Subgroups — C₄₇×#3214 #20240824

Order 6016.

434 conjugacy classes of subgroups.

The subgroup lattice has 236 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, complemented
2	C₄₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#496	C₂	1	2	yes	maximal, complemented
4	C₄₇×#489	C₂	1	2	yes	maximal, complemented
5	C₄₇×#461	C₂	1	2	yes	maximal, complemented
6	C₄₇×#459	C₂	1	2	yes	maximal, complemented
7	C₄₇×#449	C₂	1	4	yes	maximal, complemented
8	C₄₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₄₇	—	2	1	no	—
10	C₂×#41×C₄₇	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₄₇	C₂²	1	4	yes	—
12	C₂×D₈×C₄₇	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	#127×C₄₇	C₂²	1	1	yes	characteristic
16	#127×C₄₇	C₂²	1	1	yes	characteristic
17	#124×C₄₇	—	2	1	no	—
18	#124×C₄₇	C₂²	1	2	yes	—
19	#121×C₄₇	—	2	1	no	—
20	#121×C₄₇	—	2	1	no	—
21	#121×C₄₇	—	2	1	no	—
22	#121×C₄₇	C₂²	1	2	yes	—
23	C₄×D₄×C₄₇	—	2	1	no	—
24	C₄×D₄×C₄₇	—	2	1	no	—
25	C₄×D₄×C₄₇	—	2	1	no	—
26	C₄×D₄×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
29	#103×C₄₇	C₂²	1	4	yes	—
30	#102×C₄₇	C₂²	1	4	yes	—
31	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
32	#97×C₄₇	C₂²	1	2	yes	—
33	#41×C₄₇	—	4	4	no	—
34	#41×C₄₇	—	4	4	no	—
35	C₂×Q₈×C₄₇	—	2	1	no	—
36	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
37	C₂×D₄×C₄₇	—	2	2	no	—
38	C₂×D₄×C₄₇	—	2	2	no	—
39	C₂×D₄×C₄₇	—	2	1	no	—
40	C₂×D₄×C₄₇	—	2	1	no	—
41	C₂×D₄×C₄₇	—	2	1	no	—
42	C₂×D₄×C₄₇	—	2	1	no	—
43	C₂×D₄×C₄₇	C₂³	1	2	yes	—
44	C₂×D₄×C₄₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₇	—	2	2	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
50	SD₁₆×C₄₇	—	4	4	no	—
51	D₈×C₄₇	—	4	4	no	—
52	C₂×C₈×C₄₇	C₂³	1	4	yes	—
53	C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₄₇	—	2	2	no	—
58	C₂²⋊C₄×C₄₇	—	2	2	no	—
59	C₂²⋊C₄×C₄₇	—	2	2	no	—
60	C₂²⋊C₄×C₄₇	—	2	2	no	—
61	C₄²×C₄₇	D₄	1	1	yes	characteristic
62	C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄₇	—	2	2	no	—
66	C₂³×C₄₇	—	2	2	no	—
67	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
68	Q₈×C₄₇	—	2	4	no	—
69	Q₈×C₄₇	—	4	2	no	—
70	D₄×C₄₇	—	2	2	no	—
71	D₄×C₄₇	—	2	4	no	—
72	D₄×C₄₇	—	2	2	no	—
73	D₄×C₄₇	—	4	2	no	—
74	D₄×C₄₇	—	4	2	no	—
75	D₄×C₄₇	—	4	2	no	—
76	D₄×C₄₇	—	4	4	no	—
77	D₄×C₄₇	—	4	4	no	—
78	D₄×C₄₇	—	4	2	no	—
79	C₂×C₄×C₄₇	—	2	2	no	—
80	C₂×C₄×C₄₇	—	2	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	4	1	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	4	4	no	—
86	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₄₇	—	2	4	no	—
93	C₂²×C₄₇	—	2	1	no	—
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	2	1	no	—
96	C₂²×C₄₇	—	4	2	no	—
97	C₂²×C₄₇	—	4	2	no	—
98	C₂²×C₄₇	—	4	2	no	—
99	C₂²×C₄₇	—	4	2	no	—
100	C₂²×C₄₇	—	4	2	no	—
101	C₂²×C₄₇	—	4	2	no	—
102	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	—	2	1	no	—
105	C₄×C₄₇	—	2	1	no	—
106	C₄×C₄₇	—	2	1	no	—
107	C₄×C₄₇	—	4	2	no	—
108	C₄×C₄₇	—	4	2	no	—
109	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
110	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
111	#3214	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₄₇	—	4	1	no	—
113	C₂×C₄₇	—	8	2	no	—
114	C₂×C₄₇	—	8	2	no	—
115	C₂×C₄₇	#535	1	1	yes	characteristic
