Subgroups — C₂²×#15688898 #26368903

Order 12032.

942 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688898	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₄	1	8	yes	complemented
8	C₂²×C₄₇⋊C₁₆	C₂²	1	6	yes	—
9	C₂×C₈×Dic₄₇	C₄	1	6	yes	—
10	C₂×C₈×Dic₄₇	C₂²	1	6	yes	—
11	#15688898	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₄	1	8	yes	complemented
14	C₂²×C₁₆×C₄₇	C₂²	1	6	yes	—
15	C₂³×Dic₄₇	C₈	1	2	yes	—
16	C₂²×C₄₇⋊C₈	C₈	1	8	yes	—
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	6	yes	—
18	C₂×C₄×Dic₄₇	C₈	1	6	yes	—
19	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	24	yes	complemented
21	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
22	C₈×Dic₄₇	C₂×C₄	1	12	yes	—
23	C₈×Dic₄₇	C₂³	1	4	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₄₇	C₂³	1	3	yes	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	24	yes	complemented
30	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
31	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
33	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₄₇	—	2	8	no	—
37	C₂²×Dic₄₇	C₂×C₈	1	12	yes	—
38	C₂×C₈×C₄₇	C₄²	1	12	yes	—
39	C₂×C₈×C₄₇	C₂²×C₄	1	3	yes	—
40	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
41	C₂×C₈×C₄₇	C₂²×C₄	1	6	yes	—
42	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₄×Dic₄₇	C₂×C₈	1	12	yes	—
44	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄₇⋊C₈	C₂×C₈	1	24	yes	—
46	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
47	C₁₆×C₄₇	C₂²×C₄	1	8	yes	complemented
48	C₄₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
49	C₂³×C₄₇	C₄×C₈	1	4	yes	—
50	C₂³×C₄₇	M₃₂	1	8	yes	—
51	C₂³×C₄₇	C₂²×C₈	1	3	yes	—
52	C₂×C₄×C₄₇	C₄×C₈	1	12	yes	—
53	C₂×C₄×C₄₇	C₂×C₄²	1	6	yes	—
54	C₂×C₄×C₄₇	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₄₇	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
57	C₂×Dic₄₇	—	2	24	no	—
58	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
59	C₈×C₄₇	C₂×C₄²	1	6	yes	—
60	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
61	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
62	C₄₇⋊C₈	C₂²×C₈	1	8	yes	—
63	C₂²×#346	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#346	1	16	yes	complemented
65	C₂²×C₄₇	C₂×C₄×C₈	1	6	yes	—
66	C₂²×C₄₇	C₂×M₃₂	1	12	yes	—
67	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₄₇	C₂×C₄×C₈	1	6	yes	—
69	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
70	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
71	Dic₄₇	—	2	8	no	—
72	C₂³×C₁₆	—	47	1	no	—
73	C₂³×C₁₆	D₄₇	1	1	yes	characteristic
74	C₂²×C₄×C₈	—	47	1	no	—
75	C₂×#346	—	47	12	no	—
76	C₂×C₄₇	C₂×#346	1	12	yes	complemented
77	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
78	C₂×C₄₇	C₂²×M₃₂	1	2	yes	—
79	C₂³×C₈	—	47	1	no	—
80	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
81	C₂²×C₄²	—	47	1	no	—
82	C₂²×C₁₆	—	47	8	no	—
83	C₂²×C₁₆	—	47	6	no	—
84	C₂²×C₁₆	Dic₄₇	1	8	yes	complemented
85	C₂²×C₁₆	C₂×D₄₇	1	6	yes	—
86	C₂×C₄×C₈	—	47	6	no	—
87	C₂×C₄×C₈	—	47	6	no	—
88	#346	—	47	16	no	—
89	C₄₇	C₂²×#346	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	—	47	2	no	—
91	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
92	C₂²×C₈	—	47	8	no	—
93	C₂²×C₈	—	47	6	no	—
94	C₂²×C₈	C₄×D₄₇	1	3	yes	—
95	C₂²×C₈	C₄×D₄₇	1	4	yes	—
96	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
97	C₂²×C₈	C₂²×D₄₇	1	3	yes	—
98	C₂×C₄²	—	47	6	no	—
99	C₂×C₄²	—	47	6	no	—
100	C₂×C₁₆	—	47	24	no	—
101	C₂×C₁₆	—	47	4	no	—
102	C₂×C₁₆	C₂×Dic₄₇	1	24	yes	complemented
103	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
104	C₄×C₈	—	47	12	no	—
105	C₄×C₈	—	47	4	no	—
106	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
107	C₂²×C₄	—	47	12	no	—
108	C₂²×C₄	—	94	8	no	—
109	C₂²×C₄	C₈×D₄₇	1	3	yes	—
110	C₂²×C₄	C₈×D₄₇	1	4	yes	—
111	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
112	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
113	C₂×C₈	—	47	24	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	C₄×Dic₄₇	1	12	yes	—
116	C₂×C₈	C₂×C₄×D₄₇	1	3	yes	—
117	C₂×C₈	C₂×C₄×D₄₇	1	6	yes	—
118	C₂×C₈	C₂²×Dic₄₇	1	6	yes	—
119	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
120	C₄²	—	47	12	no	—
121	C₄²	—	47	4	no	—
122	C₁₆	—	47	8	no	—
123	C₁₆	C₂²×Dic₄₇	1	8	yes	complemented
124	C₂³	C₈×Dic₄₇	1	4	yes	—
125	C₂³	#13093963	1	8	yes	—
126	C₂³	C₂×C₈×D₄₇	1	3	yes	—
127	C₂×C₄	—	47	8	no	—
128	C₂×C₄	—	94	24	no	—
129	C₂×C₄	C₈×Dic₄₇	1	12	yes	—
130	C₂×C₄	C₂×C₄×Dic₄₇	1	6	yes	—
131	C₂×C₄	C₂×C₈×D₄₇	1	3	yes	—
132	C₂×C₄	C₂×C₈×D₄₇	1	6	yes	—
133	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
134	C₈	—	47	8	no	—
135	C₈	C₂×C₄×Dic₄₇	1	6	yes	—
136	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
137	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
138	C₂²	#15688898	1	16	yes	direct product factor
139	C₂²	C₂×C₈×Dic₄₇	1	6	yes	—
140	C₂²	C₂×#13093963	1	12	yes	—
141	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
142	C₄	—	94	8	no	—
143	C₄	C₂×C₈×Dic₄₇	1	6	yes	—
144	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
145	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
146	C₂	C₂×#15688898	1	12	yes	direct product factor
147	C₂	C₂²×C₈×Dic₄₇	1	1	yes	characteristic
148	C₂	C₂²×#13093963	1	2	yes	—

