Subgroups — C₄₇×#3328 #20240938

Order 6016.

374 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#556	C₂	1	1	yes	maximal, characteristic
2	D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#485	C₂	1	2	yes	maximal, complemented
5	C₄₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#470	C₂	1	2	yes	maximal, complemented
7	C₄₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#451	C₂	1	2	yes	maximal, complemented
9	C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#436	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#419	C₂	1	2	yes	maximal, complemented
12	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	—
15	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₄₇	—	2	1	no	—
17	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	#126×C₄₇	—	2	2	no	—
20	#124×C₄₇	—	2	1	no	—
21	C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
23	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
24	C₄×Q₈×C₄₇	—	2	1	no	—
25	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₄₇	—	2	1	no	—
28	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
29	#117×C₄₇	C₂²	1	2	yes	—
30	#107×C₄₇	C₂²	1	1	yes	characteristic
31	#106×C₄₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	#103×C₄₇	C₂²	1	2	yes	—
34	#103×C₄₇	C₂²	1	2	yes	—
35	#103×C₄₇	C₂²	1	2	yes	—
36	#103×C₄₇	C₂²	1	1	yes	characteristic
37	#103×C₄₇	C₂²	1	2	yes	—
38	#103×C₄₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	—	2	1	no	—
42	C₂×Q₈×C₄₇	—	2	2	no	—
43	C₂×Q₈×C₄₇	—	2	4	no	—
44	C₂×Q₈×C₄₇	—	4	2	no	—
45	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
46	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
51	Q₁₆×C₄₇	—	2	2	no	—
52	Q₁₆×C₄₇	—	4	2	no	—
53	M₁₆×C₄₇	—	4	2	no	—
54	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂³	1	2	yes	—
56	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₄⋊C₄×C₄₇	—	2	1	no	—
61	C₄⋊C₄×C₄₇	—	2	1	no	—
62	C₄⋊C₄×C₄₇	—	2	2	no	—
63	C₄⋊C₄×C₄₇	—	2	1	no	—
64	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₄₇	—	2	2	no	—
69	C₂²⋊C₄×C₄₇	—	2	2	no	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
72	C₄²×C₄₇	—	2	2	no	—
73	C₄²×C₄₇	—	2	1	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄₇	C₂×D₄	1	2	yes	—
77	Q₈×C₄₇	—	2	1	no	—
78	Q₈×C₄₇	—	2	1	no	—
79	Q₈×C₄₇	—	2	2	no	—
80	Q₈×C₄₇	—	2	1	no	—
81	Q₈×C₄₇	—	2	2	no	—
82	Q₈×C₄₇	—	4	4	no	—
83	Q₈×C₄₇	—	4	2	no	—
84	Q₈×C₄₇	#41	1	2	yes	—
85	D₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	2	1	no	—
91	C₂×C₄×C₄₇	—	2	1	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₄₇	—	2	1	no	—
100	C₈×C₄₇	—	2	1	no	—
101	C₈×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	2	no	—
105	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	2	no	—
108	C₄×C₄₇	—	2	1	no	—
109	C₄×C₄₇	—	4	2	no	—
110	C₄×C₄₇	—	4	1	no	—
111	C₄×C₄₇	—	4	2	no	—
112	C₄×C₄₇	—	4	1	no	—
113	C₄×C₄₇	—	4	1	no	—
114	C₄×C₄₇	#137	1	2	yes	—
115	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
116	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
117	#3328	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₄₇	—	4	2	no	—
119	C₂×C₄₇	#546	1	1	yes	characteristic
120	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
121	C₂×C₄₇	#578	1	1	yes	characteristic
122	#556	C₂×C₄₇	1	1	yes	characteristic
123	D₄×Q₈	C₂×C₄₇	1	1	yes	characteristic
124	#487	C₂×C₄₇	1	1	yes	characteristic, complemented
125	#485	C₂×C₄₇	1	2	yes	complemented
126	#475	C₂×C₄₇	1	1	yes	characteristic, complemented
127	#470	C₂×C₄₇	1	2	yes	complemented
128	#462	C₂×C₄₇	1	1	yes	characteristic, complemented
129	#451	C₂×C₄₇	1	2	yes	complemented
130	C₄×Q₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
131	#436	C₂×C₄₇	1	1	yes	characteristic
132	#419	C₂×C₄₇	1	2	yes	complemented
133	C₄₇	#3328	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×Q₈	C₂²×C₄₇	1	2	yes	—
135	C₂×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
136	C₂×Q₁₆	C₂²×C₄₇	1	2	yes	—
137	C₂×M₁₆	C₂²×C₄₇	1	2	yes	—
138	C₄⋊Q₈	—	2	1	no	—
139	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
141	#126	—	2	2	no	—
142	#124	—	2	1	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
145	C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
148	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
151	#117	C₂²×C₄₇	1	2	yes	—
152	#107	C₂²×C₄₇	1	1	yes	characteristic
153	#106	C₂²×C₄₇	1	2	yes	—
154	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
155	#103	C₂²×C₄₇	1	1	yes	characteristic
156	#103	C₂²×C₄₇	1	2	yes	—
157	#103	C₂²×C₄₇	1	2	yes	—
158	#103	C₂²×C₄₇	1	2	yes	—
159	#103	C₂²×C₄₇	1	1	yes	characteristic
160	#103	C₂²×C₄₇	1	2	yes	—
161	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
162	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	C₂³×C₄₇	1	2	yes	—
168	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
169	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
170	C₂×D₄	D₄×C₄₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
173	Q₁₆	—	2	2	no	—
174	Q₁₆	—	4	2	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₄₇	1	2	yes	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
188	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂³×C₄₇	1	1	yes	characteristic
198	C₂³	C₂×D₄×C₄₇	1	2	yes	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	2	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	2	no	—
204	Q₈	—	4	4	no	—
205	Q₈	—	4	2	no	—
206	Q₈	#41×C₄₇	1	2	yes	—
207	D₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#137×C₄₇	1	2	yes	—
237	C₄	C₂×#41×C₄₇	1	1	yes	characteristic
238	C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
239	C₂	—	4	2	no	—
240	C₂	C₄₇×#546	1	1	yes	characteristic
241	C₂	C₂×#137×C₄₇	1	1	yes	characteristic
242	C₂	C₄₇×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₄₇×#487	C₂	semidirect
4. C₄₇×#485	C₂	semidirect
5. C₄₇×#475	C₂	semidirect
6. C₄₇×#470	C₂	semidirect
7. C₄₇×#462	C₂	semidirect
8. C₄₇×#451	C₂	semidirect
9. C₄×Q₁₆×C₄₇	C₂	semidirect
11. C₄₇×#419	C₂	semidirect
117. #3328	C₄₇	direct (N × H)
124. #487	C₂×C₄₇	semidirect
125. #485	C₂×C₄₇	semidirect
126. #475	C₂×C₄₇	semidirect
127. #470	C₂×C₄₇	semidirect
128. #462	C₂×C₄₇	semidirect
129. #451	C₂×C₄₇	semidirect
130. C₄×Q₁₆	C₂×C₄₇	semidirect
132. #419	C₂×C₄₇	semidirect
133. C₄₇	#3328	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×#556	2
D₄×Q₈×C₄₇	2
C₄₇×#487	2
C₄₇×#485	2
C₄₇×#475	2
C₄₇×#470	2
C₄₇×#462	2
C₄₇×#451	2
C₄×Q₁₆×C₄₇	2
C₄₇×#436	2
C₄₇×#419	2
#3328	47

Subgroups — C47×#3328 #20240938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3328 #20240938