Subgroups — C₄₇×#3384 #20240994

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×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#501	C₂	1	2	yes	maximal, complemented
4	C₄₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#478	C₂	1	2	yes	maximal, complemented
6	C₄₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#476	C₂	1	2	yes	maximal, complemented
8	C₄₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#442	C₂	1	2	yes	maximal
11	C₄×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₄₇	—	2	1	no	—
13	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
17	#125×C₄₇	C₂²	1	2	yes	—
18	C₂²⋊Q₈×C₄₇	—	2	1	no	—
19	C₂²⋊Q₈×C₄₇	—	2	1	no	—
20	C₂²⋊Q₈×C₄₇	—	2	2	no	—
21	C₂²⋊Q₈×C₄₇	—	2	2	no	—
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₄₇	C₂²	1	2	yes	—
26	C₄×D₄×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	#107×C₄₇	C₂²	1	2	yes	—
29	#107×C₄₇	C₂²	1	1	yes	characteristic
30	#106×C₄₇	C₂²	1	2	yes	—
31	#106×C₄₇	C₂²	1	2	yes	—
32	#106×C₄₇	C₂²	1	2	yes	—
33	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
35	#102×C₄₇	C₂²	1	2	yes	—
36	#102×C₄₇	C₂²	1	1	yes	characteristic
37	#102×C₄₇	C₂²	1	2	yes	—
38	#102×C₄₇	C₂²	1	1	yes	characteristic
39	#97×C₄₇	C₂²	1	2	yes	—
40	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₄₇	—	2	2	no	—
42	C₂×Q₈×C₄₇	—	2	1	no	—
43	C₂×Q₈×C₄₇	—	2	1	no	—
44	C₂×Q₈×C₄₇	—	4	4	no	—
45	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	1	no	—
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	D₈×C₄₇	Q₈	1	4	yes	complemented
53	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₄₇	C₂³	1	2	yes	—
55	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₄₇	—	2	2	no	—
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	2	no	—
61	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
63	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₄₇	—	2	1	no	—
68	C₂²⋊C₄×C₄₇	—	2	2	no	—
69	C₂²⋊C₄×C₄₇	—	2	1	no	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	C₂³	1	2	yes	—
72	C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄₇	—	2	1	no	—
74	C₂³×C₄₇	—	2	1	no	—
75	Q₈×C₄₇	—	2	2	no	—
76	Q₈×C₄₇	—	4	4	no	—
77	Q₈×C₄₇	—	4	2	no	—
78	Q₈×C₄₇	—	8	1	no	—
79	D₄×C₄₇	—	2	1	no	—
80	D₄×C₄₇	—	2	1	no	—
81	D₄×C₄₇	C₂×Q₈	1	2	yes	—
82	D₄×C₄₇	C₂×Q₈	1	2	yes	complemented
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	2	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	4	2	no	—
89	C₂×C₄×C₄₇	—	4	2	no	—
90	C₂×C₄×C₄₇	—	8	2	no	—
91	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
96	C₈×C₄₇	—	2	1	no	—
97	C₈×C₄₇	—	2	2	no	—
98	C₈×C₄₇	C₂×Q₈	1	2	yes	complemented
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	2	2	no	—
101	C₂²×C₄₇	—	4	1	no	—
102	C₂²×C₄₇	—	4	1	no	—
103	C₂²×C₄₇	—	4	1	no	—
104	C₂²×C₄₇	—	4	1	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	1	no	—
110	C₄×C₄₇	—	4	2	no	—
111	C₄×C₄₇	—	4	1	no	—
112	C₄×C₄₇	—	4	2	no	—
113	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
114	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
115	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
116	#3384	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₄₇	—	4	2	no	—
118	C₂×C₄₇	—	4	2	no	—
119	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
120	C₂×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄₇	#577	1	1	yes	characteristic
122	#554	C₂×C₄₇	1	1	yes	characteristic, complemented
123	D₄×Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
124	#501	C₂×C₄₇	1	2	yes	complemented
125	#498	C₂×C₄₇	1	1	yes	characteristic, complemented
126	#478	C₂×C₄₇	1	2	yes	complemented
127	#476	C₂×C₄₇	1	2	yes	complemented
128	#476	C₂×C₄₇	1	1	yes	characteristic, complemented
129	#474	C₂×C₄₇	1	1	yes	characteristic, complemented
130	#446	C₂×C₄₇	1	1	yes	characteristic, complemented
131	#442	C₂×C₄₇	1	2	yes	—
