Subgroups — C₄₇×#3278 #20240888

Order 6016.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₄₇	C₂	1	2	yes	maximal, complemented
2	C₄₇×#550	C₂	1	1	yes	maximal, characteristic
3	D₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#468	C₂	1	2	yes	maximal, complemented
6	C₄₇×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#452	C₂	1	2	yes	maximal, complemented
9	C₄₇×#448	C₂	1	2	yes	maximal, complemented
10	C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₄₇	C₂²	1	2	yes	—
13	C₂²×Q₈×C₄₇	C₂²	1	2	yes	—
14	#137×C₄₇	—	2	8	no	—
15	C₂×Q₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₄₇	—	2	1	no	—
22	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	#127×C₄₇	—	2	1	no	—
24	#127×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²⋊Q₈×C₄₇	—	2	2	no	—
26	C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
27	#121×C₄₇	—	2	2	no	—
28	#121×C₄₇	C₂²	1	2	yes	—
29	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₄₇	—	2	1	no	—
32	C₄×D₄×C₄₇	—	2	1	no	—
33	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
34	#107×C₄₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
36	#103×C₄₇	C₂²	1	1	yes	characteristic
37	#103×C₄₇	C₂²	1	2	yes	—
38	#103×C₄₇	C₂²	1	1	yes	characteristic
39	#102×C₄₇	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
41	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
42	#41×C₄₇	—	2	4	no	—
43	#41×C₄₇	—	4	2	no	—
44	C₂×Q₈×C₄₇	—	2	2	no	—
45	C₂×Q₈×C₄₇	—	2	4	no	—
46	C₂×Q₈×C₄₇	—	4	2	no	—
47	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
48	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₇	—	2	2	no	—
51	C₂×D₄×C₄₇	—	2	2	no	—
52	C₂×D₄×C₄₇	—	2	1	no	—
53	C₂×D₄×C₄₇	—	2	1	no	—
54	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
55	C₂×D₄×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
59	Q₁₆×C₄₇	—	2	4	no	—
60	Q₁₆×C₄₇	D₄	1	4	yes	complemented
61	SD₁₆×C₄₇	—	2	4	no	—
62	SD₁₆×C₄₇	—	2	4	no	—
63	SD₁₆×C₄₇	—	4	2	no	—
64	M₁₆×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂³	1	2	yes	—
67	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₄⋊C₄×C₄₇	—	2	1	no	—
70	C₄⋊C₄×C₄₇	—	2	1	no	—
71	C₄⋊C₄×C₄₇	—	2	1	no	—
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₄₇	—	2	2	no	—
77	C₂²⋊C₄×C₄₇	—	2	2	no	—
78	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	—	2	1	no	—
81	C₄²×C₄₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄₇	—	2	2	no	—
83	C₂³×C₄₇	C₂×D₄	1	2	yes	—
84	Q₈×C₄₇	—	2	1	no	—
85	Q₈×C₄₇	—	2	1	no	—
86	Q₈×C₄₇	—	2	2	no	—
87	Q₈×C₄₇	—	2	2	no	—
88	Q₈×C₄₇	—	4	4	no	—
89	Q₈×C₄₇	C₂×D₄	1	2	yes	—
90	Q₈×C₄₇	C₂×D₄	1	2	yes	—
91	D₄×C₄₇	—	2	2	no	—
92	D₄×C₄₇	—	2	2	no	—
93	D₄×C₄₇	—	2	2	no	—
94	D₄×C₄₇	—	4	4	no	—
95	D₄×C₄₇	—	4	2	no	—
96	D₄×C₄₇	—	8	1	no	—
97	D₄×C₄₇	—	8	2	no	—
98	C₂×C₄×C₄₇	—	2	1	no	—
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	2	1	no	—
113	C₈×C₄₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	—	4	2	no	—
119	C₂²×C₄₇	—	4	2	no	—
120	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₄₇	—	2	2	no	—
122	C₄×C₄₇	—	2	2	no	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	2	1	no	—
125	C₄×C₄₇	—	4	2	no	—
126	C₄×C₄₇	—	4	1	no	—
127	C₄×C₄₇	—	4	1	no	—
128	C₄×C₄₇	#137	1	2	yes	complemented
129	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
131	#3278	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂×C₄₇	—	4	2	no	—
133	C₂×C₄₇	—	8	2	no	—
134	C₂×C₄₇	D₄²	1	1	yes	characteristic
135	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
136	C₂×C₄₇	#577	1	1	yes	characteristic
137	C₂×#137	C₂×C₄₇	1	2	yes	complemented
138	#550	C₂×C₄₇	1	1	yes	characteristic
139	D₄×Q₈	C₂×C₄₇	1	1	yes	characteristic, complemented
140	#492	C₂×C₄₇	1	1	yes	characteristic, complemented
141	#468	C₂×C₄₇	1	2	yes	complemented
142	#462	C₂×C₄₇	1	1	yes	characteristic, complemented
143	#460	C₂×C₄₇	1	1	yes	characteristic, complemented
