Subgroups — C₄₇×#3385 #20240995

Order 6016.

340 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#554	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#501	C₂	1	2	yes	maximal, complemented
4	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#479	C₂	1	2	yes	maximal, complemented
6	C₄₇×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#476	C₂	1	1	yes	maximal, characteristic
8	C₄₇×#474	C₂	1	2	yes	maximal
9	C₄₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#440	C₂	1	2	yes	maximal
11	C₄×SD₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₄₇	—	2	1	no	—
14	C₄⋊Q₈×C₄₇	—	2	2	no	—
15	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
18	#125×C₄₇	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₄₇	—	2	1	no	—
20	C₂²⋊Q₈×C₄₇	—	2	2	no	—
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
24	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₄₇	C₂²	1	2	yes	—
26	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	#107×C₄₇	C₂²	1	2	yes	—
29	#107×C₄₇	C₂²	1	2	yes	—
30	#107×C₄₇	C₂²	1	2	yes	—
31	#106×C₄₇	C₂²	1	2	yes	—
32	#106×C₄₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
34	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
35	#103×C₄₇	C₂²	1	2	yes	—
36	#103×C₄₇	C₂²	1	1	yes	characteristic
37	#102×C₄₇	C₂²	1	1	yes	characteristic
38	#102×C₄₇	C₂²	1	2	yes	—
39	#97×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	1	no	—
43	C₂×Q₈×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
45	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	—	2	1	no	—
48	C₂²×C₄×C₄₇	—	2	2	no	—
49	SD₁₆×C₄₇	Q₈	1	4	yes	complemented
50	C₂×C₈×C₄₇	C₂³	1	2	yes	—
51	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₄₇	—	2	2	no	—
54	C₄⋊C₄×C₄₇	—	2	1	no	—
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₄⋊C₄×C₄₇	—	2	1	no	—
57	C₄⋊C₄×C₄₇	—	2	1	no	—
58	C₄⋊C₄×C₄₇	—	2	2	no	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
62	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₄₇	—	2	1	no	—
67	C₂²⋊C₄×C₄₇	—	2	2	no	—
68	C₄²×C₄₇	—	2	2	no	—
69	C₄²×C₄₇	—	2	1	no	—
70	C₄²×C₄₇	C₂³	1	2	yes	—
71	C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄₇	—	2	1	no	—
73	Q₈×C₄₇	—	2	2	no	—
74	Q₈×C₄₇	—	2	1	no	—
75	Q₈×C₄₇	—	8	2	no	—
76	Q₈×C₄₇	—	8	1	no	—
77	Q₈×C₄₇	—	8	1	no	—
78	Q₈×C₄₇	C₂×Q₈	1	2	yes	—
79	D₄×C₄₇	—	2	1	no	—
80	D₄×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	2	1	no	—
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	—	4	2	no	—
87	C₂×C₄×C₄₇	—	8	2	no	—
88	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₈×C₄₇	—	2	2	no	—
94	C₈×C₄₇	—	2	1	no	—
95	C₈×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂²×C₄₇	—	2	2	no	—
97	C₂²×C₄₇	—	4	1	no	—
98	C₂²×C₄₇	—	4	1	no	—
99	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
100	C₄×C₄₇	—	2	2	no	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	2	2	no	—
103	C₄×C₄₇	—	2	1	no	—
104	C₄×C₄₇	—	4	1	no	—
105	C₄×C₄₇	—	4	2	no	—
106	C₄×C₄₇	—	4	1	no	—
107	C₄×C₄₇	—	4	2	no	—
108	C₄×C₄₇	#137	1	2	yes	—
109	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
110	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
111	#3385	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₄₇	—	4	2	no	—
113	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
114	C₂×C₄₇	C₂×#137	1	1	yes	characteristic
115	C₂×C₄₇	#576	1	1	yes	characteristic
116	Q₈²	C₂×C₄₇	1	1	yes	characteristic, complemented
117	#554	C₂×C₄₇	1	1	yes	characteristic
118	#501	C₂×C₄₇	1	2	yes	complemented