132	C₄×D₈	C₂×C₄₇	1	1	yes	characteristic
133	C₄₇	#3384	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×Q₈	—	2	1	no	—
135	C₂×D₈	C₂²×C₄₇	1	1	yes	characteristic
136	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
138	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
139	#125	C₂²×C₄₇	1	2	yes	—
140	C₂²⋊Q₈	—	2	1	no	—
141	C₂²⋊Q₈	—	2	1	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₄₇	1	2	yes	—
147	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
148	C₄×D₄	C₂²×C₄₇	1	2	yes	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	#107	C₂²×C₄₇	1	1	yes	characteristic
151	#107	C₂²×C₄₇	1	2	yes	—
152	#106	C₂²×C₄₇	1	2	yes	—
153	#106	C₂²×C₄₇	1	2	yes	—
154	#106	C₂²×C₄₇	1	2	yes	—
155	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
156	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
157	#102	C₂²×C₄₇	1	1	yes	characteristic
158	#102	C₂²×C₄₇	1	1	yes	characteristic
159	#102	C₂²×C₄₇	1	2	yes	—
160	#102	C₂²×C₄₇	1	2	yes	—
161	#97	C₂²×C₄₇	1	2	yes	—
162	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	D₄×C₄₇	1	1	yes	characteristic
168	C₂×D₄	C₂³×C₄₇	1	1	yes	characteristic
169	C₂×D₄	C₂³×C₄₇	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	D₈	Q₈×C₄₇	1	4	yes	complemented
175	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₄₇	1	2	yes	—
177	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
184	C₄⋊C₄	D₄×C₄₇	1	2	yes	—
185	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
186	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
188	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	C₂³×C₄₇	1	1	yes	characteristic
194	C₄²	C₂³×C₄₇	1	2	yes	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	Q₈	—	2	2	no	—
198	Q₈	—	4	4	no	—
199	Q₈	—	4	2	no	—
200	Q₈	—	8	1	no	—
201	D₄	—	2	1	no	—
202	D₄	—	2	1	no	—
203	D₄	C₂×Q₈×C₄₇	1	2	yes	—
204	D₄	C₂×Q₈×C₄₇	1	2	yes	complemented
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
214	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
215	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
217	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	C₂×Q₈×C₄₇	1	2	yes	complemented
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	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	1	no	—
234	C₄	—	4	2	no	—
235	C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
236	C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
237	C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
238	C₂	—	4	2	no	—
239	C₂	—	4	2	no	—
240	C₂	D₄×Q₈×C₄₇	1	1	yes	characteristic
241	C₂	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
242	C₂	C₄₇×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₄₇
prime radical	C₂×D₈×C₄₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄×Q₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#554	C₂	semidirect
2. D₄×Q₈×C₄₇	C₂	semidirect
3. C₄₇×#501	C₂	semidirect
4. C₄₇×#498	C₂	semidirect
5. C₄₇×#478	C₂	semidirect
6. C₄₇×#476	C₂	semidirect
7. C₄₇×#476	C₂	semidirect
8. C₄₇×#474	C₂	semidirect
9. C₄₇×#446	C₂	semidirect
52. D₈×C₄₇	Q₈	semidirect
82. D₄×C₄₇	C₂×Q₈	semidirect
98. C₈×C₄₇	C₂×Q₈	semidirect
116. #3384	C₄₇	direct (N × H)
122. #554	C₂×C₄₇	semidirect
123. D₄×Q₈	C₂×C₄₇	semidirect
124. #501	C₂×C₄₇	semidirect
125. #498	C₂×C₄₇	semidirect
126. #478	C₂×C₄₇	semidirect
127. #476	C₂×C₄₇	semidirect
128. #476	C₂×C₄₇	semidirect
129. #474	C₂×C₄₇	semidirect
130. #446	C₂×C₄₇	semidirect
133. C₄₇	#3384	direct (N × H)
174. D₈	Q₈×C₄₇	semidirect
204. D₄	C₂×Q₈×C₄₇	semidirect
220. C₈	C₂×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×#554	2
D₄×Q₈×C₄₇	2
C₄₇×#501	2
C₄₇×#498	2
C₄₇×#478	2
C₄₇×#476	2
C₄₇×#474	2
C₄₇×#446	2
C₄₇×#442	2
C₄×D₈×C₄₇	2
#3384	47

Subgroups — C47×#3384 #20240994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3384 #20240994