Subgroups — C₄₇×#33956 #26073654

Order 12032.

472 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3380	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2255	C₂	1	2	yes	maximal, complemented
3	C₄₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2228	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2228	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2224	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#2177	C₂	1	2	yes	maximal
9	C₄×D₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₄₇×#554	—	2	2	no	—
11	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#500	C₂²	1	2	yes	—
13	C₄₇×#499	C₂²	1	2	yes	—
14	C₄₇×#499	C₂²	1	2	yes	—
15	C₄₇×#478	—	2	2	no	—
16	C₄₇×#478	—	2	2	no	—
17	C₄₇×#476	—	2	2	no	—
18	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×D₈×C₄₇	C₂²	1	2	yes	—
20	C₄×D₈×C₄₇	C₂²	1	2	yes	—
21	C₄×D₈×C₄₇	C₂²	1	2	yes	—
22	C₄₇×#367	C₂²	1	2	yes	—
23	C₄₇×#367	C₂²	1	2	yes	—
24	C₄₇×#366	C₂²	1	1	yes	characteristic
25	C₄₇×#366	C₂²	1	1	yes	characteristic
26	C₄₇×#366	C₂²	1	2	yes	—
27	C₄₇×#366	C₂²	1	1	yes	characteristic
28	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄₇×#357	C₂²	1	2	yes	—
31	C₄₇×#357	C₂²	1	2	yes	—
32	C₄₇×#357	C₂²	1	2	yes	—
33	C₄₇×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×D₈×C₄₇	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₄₇	—	2	2	no	—
37	#125×C₄₇	—	2	2	no	—
38	#125×C₄₇	—	2	2	no	—
39	C₂²⋊Q₈×C₄₇	—	4	2	no	—
40	C₂²⋊Q₈×C₄₇	—	4	2	no	—
41	C₂²⋊Q₈×C₄₇	—	4	2	no	—
42	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
43	C₄×D₄×C₄₇	—	2	2	no	—
44	C₄×D₄×C₄₇	—	2	2	no	—
45	C₄×D₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
47	D₁₆×C₄₇	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
50	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
51	#107×C₄₇	—	2	2	no	—
52	#107×C₄₇	C₂³	1	2	yes	—
53	#107×C₄₇	C₂³	1	2	yes	—
54	#107×C₄₇	C₂³	1	2	yes	—
55	#106×C₄₇	—	2	2	no	—
56	#106×C₄₇	—	2	2	no	—
57	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
58	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
59	#102×C₄₇	—	2	2	no	—
60	#102×C₄₇	—	2	2	no	—
61	#102×C₄₇	—	2	2	no	—
62	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂×Q₈×C₄₇	—	4	2	no	—
65	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
66	C₂×D₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	—	4	2	no	—
70	D₈×C₄₇	—	2	2	no	—
71	D₈×C₄₇	C₂×Q₈	1	4	yes	—
72	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₄⋊C₄×C₄₇	—	4	2	no	—
83	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
84	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
85	C₂²⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	4	2	no	—
87	C₂²⋊C₄×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂×D₄	1	2	yes	—
90	C₁₆×C₄₇	—	2	1	no	—
91	C₁₆×C₄₇	—	2	2	no	—
92	C₁₆×C₄₇	C₂×Q₈	1	2	yes	—
93	C₂³×C₄₇	—	4	2	no	—
94	Q₈×C₄₇	—	4	1	no	—
95	Q₈×C₄₇	—	16	2	no	—
96	D₄×C₄₇	—	2	4	no	—
97	D₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	2	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	8	4	no	—
102	C₂×C₄×C₄₇	—	16	2	no	—
103	C₂×C₄×C₄₇	—	16	2	no	—
104	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₄₇	—	2	1	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
112	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
113	#33956	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₄₇	—	4	4	no	—
115	C₂²×C₄₇	—	8	2	no	—
116	C₂²×C₄₇	—	8	2	no	—
117	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	8	2	no	—
123	C₄×C₄₇	—	8	2	no	—
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	D₄×Q₈	1	1	yes	characteristic
126	C₄×C₄₇	#578	1	1	yes	characteristic
127	#3380	C₂×C₄₇	1	2	yes	complemented
128	#2255	C₂×C₄₇	1	2	yes	complemented
129	#2253	C₂×C₄₇	1	1	yes	characteristic, complemented
130	#2228	C₂×C₄₇	1	2	yes	complemented
131	#2228	C₂×C₄₇	1	2	yes	complemented
132	#2224	C₂×C₄₇	1	2	yes	complemented
133	#2183	C₂×C₄₇	1	1	yes	characteristic, complemented
134	#2177	C₂×C₄₇	1	2	yes	—
