Subgroups — C₄₇×#1803 #20239413

Order 6016.

308 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#103×C₄₇	C₂	1	2	yes	maximal
4	C₄₇×#384	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#340	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
8	C₄×Q₈×C₄₇	—	2	2	no	—
9	C₄×Q₈×C₄₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₄₇	—	2	1	no	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₄₇	—	2	2	no	—
18	#103×C₄₇	—	2	4	no	—
19	C₂×Q₈×C₄₇	—	2	1	no	—
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₄₇	D₄	1	2	yes	—
24	C₂×Q₈×C₄₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	—	2	2	no	—
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₄⋊C₄×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄⋊C₄×C₄₇	—	4	2	no	—
39	C₄⋊C₄×C₄₇	—	4	2	no	—
40	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₄²×C₄₇	—	2	4	no	—
42	C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₄²×C₄₇	D₄	1	2	yes	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₄₇	—	2	2	no	—
46	Q₈×C₄₇	—	2	2	no	—
47	Q₈×C₄₇	—	2	2	no	—
48	Q₈×C₄₇	C₄⋊C₄	1	4	yes	complemented
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
63	C₂×C₄×C₄₇	Q₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
70	C₈×C₄₇	—	4	2	no	—
71	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄₇	#121	1	1	yes	characteristic
73	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×C₄₇	#137	1	1	yes	characteristic
78	C₄×C₄₇	—	2	4	no	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	4	2	no	—
81	C₄×C₄₇	—	8	2	no	—
82	C₄×C₄₇	#103	1	4	yes	complemented
83	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₄₇	#117	1	1	yes	characteristic
85	C₄×C₄₇	#121	1	1	yes	characteristic
86	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
87	#1803	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂×C₄₇	#380	1	1	yes	characteristic
89	C₂×C₄₇	C₂×#103	1	1	yes	characteristic
90	C₂×C₄₇	#417	1	1	yes	characteristic
91	C₂×C₄₇	#460	1	1	yes	characteristic
92	C₂×C₄₇	#462	1	1	yes	characteristic
93	C₂×C₄₇	#475	1	1	yes	characteristic
94	C₂×C₄₇	#477	1	1	yes	characteristic
95	C₂×C₄×Q₈	C₂×C₄₇	1	1	yes	characteristic
96	C₂×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic
97	C₂×#103	C₂×C₄₇	1	2	yes	—
98	#384	C₂×C₄₇	1	1	yes	characteristic
99	#340	C₂×C₄₇	1	2	yes	—
100	C₄₇	#1803	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×Q₈	C₂²×C₄₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
103	C₄×Q₈	—	2	2	no	—
104	C₄×Q₈	C₄×C₄₇	1	4	yes	complemented
105	C₂×C₄⋊C₄	—	2	1	no	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	—	2	1	no	—
108	C₂×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	#103	—	2	4	no	—
114	C₂×Q₈	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	1	no	—
117	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
118	C₂×Q₈	D₄×C₄₇	1	2	yes	—
119	C₂×Q₈	Q₈×C₄₇	1	2	yes	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	1	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
124	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
125	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
136	C₄²	—	2	4	no	—
137	C₄²	C₂×C₄×C₄₇	1	2	yes	—
138	C₄²	D₄×C₄₇	1	2	yes	—
139	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	2	2	no	—
143	Q₈	C₄⋊C₄×C₄₇	1	4	yes	complemented
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
156	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
157	C₂×C₄	SD₁₆×C₄₇	1	2	yes	—
158	C₂×C₄	Q₁₆×C₄₇	1	2	yes	—
159	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
160	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
161	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
162	C₂×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
163	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
164	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
167	C₂²	#121×C₄₇	1	1	yes	characteristic
168	C₂²	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
169	C₂²	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
170	C₂²	C₂×Q₁₆×C₄₇	1	1	yes	characteristic
171	C₂²	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
172	C₂²	#137×C₄₇	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	8	2	no	—
177	C₄	#103×C₄₇	1	4	yes	complemented
178	C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
179	C₄	#117×C₄₇	1	1	yes	characteristic
180	C₄	#121×C₄₇	1	1	yes	characteristic
181	C₄	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
182	C₂	C₄₇×#380	1	1	yes	characteristic
183	C₂	C₂×#103×C₄₇	1	1	yes	characteristic
184	C₂	C₄₇×#417	1	1	yes	characteristic
185	C₂	C₄₇×#460	1	1	yes	characteristic
186	C₂	C₄₇×#462	1	1	yes	characteristic
187	C₂	C₄₇×#475	1	1	yes	characteristic
188	C₂	C₄₇×#477	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄×Q₈×C₄₇	C₄	semidirect
48. Q₈×C₄₇	C₄⋊C₄	semidirect
82. C₄×C₄₇	#103	semidirect
87. #1803	C₄₇	direct (N × H)
100. C₄₇	#1803	direct (N × H)
104. C₄×Q₈	C₄×C₄₇	semidirect
143. Q₈	C₄⋊C₄×C₄₇	semidirect
177. C₄	#103×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₄₇	2
C₂×C₄⋊C₈×C₄₇	2
C₂×#103×C₄₇	2
C₄₇×#384	2
C₄₇×#340	2
#1803	47

Subgroups — C47×#1803 #20239413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1803 #20239413