Subgroups — C₄₇×#13272 #26052970

Order 12032.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#498	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#364	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1772	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1386	C₂	1	2	yes	maximal
5	C₄₇×#1383	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₄₇	—	2	1	no	—
7	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
8	C₄₇×#498	—	2	2	no	—
9	C₄₇×#498	C₄	1	4	yes	complemented
10	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#427	C₂²	1	1	yes	characteristic
12	C₂×#106×C₄₇	—	2	1	no	—
13	C₂×#106×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#364	—	2	2	no	—
17	C₄₇×#339	—	2	1	no	—
18	C₂²×Q₈×C₄₇	—	4	1	no	—
19	C₂×M₁₆×C₄₇	—	2	1	no	—
20	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₄⋊Q₈×C₄₇	—	2	4	no	—
23	C₄⋊Q₈×C₄₇	—	4	2	no	—
24	#117×C₄₇	—	2	1	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
27	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
28	M₃₂×C₄₇	—	2	2	no	—
29	M₃₂×C₄₇	—	2	2	no	—
30	C₂×C₁₆×C₄₇	—	2	2	no	—
31	#108×C₄₇	—	2	1	no	—
32	#108×C₄₇	Q₈	1	2	yes	complemented
33	#107×C₄₇	Q₈	1	2	yes	complemented
34	#106×C₄₇	—	2	4	no	—
35	#106×C₄₇	—	2	2	no	—
36	#106×C₄₇	—	2	1	no	—
37	#106×C₄₇	C₂×C₄	1	4	yes	—
38	#97×C₄₇	—	4	1	no	—
39	C₄×C₈×C₄₇	—	4	1	no	—
40	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₄₇	D₄	1	2	yes	—
42	C₂×Q₈×C₄₇	—	4	4	no	—
43	C₂×Q₈×C₄₇	—	8	4	no	—
44	C₂²×C₄×C₄₇	—	2	1	no	—
45	C₂²×C₄×C₄₇	—	4	2	no	—
46	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₄₇	—	2	2	no	—
48	M₁₆×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	—	4	1	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
59	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₄₇	—	2	4	no	—
61	C₄⋊C₄×C₄₇	—	2	2	no	—
62	C₄⋊C₄×C₄₇	—	4	4	no	—
63	C₄⋊C₄×C₄₇	—	4	2	no	—
64	C₂²⋊C₄×C₄₇	—	4	1	no	—
65	C₄²×C₄₇	—	2	1	no	—
66	C₄²×C₄₇	—	4	1	no	—
67	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₁₆×C₄₇	—	4	2	no	—
69	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Q₈×C₄₇	—	8	8	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	4	4	no	—
74	C₂×C₄×C₄₇	—	4	2	no	—
75	C₂×C₄×C₄₇	—	8	4	no	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₈×C₄₇	—	4	2	no	—
83	C₈×C₄₇	#103	1	2	yes	—
84	C₈×C₄₇	#103	1	2	yes	—
85	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₄₇	#124	1	1	yes	characteristic
88	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
89	#13272	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₄₇	—	2	2	no	—
91	C₂²×C₄₇	#418	1	1	yes	characteristic
92	C₂²×C₄₇	#476	1	1	yes	characteristic
93	C₂²×C₄₇	#477	1	1	yes	characteristic
94	C₄×C₄₇	—	2	4	no	—
95	C₄×C₄₇	—	8	4	no	—
96	C₄×C₄₇	—	8	2	no	—
97	C₄×C₄₇	#391	1	1	yes	characteristic
98	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
99	C₄×C₄₇	#476	1	1	yes	characteristic
100	C₄×C₄₇	#477	1	1	yes	characteristic
101	C₂×#498	C₂×C₄₇	1	1	yes	characteristic
102	C₂×#364	C₂×C₄₇	1	1	yes	characteristic
103	#1772	C₂×C₄₇	1	1	yes	characteristic
104	#1386	C₂×C₄₇	1	2	yes	—
105	#1383	C₂×C₄₇	1	2	yes	—
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	#1987	1	1	yes	characteristic
109	C₂×C₄₇	#2230	1	2	yes	—
110	C₂×C₄⋊Q₈	—	2	1	no	—
