Subgroups — C₄₇×#33435 #26073133

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#2168	C₂	1	8	yes	maximal, complemented
4	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2113	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#131×C₄₇	C₂²	1	2	yes	—
8	C₄₇×#532	C₄	1	1	yes	characteristic
9	C₄₇×#504	—	2	8	no	—
10	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
11	C₂×M₃₂×C₄₇	C₂²	1	4	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₄₇	C₂²	1	4	yes	—
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#436	C₄	1	4	yes	—
16	C₈×D₄×C₄₇	C₂²	1	4	yes	—
17	C₄₇×#423	C₄	1	1	yes	characteristic
18	C₄₇×#408	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	—
21	C₄₇×C₂²⋊C₁₆	C₂²	1	8	yes	—
22	C₄₇×#346	C₂²	1	4	yes	—
23	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
24	#131×C₄₇	—	2	4	no	—
25	#131×C₄₇	—	2	4	no	—
26	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₇	C₈	1	1	yes	characteristic
32	#127×C₄₇	C₈	1	1	yes	characteristic
33	#124×C₄₇	C₈	1	2	yes	—
34	#121×C₄₇	C₈	1	4	yes	—
35	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₄₇	—	2	4	no	—
38	M₃₂×C₄₇	—	2	8	no	—
39	C₂×C₁₆×C₄₇	—	2	2	no	—
40	C₂×C₁₆×C₄₇	—	2	8	no	—
41	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
42	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
43	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
44	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
45	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂³	1	2	yes	—
49	#41×C₄₇	—	2	4	no	—
50	#41×C₄₇	—	2	4	no	—
51	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
52	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
53	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
54	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	M₁₆×C₄₇	—	2	4	no	—
59	M₁₆×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	4	no	—
62	C₂×C₈×C₄₇	—	2	4	no	—
63	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₄₇	#41	1	2	yes	—
68	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄×C₄₇	C₂×C₈	1	8	yes	—
71	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
74	C₁₆×C₄₇	—	2	4	no	—
75	C₁₆×C₄₇	—	2	2	no	—
76	C₁₆×C₄₇	C₂×D₄	1	4	yes	—
77	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
78	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
79	Q₈×C₄₇	—	2	4	no	—
80	D₄×C₄₇	—	2	8	no	—
81	D₄×C₄₇	—	2	4	no	—
82	D₄×C₄₇	—	4	4	no	—
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	4	2	no	—
86	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	#131	1	2	yes	—
92	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	C₈×C₄₇	—	2	4	no	—
94	C₈×C₄₇	—	2	2	no	—
95	C₈×C₄₇	C₄×D₄	1	2	yes	—
96	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
98	#33435	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₄₇	—	2	2	no	—
100	C₂²×C₄₇	—	2	8	no	—
101	C₂²×C₄₇	—	2	4	no	—
102	C₂²×C₄₇	C₈×D₄	1	2	yes	—
103	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
104	C₄×C₄₇	—	2	2	no	—
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	4	no	—
107	C₄×C₄₇	C₈×D₄	1	2	yes	—
108	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
110	C₂×#504	C₂×C₄₇	1	2	yes	complemented
111	#2931	C₂×C₄₇	1	1	yes	characteristic
112	#2168	C₂×C₄₇	1	8	yes	complemented
113	#2150	C₂×C₄₇	1	1	yes	characteristic, complemented
114	#2113	C₂×C₄₇	1	2	yes	complemented
115	C₂×#346	C₂×C₄₇	1	1	yes	characteristic, complemented
116	C₂×C₄₇	—	2	2	no	—
117	C₂×C₄₇	—	4	4	no	—
118	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
119	C₂×C₄₇	#3407	1	2	yes	—
120	C₂×#131	C₂²×C₄₇	1	2	yes	—
121	#532	C₄×C₄₇	1	1	yes	characteristic
122	#504	—	2	8	no	—
123	C₂×M₃₂	C₂²×C₄₇	1	2	yes	—
124	C₂×M₃₂	C₂²×C₄₇	1	4	yes	—
125	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₄₇	1	4	yes	—
127	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
128	#436	C₄×C₄₇	1	4	yes	—
