Subgroups — C₄₇×#33434 #26073132

Order 12032.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#504	C₂	1	2	yes	maximal, complemented
2	C₄₇×#2931	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2169	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆×C₄₇	C₂	1	4	yes	maximal, complemented
5	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2113	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₄₇	C₂²	1	2	yes	—
9	C₄₇×#532	C₄	1	1	yes	characteristic
10	C₄₇×#504	—	2	8	no	—
11	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
12	C₂×M₃₂×C₄₇	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#436	C₄	1	4	yes	—
17	C₈×D₄×C₄₇	C₂²	1	4	yes	—
18	C₄₇×#423	C₄	1	1	yes	characteristic
19	C₄₇×#408	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₄₇×C₂²⋊C₁₆	C₂²	1	8	yes	—
23	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
24	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
25	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
26	#131×C₄₇	—	2	4	no	—
27	#131×C₄₇	—	2	4	no	—
28	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₇	C₈	1	1	yes	characteristic
34	#127×C₄₇	C₈	1	1	yes	characteristic
35	#124×C₄₇	C₈	1	2	yes	—
36	#121×C₄₇	C₈	1	4	yes	—
37	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
39	M₃₂×C₄₇	—	2	4	no	—
40	M₃₂×C₄₇	—	2	8	no	—
41	C₂×C₁₆×C₄₇	—	2	2	no	—
42	C₂×C₁₆×C₄₇	—	2	8	no	—
43	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
44	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
45	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
46	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
49	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	#41×C₄₇	—	2	4	no	—
53	#41×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
55	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
57	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	M₁₆×C₄₇	—	2	4	no	—
62	M₁₆×C₄₇	—	2	8	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₇	#41	1	2	yes	—
71	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
73	C₂²⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
74	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₁₆×C₄₇	—	2	4	no	—
78	C₁₆×C₄₇	—	2	2	no	—
79	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
80	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
81	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
82	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
83	Q₈×C₄₇	—	2	4	no	—
84	D₄×C₄₇	—	2	8	no	—
85	D₄×C₄₇	—	2	4	no	—
86	D₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	2	4	no	—
90	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
93	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₄₇	#131	1	2	yes	—
96	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₄₇	—	2	4	no	—
98	C₈×C₄₇	—	2	2	no	—
99	C₈×C₄₇	C₄×D₄	1	2	yes	—
100	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
102	#33434	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₄₇	—	2	2	no	—
104	C₂²×C₄₇	—	2	8	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	C₈×D₄	1	2	yes	—
107	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
108	C₄×C₄₇	—	2	2	no	—
109	C₄×C₄₇	—	2	4	no	—
110	C₄×C₄₇	C₈×D₄	1	2	yes	—
111	C₄×C₄₇	#504	1	4	yes	complemented
112	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
114	C₂×#504	C₂×C₄₇	1	2	yes	complemented
115	#2931	C₂×C₄₇	1	1	yes	characteristic
116	#2169	C₂×C₄₇	1	4	yes	complemented
117	D₄×C₁₆	C₂×C₄₇	1	4	yes	complemented
118	#2150	C₂×C₄₇	1	1	yes	characteristic, complemented
119	#2113	C₂×C₄₇	1	2	yes	complemented
120	C₂×C₄×C₁₆	C₂×C₄₇	1	1	yes	characteristic, complemented
121	C₂×C₄₇	—	2	2	no	—
122	C₂×C₄₇	—	4	4	no	—
123	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
124	C₂×C₄₇	C₂×#504	1	2	yes	—
125	C₂×#131	C₂²×C₄₇	1	2	yes	—
126	#532	C₄×C₄₇	1	1	yes	characteristic
127	#504	—	2	8	no	—
128	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
129	C₂×M₃₂	C₂²×C₄₇	1	4	yes	—
130	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₄₇	1	4	yes	—
132	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
133	#436	C₄×C₄₇	1	4	yes	—