111	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
112	#498	—	2	2	no	—
113	#498	C₄×C₄₇	1	4	yes	complemented
114	C₂×#108	C₂²×C₄₇	1	1	yes	characteristic
115	#427	C₂²×C₄₇	1	1	yes	characteristic
116	C₂×#106	—	2	1	no	—
117	C₂×#106	C₂²×C₄₇	1	2	yes	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
120	#364	—	2	2	no	—
121	#339	—	2	1	no	—
122	C₄₇	#13272	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×Q₈	—	4	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂²×C₈	D₄×C₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	#117	—	2	1	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
133	M₃₂	—	2	2	no	—
134	M₃₂	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	#108	—	2	1	no	—
137	#108	Q₈×C₄₇	1	2	yes	complemented
138	#107	Q₈×C₄₇	1	2	yes	complemented
139	#106	—	2	4	no	—
140	#106	—	2	2	no	—
141	#106	—	2	1	no	—
142	#106	C₂×C₄×C₄₇	1	4	yes	—
143	#97	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	C₂×C₄×C₄₇	1	2	yes	—
146	C₄×C₈	D₄×C₄₇	1	2	yes	—
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	8	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	C₂²⋊C₄×C₄₇	1	2	yes	—
156	C₂×C₈	SD₁₆×C₄₇	1	1	yes	characteristic
157	C₂×C₈	SD₁₆×C₄₇	1	1	yes	characteristic
158	C₂×C₈	Q₁₆×C₄₇	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
161	C₂×C₈	C₂×Q₈×C₄₇	1	1	yes	characteristic
162	C₂×C₈	C₂×Q₈×C₄₇	1	1	yes	characteristic
163	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
164	C₂×C₈	#41×C₄₇	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₂²⋊C₄×C₄₇	1	2	yes	—
173	C₁₆	—	4	2	no	—
174	C₂³	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
182	C₂×C₄	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
183	C₂×C₄	C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₁₆×C₄₇	1	1	yes	characteristic
186	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	#103×C₄₇	1	2	yes	—
189	C₈	#103×C₄₇	1	2	yes	—
190	C₈	C₄×Q₈×C₄₇	1	1	yes	characteristic
191	C₈	C₄×Q₈×C₄₇	1	1	yes	characteristic
192	C₈	#124×C₄₇	1	1	yes	characteristic
193	C₈	C₄⋊Q₈×C₄₇	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	C₄₇×#418	1	1	yes	characteristic
196	C₂²	C₄₇×#476	1	1	yes	characteristic
197	C₂²	C₄₇×#477	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	—	8	2	no	—
201	C₄	C₄₇×#391	1	1	yes	characteristic
202	C₄	C₂×#103×C₄₇	1	1	yes	characteristic
203	C₄	C₄₇×#476	1	1	yes	characteristic
204	C₄	C₄₇×#477	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₄₇×#1987	1	1	yes	characteristic
208	C₂	C₄₇×#2230	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	#476
commutator	C₂×C₈	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₈	C₂³×C₄₇
prime radical	C₂³×C₄₇	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ C₂×C₄×C₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₄₇×#498	C₄	semidirect
32. #108×C₄₇	Q₈	semidirect
33. #107×C₄₇	Q₈	semidirect
89. #13272	C₄₇	direct (N × H)
113. #498	C₄×C₄₇	semidirect
122. C₄₇	#13272	direct (N × H)
137. #108	Q₈×C₄₇	semidirect
138. #107	Q₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#498	2
C₂×C₄₇×#364	2
C₄₇×#1772	2
C₄₇×#1386	2
C₄₇×#1383	2
#13272	47

Subgroups — C47×#13272 #26052970

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#13272 #26052970