129	C₈×D₄	C₂²×C₄₇	1	4	yes	—
130	#423	C₄×C₄₇	1	1	yes	characteristic
131	#408	C₄×C₄₇	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₄₇	1	4	yes	—
134	C₂²⋊C₁₆	C₂²×C₄₇	1	8	yes	—
135	#346	C₂²×C₄₇	1	4	yes	—
136	C₄₇	#33435	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#41	C₂×C₄×C₄₇	1	2	yes	—
138	#131	—	2	4	no	—
139	#131	—	2	4	no	—
140	C₂×M₁₆	C₂×C₄×C₄₇	1	2	yes	—
141	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
142	C₂²×C₈	C₂×C₄×C₄₇	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
144	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
145	C₄⋊Q₈	C₈×C₄₇	1	1	yes	characteristic
146	#127	C₈×C₄₇	1	1	yes	characteristic
147	#124	C₈×C₄₇	1	2	yes	—
148	#121	C₈×C₄₇	1	4	yes	—
149	C₄×D₄	C₂×C₄×C₄₇	1	4	yes	—
150	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
151	M₃₂	—	2	4	no	—
152	M₃₂	—	2	8	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	8	no	—
155	C₂×C₁₆	D₄×C₄₇	1	4	yes	complemented
156	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
157	C₂×C₁₆	C₂³×C₄₇	1	4	yes	—
158	C₂×C₁₆	C₂³×C₄₇	1	2	yes	—
159	C₄⋊C₈	C₂×C₄×C₄₇	1	4	yes	—
160	C₂²⋊C₈	C₂×C₄×C₄₇	1	8	yes	—
161	C₄×C₈	C₂×C₄×C₄₇	1	2	yes	—
162	C₄×C₈	C₂³×C₄₇	1	2	yes	—
163	#41	—	2	4	no	—
164	#41	—	2	4	no	—
165	C₂×Q₈	C₂×C₈×C₄₇	1	2	yes	—
166	C₂×D₄	C₂×C₈×C₄₇	1	4	yes	—
167	C₂×D₄	C₂×C₈×C₄₇	1	2	yes	—
168	C₂×D₄	C₂×C₈×C₄₇	1	4	yes	—
169	C₂²×C₄	C₂×C₈×C₄₇	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
172	M₁₆	—	2	4	no	—
173	M₁₆	—	2	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	C₂²×C₄×C₄₇	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
179	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
180	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
181	C₂×C₈	#41×C₄₇	1	2	yes	—
182	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
183	C₄⋊C₄	C₂×C₈×C₄₇	1	4	yes	—
184	C₂²⋊C₄	C₂×C₈×C₄₇	1	8	yes	—
185	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
186	C₄²	C₂×C₈×C₄₇	1	1	yes	characteristic
187	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
188	C₁₆	—	2	4	no	—
189	C₁₆	—	2	2	no	—
190	C₁₆	C₂×D₄×C₄₇	1	4	yes	—
191	C₂³	C₂²×C₈×C₄₇	1	4	yes	—
192	C₂³	C₂²×C₈×C₄₇	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	2	4	no	—
196	D₄	—	4	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
201	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
202	C₂×C₄	C₂²×C₈×C₄₇	1	2	yes	—
203	C₂×C₄	C₂²×C₈×C₄₇	1	4	yes	—
204	C₂×C₄	C₂²×C₈×C₄₇	1	1	yes	characteristic
205	C₂×C₄	#131×C₄₇	1	2	yes	—
206	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
207	C₈	—	2	4	no	—
208	C₈	—	2	2	no	—
209	C₈	C₄×D₄×C₄₇	1	2	yes	—
210	C₈	C₂²×D₄×C₄₇	1	1	yes	characteristic
211	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	2	8	no	—
214	C₂²	—	2	4	no	—
215	C₂²	C₈×D₄×C₄₇	1	2	yes	—
216	C₂²	C₂³×C₈×C₄₇	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	4	no	—
220	C₄	C₈×D₄×C₄₇	1	2	yes	—
221	C₄	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
222	C₄	C₂×#131×C₄₇	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	4	4	no	—
225	C₂	C₂×C₈×D₄×C₄₇	1	1	yes	characteristic
226	C₂	C₄₇×#3407	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₄₇	C₂⁴
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×C₈×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×#2168	C₂	semidirect
4. C₄₇×#2150	C₂	semidirect
5. C₄₇×#2113	C₂	semidirect
6. C₂×C₄₇×#346	C₂	semidirect
41. C₂×C₁₆×C₄₇	D₄	semidirect
98. #33435	C₄₇	direct (N × H)
110. C₂×#504	C₂×C₄₇	semidirect
112. #2168	C₂×C₄₇	semidirect
113. #2150	C₂×C₄₇	semidirect
114. #2113	C₂×C₄₇	semidirect
115. C₂×#346	C₂×C₄₇	semidirect
136. C₄₇	#33435	direct (N × H)
155. C₂×C₁₆	D₄×C₄₇	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×#2168	2
C₄₇×#2150	2
C₄₇×#2113	2
C₂×C₄₇×#346	2
#33435	47

Subgroups — C47×#33435 #26073133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33435 #26073133