Subgroups — C₄₇×#33474 #26073172

Order 12032.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2960	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2183	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₆×C₄₇	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2151	C₂	1	2	yes	maximal
5	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic
6	C₄₇×#2108	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#533	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#446	C₄	1	4	yes	—
13	C₈×Q₈×C₄₇	C₂²	1	4	yes	—
14	C₄₇×#424	C₄	1	2	yes	—
15	C₄₇×#423	C₄	1	1	yes	characteristic
16	C₄₇×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
19	C₄₇×C₄⋊C₁₆	C₂²	1	8	yes	—
20	C₄₇×#346	C₂²	1	4	yes	—
21	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
22	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
23	C₄×C₁₆×C₄₇	C₂²	1	4	yes	—
24	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	C₈	1	2	yes	—
28	#125×C₄₇	C₈	1	2	yes	—
29	C₂²⋊Q₈×C₄₇	C₈	1	4	yes	—
30	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
31	#117×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₄₇	—	2	4	no	—
34	C₂×C₁₆×C₄₇	—	2	2	no	—
35	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
36	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
37	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
38	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
42	#97×C₄₇	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	C₂³	1	2	yes	—
45	C₄×C₈×C₄₇	C₂³	1	4	yes	—
46	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	M₁₆×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	—	2	2	no	—
51	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₄₇	#41	1	2	yes	—
56	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
58	C₄⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
59	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
60	C₂²⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
61	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
65	C₁₆×C₄₇	—	2	2	no	—
66	C₁₆×C₄₇	—	2	4	no	—
67	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
68	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
70	Q₈×C₄₇	—	2	8	no	—
71	C₂×C₄×C₄₇	—	2	4	no	—
72	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
73	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₄₇	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₄₇	#131	1	2	yes	—
78	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
79	C₈×C₄₇	—	2	2	no	—
80	C₈×C₄₇	—	2	4	no	—
81	C₈×C₄₇	C₄×Q₈	1	2	yes	—
82	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
83	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
84	#33474	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₄₇	—	2	2	no	—
86	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
87	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
88	C₄×C₄₇	—	2	2	no	—
89	C₄×C₄₇	—	2	8	no	—
90	C₄×C₄₇	C₈×Q₈	1	2	yes	—
91	C₄×C₄₇	#504	1	4	yes	—
92	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
94	#2960	C₂×C₄₇	1	1	yes	characteristic
95	#2183	C₂×C₄₇	1	4	yes	complemented
96	Q₈×C₁₆	C₂×C₄₇	1	4	yes	complemented
97	#2151	C₂×C₄₇	1	2	yes	—
98	#2150	C₂×C₄₇	1	1	yes	characteristic
99	#2108	C₂×C₄₇	1	2	yes	—
100	C₂×C₄×C₁₆	C₂×C₄₇	1	1	yes	characteristic
101	C₂×C₄₇	—	2	2	no	—
102	C₂×C₄₇	C₂×C₈×Q₈	1	1	yes	characteristic
103	C₂×C₄₇	C₂×#504	1	2	yes	—
104	#533	C₄×C₄₇	1	1	yes	characteristic
105	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
106	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