119	#500	C₂×C₄₇	1	1	yes	characteristic, complemented
120	#479	C₂×C₄₇	1	2	yes	complemented
121	#477	C₂×C₄₇	1	1	yes	characteristic, complemented
122	#476	C₂×C₄₇	1	1	yes	characteristic
123	#474	C₂×C₄₇	1	2	yes	—
124	#446	C₂×C₄₇	1	1	yes	characteristic, complemented
125	#440	C₂×C₄₇	1	2	yes	—
126	C₄×SD₁₆	C₂×C₄₇	1	1	yes	characteristic
127	C₄₇	#3385	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	1	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
132	C₄⋊Q₈	C₂²×C₄₇	1	2	yes	—
133	C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
134	#125	C₂²×C₄₇	1	2	yes	—
135	C₂²⋊Q₈	—	2	1	no	—
136	C₂²⋊Q₈	—	2	2	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
139	C₄×Q₈	C₂²×C₄₇	1	2	yes	—
140	C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
141	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
142	C₄×D₄	C₂²×C₄₇	1	2	yes	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	#107	C₂²×C₄₇	1	2	yes	—
145	#107	C₂²×C₄₇	1	2	yes	—
146	#107	C₂²×C₄₇	1	2	yes	—
147	#106	C₂²×C₄₇	1	1	yes	characteristic
148	#106	C₂²×C₄₇	1	2	yes	—
149	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
150	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
151	#103	C₂²×C₄₇	1	1	yes	characteristic
152	#103	C₂²×C₄₇	1	2	yes	—
153	#102	C₂²×C₄₇	1	1	yes	characteristic
154	#102	C₂²×C₄₇	1	2	yes	—
155	#97	C₂²×C₄₇	1	2	yes	—
156	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	D₄×C₄₇	1	1	yes	characteristic
161	C₂×Q₈	C₂³×C₄₇	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	2	no	—
165	SD₁₆	Q₈×C₄₇	1	4	yes	complemented
166	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₄₇	1	2	yes	—
168	C₂×C₈	C₂³×C₄₇	1	1	yes	characteristic
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	1	no	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
177	C₄⋊C₄	D₄×C₄₇	1	2	yes	—
178	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
180	C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂³×C₄₇	1	1	yes	characteristic
187	C₄²	C₂³×C₄₇	1	2	yes	—
188	C₂³	—	2	1	no	—
189	Q₈	—	2	2	no	—
190	Q₈	—	2	1	no	—
191	Q₈	—	8	2	no	—
192	Q₈	—	8	1	no	—
193	Q₈	—	8	1	no	—
194	Q₈	C₂×Q₈×C₄₇	1	2	yes	—
195	D₄	—	2	1	no	—
196	D₄	C₂×Q₈×C₄₇	1	2	yes	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	C₂×Q₈×C₄₇	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	C₂²×D₄×C₄₇	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	4	2	no	—
224	C₄	#137×C₄₇	1	2	yes	—
225	C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
226	C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	D₄×Q₈×C₄₇	1	1	yes	characteristic
229	C₂	C₂×#137×C₄₇	1	1	yes	characteristic
230	C₂	C₄₇×#576	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	C₂³
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. Q₈²×C₄₇	C₂	semidirect
3. C₄₇×#501	C₂	semidirect
4. C₄₇×#500	C₂	semidirect
5. C₄₇×#479	C₂	semidirect
6. C₄₇×#477	C₂	semidirect
9. C₄₇×#446	C₂	semidirect
49. SD₁₆×C₄₇	Q₈	semidirect
111. #3385	C₄₇	direct (N × H)
116. Q₈²	C₂×C₄₇	semidirect
118. #501	C₂×C₄₇	semidirect
119. #500	C₂×C₄₇	semidirect
120. #479	C₂×C₄₇	semidirect
121. #477	C₂×C₄₇	semidirect
124. #446	C₂×C₄₇	semidirect
127. C₄₇	#3385	direct (N × H)
165. SD₁₆	Q₈×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈²×C₄₇	2
C₄₇×#554	2
C₄₇×#501	2
C₄₇×#500	2
C₄₇×#479	2
C₄₇×#477	2
C₄₇×#476	2
C₄₇×#474	2
C₄₇×#446	2
C₄₇×#440	2
C₄×SD₁₆×C₄₇	2
#3385	47

Subgroups — C47×#3385 #20240995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#3385 #20240995