144	#452	C₂×C₄₇	1	2	yes	complemented
145	#448	C₂×C₄₇	1	2	yes	complemented
146	C₄×Q₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
147	#436	C₂×C₄₇	1	1	yes	characteristic, complemented
148	C₄₇	#3278	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×#41	C₂²×C₄₇	1	2	yes	—
150	C₂²×Q₈	C₂²×C₄₇	1	2	yes	—
151	#137	—	2	8	no	—
152	C₂×Q₁₆	C₂²×C₄₇	1	2	yes	—
153	C₂×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic
154	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	—
155	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	—
156	C₂×SD₁₆	C₂²×C₄₇	1	2	yes	—
157	C₂×M₁₆	C₂²×C₄₇	1	2	yes	—
158	C₄⋊Q₈	—	2	1	no	—
159	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
160	#127	—	2	1	no	—
161	#127	C₂²×C₄₇	1	1	yes	characteristic
162	C₂²⋊Q₈	—	2	2	no	—
163	C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
164	#121	—	2	2	no	—
165	#121	C₂²×C₄₇	1	2	yes	—
166	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
167	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
171	#107	C₂²×C₄₇	1	1	yes	characteristic
172	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
173	#103	C₂²×C₄₇	1	1	yes	characteristic
174	#103	C₂²×C₄₇	1	1	yes	characteristic
175	#103	C₂²×C₄₇	1	2	yes	—
176	#102	C₂²×C₄₇	1	2	yes	—
177	C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
178	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
179	#41	—	2	4	no	—
180	#41	—	4	2	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	C₂³×C₄₇	1	2	yes	—
185	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
186	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	D₄×C₄₇	1	1	yes	characteristic
192	C₂×D₄	C₂³×C₄₇	1	2	yes	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₂³×C₄₇	1	2	yes	—
196	Q₁₆	—	2	4	no	—
197	Q₁₆	D₄×C₄₇	1	4	yes	complemented
198	SD₁₆	—	2	4	no	—
199	SD₁₆	—	2	4	no	—
200	SD₁₆	—	4	2	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₄₇	1	2	yes	—
204	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₄₇	1	2	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂³×C₄₇	1	1	yes	characteristic
219	C₂³	—	2	2	no	—
220	C₂³	C₂×D₄×C₄₇	1	2	yes	—
221	Q₈	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	2	2	no	—
225	Q₈	—	4	4	no	—
226	Q₈	C₂×D₄×C₄₇	1	2	yes	—
227	Q₈	C₂×D₄×C₄₇	1	2	yes	—
228	D₄	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×D₄×C₄₇	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	#137×C₄₇	1	2	yes	complemented
266	C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
267	C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
268	C₂	—	4	2	no	—
269	C₂	—	8	2	no	—
270	C₂	D₄²×C₄₇	1	1	yes	characteristic
271	C₂	C₂×#137×C₄₇	1	1	yes	characteristic
272	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	C₄₇×#550	C₂
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
1. C₂×#137×C₄₇	C₂	semidirect
3. D₄×Q₈×C₄₇	C₂	semidirect
4. C₄₇×#492	C₂	semidirect
5. C₄₇×#468	C₂	semidirect
6. C₄₇×#462	C₂	semidirect
7. C₄₇×#460	C₂	semidirect
8. C₄₇×#452	C₂	semidirect
9. C₄₇×#448	C₂	semidirect
10. C₄×Q₁₆×C₄₇	C₂	semidirect
11. C₄₇×#436	C₂	semidirect
60. Q₁₆×C₄₇	D₄	semidirect
128. C₄×C₄₇	#137	semidirect
131. #3278	C₄₇	direct (N × H)
137. C₂×#137	C₂×C₄₇	semidirect
139. D₄×Q₈	C₂×C₄₇	semidirect
140. #492	C₂×C₄₇	semidirect
141. #468	C₂×C₄₇	semidirect
142. #462	C₂×C₄₇	semidirect
143. #460	C₂×C₄₇	semidirect
144. #452	C₂×C₄₇	semidirect
145. #448	C₂×C₄₇	semidirect
146. C₄×Q₁₆	C₂×C₄₇	semidirect
147. #436	C₂×C₄₇	semidirect
148. C₄₇	#3278	direct (N × H)
197. Q₁₆	D₄×C₄₇	semidirect
265. C₄	#137×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₄₇	2
C₄₇×#550	2
D₄×Q₈×C₄₇	2
C₄₇×#492	2
C₄₇×#468	2
C₄₇×#462	2
C₄₇×#460	2
C₄₇×#452	2
C₄₇×#448	2
C₄×Q₁₆×C₄₇	2
C₄₇×#436	2
#3278	47

Subgroups — C47×#3278 #20240888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3278 #20240888