116	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄₇	#575	1	1	yes	characteristic
118	#532	C₂×C₄₇	1	1	yes	characteristic, complemented
119	#532	C₂×C₄₇	1	1	yes	characteristic, complemented
120	#496	C₂×C₄₇	1	2	yes	complemented
121	#489	C₂×C₄₇	1	2	yes	complemented
122	#461	C₂×C₄₇	1	2	yes	complemented
123	#459	C₂×C₄₇	1	2	yes	complemented
124	#449	C₂×C₄₇	1	4	yes	complemented
125	#432	C₂×C₄₇	1	1	yes	characteristic, complemented
126	C₄₇	#3214	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#41	—	2	1	no	—
128	C₂×#41	C₂²×C₄₇	1	2	yes	—
129	C₂×SD₁₆	C₂²×C₄₇	1	4	yes	—
130	C₂×D₈	C₂²×C₄₇	1	4	yes	—
131	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
133	#127	C₂²×C₄₇	1	1	yes	characteristic
134	#127	C₂²×C₄₇	1	1	yes	characteristic
135	#124	—	2	1	no	—
136	#124	C₂²×C₄₇	1	2	yes	—
137	#121	—	2	1	no	—
138	#121	—	2	1	no	—
139	#121	—	2	1	no	—
140	#121	C₂²×C₄₇	1	2	yes	—
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	2	1	no	—
144	C₄×D₄	C₂²×C₄₇	1	2	yes	—
145	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
147	#103	C₂²×C₄₇	1	4	yes	—
148	#102	C₂²×C₄₇	1	4	yes	—
149	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
150	#97	C₂²×C₄₇	1	2	yes	—
151	#41	—	4	4	no	—
152	#41	—	4	4	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	C₂³×C₄₇	1	2	yes	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	1	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	C₂³×C₄₇	1	2	yes	—
162	C₂×D₄	C₂³×C₄₇	1	2	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
166	C₂²×C₄	D₄×C₄₇	1	1	yes	characteristic
167	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
168	SD₁₆	—	4	4	no	—
169	D₈	—	4	4	no	—
170	C₂×C₈	C₂³×C₄₇	1	4	yes	—
171	C₄⋊C₄	—	2	1	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	D₄×C₄₇	1	1	yes	characteristic
180	C₄²	D₄×C₄₇	1	1	yes	characteristic
181	C₄²	C₂³×C₄₇	1	1	yes	characteristic
182	C₄²	C₂³×C₄₇	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
186	Q₈	—	2	4	no	—
187	Q₈	—	4	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	4	2	no	—
192	D₄	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	4	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
228	C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
229	C₂	—	4	1	no	—
230	C₂	—	8	2	no	—
231	C₂	—	8	2	no	—
232	C₂	C₄₇×#535	1	1	yes	characteristic
233	C₂	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
234	C₂	C₄₇×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
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
1. C₄₇×#532	C₂	semidirect
2. C₄₇×#532	C₂	semidirect
3. C₄₇×#496	C₂	semidirect
4. C₄₇×#489	C₂	semidirect
5. C₄₇×#461	C₂	semidirect
6. C₄₇×#459	C₂	semidirect
7. C₄₇×#449	C₂	semidirect
8. C₄₇×#432	C₂	semidirect
111. #3214	C₄₇	direct (N × H)
118. #532	C₂×C₄₇	semidirect
119. #532	C₂×C₄₇	semidirect
120. #496	C₂×C₄₇	semidirect
121. #489	C₂×C₄₇	semidirect
122. #461	C₂×C₄₇	semidirect
123. #459	C₂×C₄₇	semidirect
124. #449	C₂×C₄₇	semidirect
125. #432	C₂×C₄₇	semidirect
126. C₄₇	#3214	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#532	2
C₄₇×#496	2
C₄₇×#489	2
C₄₇×#461	2
C₄₇×#459	2
C₄₇×#449	2
C₄₇×#432	2
#3214	47

Subgroups — C47×#3214 #20240824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3214 #20240824