂²×C₄₇	1	1	yes	characteristic
94	C₂²⋊Q₈	C₂²×C₄₇	1	64	yes	direct product factor
95	C₂×C₄⋊C₄	C₂²×C₄₇	1	12	yes	complemented
96	C₂×C₄⋊C₄	C₂²×C₄₇	1	24	yes	complemented
97	C₂×C₂²⋊C₄	C₂²×C₄₇	1	24	yes	—
98	C₂⁴	—	2	4	no	—
99	C₂⁴	—	2	4	no	—
100	C₂⁴	Q₈×C₄₇	1	8	yes	complemented
101	C₂⁴	C₂³×C₄₇	1	6	yes	—
102	C₂⁴	C₂³×C₄₇	1	1	yes	characteristic
103	C₂×Q₈	—	2	48	no	—
104	C₂×Q₈	C₂³×C₄₇	1	16	yes	complemented
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	8	no	—
107	C₂²×C₄	—	2	8	no	—
108	C₂²×C₄	—	2	48	no	—
109	C₂²×C₄	D₄×C₄₇	1	8	yes	—
110	C₂²×C₄	C₂³×C₄₇	1	12	yes	—
111	C₂²×C₄	C₂³×C₄₇	1	12	yes	—
112	C₂²×C₄	C₂³×C₄₇	1	6	yes	—
113	C₂²×C₄	C₂³×C₄₇	1	16	yes	—
114	C₂²×C₄	C₂³×C₄₇	1	6	yes	—
115	C₄⋊C₄	C₂³×C₄₇	1	16	yes	complemented
116	C₄⋊C₄	C₂³×C₄₇	1	32	yes	complemented
117	C₂²⋊C₄	C₂³×C₄₇	1	32	yes	—
118	C₂³	—	2	32	no	—
119	C₂³	—	2	12	no	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂×D₄×C₄₇	1	4	yes	—
122	C₂³	C₂×Q₈×C₄₇	1	4	yes	—
123	C₂³	C₂×Q₈×C₄₇	1	24	yes	complemented
124	C₂³	#41×C₄₇	1	4	yes	—
125	C₂³	C₂⁴×C₄₇	1	4	yes	—
126	C₂³	C₂⁴×C₄₇	1	3	yes	—
127	Q₈	—	2	32	no	—
128	C₂×C₄	—	2	12	no	—
129	C₂×C₄	—	2	24	no	—
130	C₂×C₄	—	2	24	no	—
131	C₂×C₄	—	2	32	no	—
132	C₂×C₄	C₂×D₄×C₄₇	1	24	yes	—
133	C₂×C₄	C₂⁴×C₄₇	1	8	yes	—
134	C₂×C₄	C₂⁴×C₄₇	1	8	yes	—
135	C₂×C₄	C₂⁴×C₄₇	1	4	yes	—
136	C₂×C₄	C₂⁴×C₄₇	1	4	yes	—
137	C₂²	—	2	48	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	2	4	no	—
140	C₂²	C₂²⋊Q₈×C₄₇	1	16	yes	direct product factor
141	C₂²	C₂²×D₄×C₄₇	1	6	yes	—
142	C₂²	C₂²×Q₈×C₄₇	1	6	yes	—
143	C₂²	C₂²×Q₈×C₄₇	1	8	yes	complemented
144	C₂²	C₂×#41×C₄₇	1	6	yes	—
145	C₂²	C₂⁵×C₄₇	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	2	8	no	—
148	C₄	—	2	8	no	—
149	C₄	C₂²×D₄×C₄₇	1	8	yes	—
150	C₂	—	2	8	no	—
151	C₂	C₂×C₂²⋊Q₈×C₄₇	1	12	yes	direct product factor
152	C₂	C₂³×D₄×C₄₇	1	1	yes	characteristic
153	C₂	C₂³×Q₈×C₄₇	1	1	yes	characteristic
154	C₂	C₂²×#41×C₄₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×C₄₇	C₂	semidirect
3. C₂×C₂²⋊Q₈×C₄₇	C₂	direct (N × H)
4. C₂²×C₄⋊C₄×C₄₇	C₂	semidirect
5. C₂²×C₄⋊C₄×C₄₇	C₂	semidirect
9. C₂²×Q₈×C₄₇	C₂²	semidirect
16. C₂²⋊Q₈×C₄₇	C₂²	direct (N × H)
17. C₂×C₄⋊C₄×C₄₇	C₂²	semidirect
18. C₂×C₄⋊C₄×C₄₇	C₂²	semidirect
22. C₂⁴×C₄₇	Q₈	semidirect
26. C₂×Q₈×C₄₇	C₂³	semidirect
37. C₄⋊C₄×C₄₇	C₂³	semidirect
38. C₄⋊C₄×C₄₇	C₂³	semidirect
45. C₂³×C₄₇	C₂×Q₈	semidirect
62. C₂²×C₄₇	C₂²⋊Q₈	direct (N × H)
65. C₂²×C₄₇	C₂²×Q₈	semidirect
72. C₂²×C₂²⋊Q₈	C₄₇	direct (N × H)
74. C₂×C₄₇	C₂×C₂²⋊Q₈	direct (N × H)
78. C₂³×Q₈	C₂×C₄₇	semidirect
80. C₂×C₂²⋊Q₈	C₂×C₄₇	direct (N × H)
81. C₂²×C₄⋊C₄	C₂×C₄₇	semidirect
82. C₂²×C₄⋊C₄	C₂×C₄₇	semidirect
84. C₄₇	C₂²×C₂²⋊Q₈	direct (N × H)
87. C₂²×Q₈	C₂²×C₄₇	semidirect
94. C₂²⋊Q₈	C₂²×C₄₇	direct (N × H)
95. C₂×C₄⋊C₄	C₂²×C₄₇	semidirect
96. C₂×C₄⋊C₄	C₂²×C₄₇	semidirect
100. C₂⁴	Q₈×C₄₇	semidirect
104. C₂×Q₈	C₂³×C₄₇	semidirect
115. C₄⋊C₄	C₂³×C₄₇	semidirect
116. C₄⋊C₄	C₂³×C₄₇	semidirect
123. C₂³	C₂×Q₈×C₄₇	semidirect
140. C₂²	C₂²⋊Q₈×C₄₇	direct (N × H)
143. C₂²	C₂²×Q₈×C₄₇	semidirect
151. C₂	C₂×C₂²⋊Q₈×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
C₂³×Q₈×C₄₇	2
C₂⁴×C₄×C₄₇	2
C₂×C₂²⋊Q₈×C₄₇	2
C₂²×C₄⋊C₄×C₄₇	2
C₂²×C₂²⋊C₄×C₄₇	2
C₂²×C₂²⋊Q₈	47

Subgroups — C22×C22⋊Q8×C47 #20241043

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊Q₈×C₄₇ #20241043