107	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
108	#446	C₄×C₄₇	1	4	yes	—
109	C₈×Q₈	C₂²×C₄₇	1	4	yes	—
110	#424	C₄×C₄₇	1	2	yes	—
111	#423	C₄×C₄₇	1	1	yes	characteristic
112	#405	C₂²×C₄₇	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
114	C₄⋊C₁₆	C₂²×C₄₇	1	4	yes	—
115	C₄⋊C₁₆	C₂²×C₄₇	1	8	yes	—
116	#346	C₂²×C₄₇	1	4	yes	—
117	C₄×C₁₆	C₂²×C₄₇	1	2	yes	—
118	C₄×C₁₆	C₂²×C₄₇	1	2	yes	—
119	C₄×C₁₆	C₂²×C₄₇	1	4	yes	—
120	C₄₇	#33474	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
122	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
124	C₄⋊Q₈	C₈×C₄₇	1	2	yes	—
125	#125	C₈×C₄₇	1	2	yes	—
126	C₂²⋊Q₈	C₈×C₄₇	1	4	yes	—
127	C₄×Q₈	C₂×C₄×C₄₇	1	4	yes	—
128	#117	C₂×C₄×C₄₇	1	2	yes	—
129	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
130	M₃₂	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	Q₈×C₄₇	1	4	yes	complemented
133	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
134	C₂×C₁₆	C₂³×C₄₇	1	4	yes	—
135	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
137	C₄⋊C₈	C₂×C₄×C₄₇	1	4	yes	—
138	C₄⋊C₈	C₂×C₄×C₄₇	1	8	yes	—
139	#97	C₂×C₄×C₄₇	1	4	yes	—
140	C₄×C₈	C₂×C₄×C₄₇	1	2	yes	—
141	C₄×C₈	C₂³×C₄₇	1	2	yes	—
142	C₄×C₈	C₂³×C₄₇	1	4	yes	—
143	C₂×Q₈	C₂×C₈×C₄₇	1	4	yes	—
144	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂²×C₄×C₄₇	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
150	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₄₇	1	2	yes	—
152	C₂×C₈	#41×C₄₇	1	2	yes	—
153	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₈×C₄₇	1	4	yes	—
155	C₄⋊C₄	C₂×C₈×C₄₇	1	8	yes	—
156	C₄⋊C₄	C₂×C₈×C₄₇	1	4	yes	—
157	C₂²⋊C₄	C₂×C₈×C₄₇	1	4	yes	—
158	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
159	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
160	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
161	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
162	C₁₆	—	2	2	no	—
163	C₁₆	—	2	4	no	—
164	C₁₆	C₂×Q₈×C₄₇	1	2	yes	—
165	C₁₆	C₂×Q₈×C₄₇	1	2	yes	—
166	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
167	Q₈	—	2	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
170	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
171	C₂×C₄	C₂²×C₈×C₄₇	1	8	yes	—
172	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
173	C₂×C₄	C₂²×C₈×C₄₇	1	1	yes	characteristic
174	C₂×C₄	#131×C₄₇	1	2	yes	—
175	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	4	no	—
178	C₈	C₄×Q₈×C₄₇	1	2	yes	—
179	C₈	C₂²×Q₈×C₄₇	1	1	yes	characteristic
180	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Q₈×C₄₇	1	2	yes	—
183	C₂²	C₂³×C₈×C₄₇	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	C₈×Q₈×C₄₇	1	2	yes	—
187	C₄	C₄₇×#504	1	4	yes	—
188	C₄	C₂×C₄×Q₈×C₄₇	1	1	yes	characteristic
189	C₄	C₂×#131×C₄₇	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Q₈×C₄₇	1	1	yes	characteristic
192	C₂	C₂×C₄₇×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₄₇	C₂³
commutator	C₂²	C₂³×C₈×C₄₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
socle	C₂²×C₄₇	C₂³×C₈
prime radical	C₂³×C₄₇	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2183	C₂	semidirect
3. Q₈×C₁₆×C₄₇	C₂	semidirect
35. C₂×C₁₆×C₄₇	Q₈	semidirect
84. #33474	C₄₇	direct (N × H)
95. #2183	C₂×C₄₇	semidirect
96. Q₈×C₁₆	C₂×C₄₇	semidirect
120. C₄₇	#33474	direct (N × H)
132. C₂×C₁₆	Q₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#2960	2
C₄₇×#2183	2
Q₈×C₁₆×C₄₇	2
C₄₇×#2151	2
C₄₇×#2150	2
C₄₇×#2108	2
C₂×C₄×C₁₆×C₄₇	2
#33474	47

Subgroups — C47×#33474 #26073172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33474 #26073172