Subgroups — C₄₇×#3206 #20240816

Order 6016.

428 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#492	C₂	1	2	yes	maximal, complemented
4	C₄₇×#487	C₂	1	2	yes	maximal, complemented
5	C₄₇×#461	C₂	1	4	yes	maximal, complemented
6	C₄₇×#448	C₂	1	4	yes	maximal, complemented
7	C₄₇×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₄₇	—	2	1	no	—
9	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
10	C₂×SD₁₆×C₄₇	C₂²	1	8	yes	—
11	C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	—	2	1	no	—
13	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	#127×C₄₇	C₂²	1	1	yes	characteristic
17	#124×C₄₇	—	2	1	no	—
18	#121×C₄₇	—	2	1	no	—
19	#121×C₄₇	C₂²	1	2	yes	—
20	C₄×D₄×C₄₇	—	2	1	no	—
21	C₄×D₄×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
23	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
26	#103×C₄₇	C₂²	1	8	yes	—
27	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
28	C₄×C₈×C₄₇	C₂²	1	2	yes	—
29	#41×C₄₇	—	4	4	no	—
30	C₂×Q₈×C₄₇	—	2	1	no	—
31	C₂×Q₈×C₄₇	—	2	2	no	—
32	C₂×Q₈×C₄₇	—	4	4	no	—
33	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
34	C₂×D₄×C₄₇	—	2	2	no	—
35	C₂×D₄×C₄₇	—	2	1	no	—
36	C₂×D₄×C₄₇	—	2	1	no	—
37	C₂×D₄×C₄₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₇	—	2	2	no	—
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	SD₁₆×C₄₇	—	4	8	no	—
44	C₂×C₈×C₄₇	C₂³	1	4	yes	—
45	C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₄⋊C₄×C₄₇	—	2	1	no	—
50	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₄₇	—	2	2	no	—
52	C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₄²×C₄₇	D₄	1	1	yes	characteristic
54	C₄²×C₄₇	D₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄₇	—	2	2	no	—
58	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
59	Q₈×C₄₇	—	2	8	no	—
60	Q₈×C₄₇	—	4	4	no	—
61	Q₈×C₄₇	—	4	2	no	—
62	D₄×C₄₇	—	2	4	no	—
63	D₄×C₄₇	—	4	4	no	—
64	D₄×C₄₇	—	4	2	no	—
65	D₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	—	2	2	no	—
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	4	no	—
70	C₂×C₄×C₄₇	—	4	4	no	—
71	C₂×C₄×C₄₇	—	4	4	no	—
72	C₂×C₄×C₄₇	SD₁₆	1	4	yes	complemented
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	C₈×C₄₇	—	2	4	no	—
80	C₂²×C₄₇	—	2	1	no	—
81	C₂²×C₄₇	—	2	1	no	—
82	C₂²×C₄₇	—	4	2	no	—
83	C₂²×C₄₇	—	4	2	no	—
84	C₂²×C₄₇	—	4	2	no	—
85	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
86	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	—	2	1	no	—
89	C₄×C₄₇	—	4	4	no	—
90	C₄×C₄₇	—	4	2	no	—
91	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
92	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
93	C₄×C₄₇	#137	1	2	yes	—
94	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
95	#3206	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂×C₄₇	—	2	2	no	—
97	C₂×C₄₇	—	8	2	no	—
98	C₂×C₄₇	#535	1	1	yes	characteristic
99	C₂×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
100	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
101	#532	C₂×C₄₇	1	1	yes	characteristic
102	C₂×C₄⋊Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
103	#492	C₂×C₄₇	1	2	yes	complemented
104	#487	C₂×C₄₇	1	2	yes	complemented
105	#461	C₂×C₄₇	1	4	yes	complemented
106	#448	C₂×C₄₇	1	4	yes	complemented
107	#431	C₂×C₄₇	1	1	yes	characteristic, complemented
108	C₄₇	#3206	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×#41	—	2	1	no	—
110	C₂²×Q₈	C₂²×C₄₇	1	2	yes	—
111	C₂×SD₁₆	C₂²×C₄₇	1	8	yes	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
115	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
116	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
117	#127	C₂²×C₄₇	1	1	yes	characteristic
118	#124	—	2	1	no	—
119	#121	—	2	1	no	—
120	#121	C₂²×C₄₇	1	2	yes	—
121	C₄×D₄	—	2	1	no	—
122	C₄×D₄	C₂²×C₄₇	1	2	yes	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
126	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
127	#103	C₂²×C₄₇	1	8	yes	—
128	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
129	C₄×C₈	C₂²×C₄₇	1	2	yes	—
130	#41	—	4	4	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	C₂³×C₄₇	1	4	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	C₂³×C₄₇	1	2	yes	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
142	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
144	SD₁₆	—	4	8	no	—
145	C₂×C₈	C₂³×C₄₇	1	4	yes	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₄²	D₄×C₄₇	1	1	yes	characteristic
155	C₄²	D₄×C₄₇	1	1	yes	characteristic
156	C₄²	C₂³×C₄₇	1	1	yes	characteristic
157	C₄²	C₂³×C₄₇	1	1	yes	characteristic
158	C₂³	—	2	2	no	—
159	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
160	Q₈	—	2	8	no	—
161	Q₈	—	4	4	no	—
162	Q₈	—	4	2	no	—
163	D₄	—	2	4	no	—
164	D₄	—	4	4	no	—
165	D₄	—	4	2	no	—
166	D₄	—	4	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	SD₁₆×C₄₇	1	4	yes	complemented
174	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
176	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
177	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
178	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
179	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
180	C₈	—	2	4	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×SD₁₆×C₄₇	1	2	yes	—
187	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	4	no	—
191	C₄	—	4	2	no	—
192	C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
193	C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
194	C₄	#137×C₄₇	1	2	yes	—
195	C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
196	C₂	—	2	2	no	—
197	C₂	—	8	2	no	—
198	C₂	C₄₇×#535	1	1	yes	characteristic
199	C₂	C₂²×SD₁₆×C₄₇	1	1	yes	characteristic
200	C₂	C₂×#137×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Q₈×C₄₇	C₂	semidirect
3. C₄₇×#492	C₂	semidirect
4. C₄₇×#487	C₂	semidirect
5. C₄₇×#461	C₂	semidirect
6. C₄₇×#448	C₂	semidirect
7. C₄₇×#431	C₂	semidirect
72. C₂×C₄×C₄₇	SD₁₆	semidirect
95. #3206	C₄₇	direct (N × H)
102. C₂×C₄⋊Q₈	C₂×C₄₇	semidirect
103. #492	C₂×C₄₇	semidirect
104. #487	C₂×C₄₇	semidirect
105. #461	C₂×C₄₇	semidirect
106. #448	C₂×C₄₇	semidirect
107. #431	C₂×C₄₇	semidirect
108. C₄₇	#3206	direct (N × H)
173. C₂×C₄	SD₁₆×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#532	2
C₂×C₄⋊Q₈×C₄₇	2
C₄₇×#492	2
C₄₇×#487	2
C₄₇×#461	2
C₄₇×#448	2
C₄₇×#431	2
#3206	47

Subgroups — C47×#3206 #20240816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3206 #20240816