	subgroup	G/H
center	C₂³×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂³×C₁₆×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×C₁₆	D₄₇
hypocenter	C₄₇	C₂²×#346
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#15688898	C₂	direct (N × H)
7. C₂²×C₄₇⋊C₁₆	C₄	semidirect
11. #15688898	C₂²	direct (N × H)
13. C₂²×C₁₆×C₄₇	C₄	semidirect
20. C₂×C₄₇⋊C₁₆	C₂×C₄	semidirect
29. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
47. C₁₆×C₄₇	C₂²×C₄	semidirect
48. C₄₇⋊C₁₆	C₂²×C₄	semidirect
64. C₂²×C₄₇	#346	semidirect
76. C₂×C₄₇	C₂×#346	semidirect
84. C₂²×C₁₆	Dic₄₇	semidirect
89. C₄₇	C₂²×#346	semidirect
102. C₂×C₁₆	C₂×Dic₄₇	semidirect
123. C₁₆	C₂²×Dic₄₇	semidirect
138. C₂²	#15688898	direct (N × H)
146. C₂	C₂×#15688898	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×C₄₇⋊C₁₆	2
C₂²×C₈×Dic₄₇	2
C₂×#15688898	2
C₂³×C₁₆×C₄₇	2
C₂²×#346	47

Subgroups — C22×#15688898 #26368903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15688898 #26368903