134	C₈×D₄	C₂²×C₄₇	1	4	yes	—
135	#423	C₄×C₄₇	1	1	yes	characteristic
136	#408	C₄×C₄₇	1	2	yes	—
137	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂²×C₄₇	1	4	yes	—
139	C₂²⋊C₁₆	C₂²×C₄₇	1	8	yes	—
140	C₄×C₁₆	C₂²×C₄₇	1	2	yes	—
141	C₄×C₁₆	C₂²×C₄₇	1	2	yes	—
142	C₄₇	#33434	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#41	C₂×C₄×C₄₇	1	2	yes	—
144	#131	—	2	4	no	—
145	#131	—	2	4	no	—
146	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
147	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
148	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
150	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
151	C₄⋊Q₈	C₈×C₄₇	1	1	yes	characteristic
152	#127	C₈×C₄₇	1	1	yes	characteristic
153	#124	C₈×C₄₇	1	2	yes	—
154	#121	C₈×C₄₇	1	4	yes	—
155	C₄×D₄	C₂×C₄×C₄₇	1	4	yes	—
156	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	2	8	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	8	no	—
161	C₂×C₁₆	D₄×C₄₇	1	4	yes	complemented
162	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
163	C₂×C₁₆	C₂³×C₄₇	1	4	yes	—
164	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
165	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
166	C₄⋊C₈	C₂×C₄×C₄₇	1	4	yes	—
167	C₂²⋊C₈	C₂×C₄×C₄₇	1	8	yes	—
168	C₄×C₈	C₂×C₄×C₄₇	1	2	yes	—
169	C₄×C₈	C₂³×C₄₇	1	2	yes	—
170	#41	—	2	4	no	—
171	#41	—	2	4	no	—
172	C₂×Q₈	C₂×C₈×C₄₇	1	2	yes	—
173	C₂×D₄	C₂×C₈×C₄₇	1	4	yes	—
174	C₂×D₄	C₂×C₈×C₄₇	1	4	yes	—
175	C₂×D₄	C₂×C₈×C₄₇	1	2	yes	—
176	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
178	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₄₇	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
186	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
188	C₂×C₈	#41×C₄₇	1	2	yes	—
189	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
190	C₄⋊C₄	C₂×C₈×C₄₇	1	4	yes	—
191	C₂²⋊C₄	C₂×C₈×C₄₇	1	8	yes	—
192	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
193	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
194	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	2	2	no	—
197	C₁₆	C₂×D₄×C₄₇	1	2	yes	—
198	C₁₆	C₂×D₄×C₄₇	1	2	yes	—
199	C₂³	C₂²×C₈×C₄₇	1	4	yes	—
200	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	D₄	—	2	8	no	—
203	D₄	—	2	8	no	—
204	D₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
209	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₄₇	1	4	yes	—
211	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
212	C₂×C₄	C₂²×C₈×C₄₇	1	1	yes	characteristic
213	C₂×C₄	#131×C₄₇	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₈	—	2	2	no	—
217	C₈	C₄×D₄×C₄₇	1	2	yes	—
218	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
219	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	4	no	—
223	C₂²	C₈×D₄×C₄₇	1	2	yes	—
224	C₂²	C₂³×C₈×C₄₇	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	C₈×D₄×C₄₇	1	2	yes	—
228	C₄	C₄₇×#504	1	4	yes	complemented
229	C₄	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
230	C₄	C₂×#131×C₄₇	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	4	no	—
233	C₂	C₂×C₈×D₄×C₄₇	1	1	yes	characteristic
234	C₂	C₂×C₄₇×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆×C₄₇	C₂³
commutator	C₂²	C₂³×C₈×C₄₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
socle	C₂²×C₄₇	C₂³×C₈
prime radical	C₄₇×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#504	C₂	semidirect
3. C₄₇×#2169	C₂	semidirect
4. D₄×C₁₆×C₄₇	C₂	semidirect
5. C₄₇×#2150	C₂	semidirect
6. C₄₇×#2113	C₂	semidirect
7. C₂×C₄×C₁₆×C₄₇	C₂	semidirect
43. C₂×C₁₆×C₄₇	D₄	semidirect
102. #33434	C₄₇	direct (N × H)
111. C₄×C₄₇	#504	semidirect
114. C₂×#504	C₂×C₄₇	semidirect
116. #2169	C₂×C₄₇	semidirect
117. D₄×C₁₆	C₂×C₄₇	semidirect
118. #2150	C₂×C₄₇	semidirect
119. #2113	C₂×C₄₇	semidirect
120. C₂×C₄×C₁₆	C₂×C₄₇	semidirect
142. C₄₇	#33434	direct (N × H)
161. C₂×C₁₆	D₄×C₄₇	semidirect
228. C₄	C₄₇×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#504	2
C₄₇×#2931	2
C₄₇×#2169	2
D₄×C₁₆×C₄₇	2
C₄₇×#2150	2
C₄₇×#2113	2
C₂×C₄×C₁₆×C₄₇	2
#33434	47

Subgroups — C47×#33434 #26073132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33434 #26073132