135	C₄×D₁₆	C₂×C₄₇	1	1	yes	characteristic
136	C₂×C₄₇	—	8	4	no	—
137	C₂×C₄₇	Q₈×D₈	1	1	yes	characteristic
138	C₂×C₄₇	#3414	1	1	yes	characteristic
139	C₂×C₄₇	#3415	1	1	yes	characteristic
140	#554	—	2	2	no	—
141	C₂×D₁₆	C₂²×C₄₇	1	1	yes	characteristic
142	#500	C₂²×C₄₇	1	2	yes	—
143	#499	C₂²×C₄₇	1	2	yes	—
144	#499	C₂²×C₄₇	1	2	yes	—
145	#478	—	2	2	no	—
146	#478	—	2	2	no	—
147	#476	—	2	2	no	—
148	C₈×Q₈	C₂²×C₄₇	1	1	yes	characteristic
149	C₄×D₈	C₂²×C₄₇	1	2	yes	—
150	C₄×D₈	C₂²×C₄₇	1	2	yes	—
151	C₄×D₈	C₂²×C₄₇	1	2	yes	—
152	#367	C₂²×C₄₇	1	2	yes	—
153	#367	C₂²×C₄₇	1	2	yes	—
154	#366	C₂²×C₄₇	1	1	yes	characteristic
155	#366	C₂²×C₄₇	1	1	yes	characteristic
156	#366	C₂²×C₄₇	1	2	yes	—
157	#366	C₂²×C₄₇	1	1	yes	characteristic
158	C₄⋊C₁₆	C₂²×C₄₇	1	2	yes	—
159	C₄⋊C₁₆	C₂²×C₄₇	1	1	yes	characteristic
160	#357	C₂²×C₄₇	1	2	yes	—
161	#357	C₂²×C₄₇	1	2	yes	—
162	#357	C₂²×C₄₇	1	2	yes	—
163	#346	C₂²×C₄₇	1	2	yes	—
164	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
165	C₄₇	#33956	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×D₈	C₂³×C₄₇	1	2	yes	—
167	C₄⋊Q₈	—	2	2	no	—
168	#125	—	2	2	no	—
169	#125	—	2	2	no	—
170	C₂²⋊Q₈	—	4	2	no	—
171	C₂²⋊Q₈	—	4	2	no	—
172	C₂²⋊Q₈	—	4	2	no	—
173	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	2	no	—
178	D₁₆	Q₈×C₄₇	1	4	yes	complemented
179	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
180	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
181	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
182	#107	—	2	2	no	—
183	#107	C₂³×C₄₇	1	2	yes	—
184	#107	C₂³×C₄₇	1	2	yes	—
185	#107	C₂³×C₄₇	1	2	yes	—
186	#106	—	2	2	no	—
187	#106	—	2	2	no	—
188	C₄⋊C₈	D₄×C₄₇	1	2	yes	—
189	C₄⋊C₈	D₄×C₄₇	1	1	yes	characteristic
190	#102	—	2	2	no	—
191	#102	—	2	2	no	—
192	#102	—	2	2	no	—
193	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₄₇	1	2	yes	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	D₈×C₄₇	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	D₈	—	2	2	no	—
202	D₈	C₂×Q₈×C₄₇	1	4	yes	—
203	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
205	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	D₈×C₄₇	1	2	yes	—
215	C₄⋊C₄	D₈×C₄₇	1	1	yes	characteristic
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₄₇	1	2	yes	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	C₂×Q₈×C₄₇	1	2	yes	—
224	C₂³	—	4	2	no	—
225	Q₈	—	4	1	no	—
226	Q₈	—	16	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	16	2	no	—
234	C₂×C₄	—	16	2	no	—
235	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
237	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
238	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂²×Q₈×C₄₇	1	1	yes	characteristic
243	C₈	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
244	C₂²	—	4	4	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	D₄×Q₈×C₄₇	1	1	yes	characteristic
256	C₄	C₄₇×#578	1	1	yes	characteristic
257	C₂	—	8	4	no	—
258	C₂	Q₈×D₈×C₄₇	1	1	yes	characteristic
259	C₂	C₄₇×#3414	1	1	yes	characteristic
260	C₂	C₄₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₂×D₁₆×C₄₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄×Q₈×C₄₇ ◁ C₈×Q₈×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3380	C₂	semidirect
2. C₄₇×#2255	C₂	semidirect
3. C₄₇×#2253	C₂	semidirect
4. C₄₇×#2228	C₂	semidirect
5. C₄₇×#2228	C₂	semidirect
6. C₄₇×#2224	C₂	semidirect
7. C₄₇×#2183	C₂	semidirect
47. D₁₆×C₄₇	Q₈	semidirect
113. #33956	C₄₇	direct (N × H)
127. #3380	C₂×C₄₇	semidirect
128. #2255	C₂×C₄₇	semidirect
129. #2253	C₂×C₄₇	semidirect
130. #2228	C₂×C₄₇	semidirect
131. #2228	C₂×C₄₇	semidirect
132. #2224	C₂×C₄₇	semidirect
133. #2183	C₂×C₄₇	semidirect
165. C₄₇	#33956	direct (N × H)
178. D₁₆	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×#3380	2
C₄₇×#2255	2
C₄₇×#2253	2
C₄₇×#2228	2
C₄₇×#2224	2
C₄₇×#2183	2
C₄₇×#2177	2
C₄×D₁₆×C₄₇	2
#33956	47

Subgroups — C47×#33956 #26073654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33956 #26073654