Subgroups — C₄₇×#33741 #26073439

Order 12032.

484 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3177	C₂	1	2	yes	maximal
2	C₄₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2245	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2219	C₂	1	2	yes	maximal
6	C₄₇×#2216	C₂	1	2	yes	maximal
7	C₄₇×#2176	C₂	1	2	yes	maximal, complemented
8	C₄×Q₃₂×C₄₇	C₂	1	2	yes	maximal, complemented
9	C₄₇×#2108	C₂	1	1	yes	maximal, characteristic
10	C₄₇×#541	—	2	2	no	—
11	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
12	C₂×Q₃₂×C₄₇	C₂²	1	2	yes	—
13	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#499	C₂²	1	2	yes	—
16	C₄₇×#494	C₂²	1	2	yes	—
17	C₄₇×#491	—	2	2	no	—
18	C₄₇×#470	—	2	2	no	—
19	C₄₇×#467	C₂²	1	4	yes	—
20	C₄₇×#441	—	2	2	no	—
21	C₄×Q₁₆×C₄₇	C₂²	1	4	yes	—
22	C₄₇×#405	C₂²	1	1	yes	characteristic
23	C₄₇×#367	C₂²	1	2	yes	—
24	C₄₇×#366	C₂²	1	2	yes	—
25	C₄₇×#358	C₂²	1	4	yes	—
26	C₄₇×#358	C₂²	1	4	yes	—
27	C₄₇×#346	C₂²	1	1	yes	characteristic
28	C₄₇×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×Q₁₆×C₄₇	C₂³	1	4	yes	—
33	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊Q₈×C₄₇	—	2	2	no	—
36	#126×C₄₇	—	4	4	no	—
37	#125×C₄₇	—	2	2	no	—
38	#125×C₄₇	—	4	4	no	—
39	C₂²⋊Q₈×C₄₇	—	2	4	no	—
40	C₄×Q₈×C₄₇	—	2	4	no	—
41	#117×C₄₇	D₄	1	1	yes	characteristic
42	Q₃₂×C₄₇	—	2	4	no	—
43	Q₃₂×C₄₇	—	2	4	no	—
44	M₃₂×C₄₇	—	2	2	no	—
45	C₂×C₁₆×C₄₇	—	2	2	no	—
46	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
50	#107×C₄₇	C₂³	1	4	yes	—
51	#106×C₄₇	—	2	2	no	—
52	#103×C₄₇	—	2	4	no	—
53	#103×C₄₇	—	2	4	no	—
54	#97×C₄₇	D₄	1	1	yes	characteristic
55	#97×C₄₇	D₄	1	1	yes	characteristic
56	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₄₇	—	2	4	no	—
61	Q₁₆×C₄₇	—	2	4	no	—
62	Q₁₆×C₄₇	—	4	4	no	—
63	M₁₆×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	1	no	—
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₄₇	—	2	4	no	—
71	C₄⋊C₄×C₄₇	—	4	4	no	—
72	C₄⋊C₄×C₄₇	—	4	4	no	—
73	C₄⋊C₄×C₄₇	—	4	4	no	—
74	C₄⋊C₄×C₄₇	—	4	4	no	—
75	C₄⋊C₄×C₄₇	D₈	1	2	yes	—
76	C₂²⋊C₄×C₄₇	—	4	4	no	—
77	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
78	C₄²×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₄₇	—	2	1	no	—
82	C₁₆×C₄₇	—	2	1	no	—
83	C₁₆×C₄₇	#41	1	2	yes	—
84	C₁₆×C₄₇	#41	1	2	yes	—
85	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
86	Q₈×C₄₇	—	4	4	no	—
87	Q₈×C₄₇	—	4	4	no	—
88	C₂×C₄×C₄₇	—	2	1	no	—
89	C₂×C₄×C₄₇	—	2	1	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₄₇	—	2	1	no	—
97	C₈×C₄₇	—	2	1	no	—
98	C₈×C₄₇	#135	1	2	yes	—
99	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
101	#33741	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₄₇	—	2	2	no	—
103	C₂²×C₄₇	—	2	1	no	—
104	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₄₇	—	2	2	no	—
106	C₄×C₄₇	—	2	2	no	—
107	C₄×C₄₇	—	2	1	no	—
108	C₄×C₄₇	—	8	4	no	—
109	C₄×C₄₇	—	8	4	no	—
110	C₄×C₄₇	#532	1	1	yes	characteristic
111	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
112	#3177	C₂×C₄₇	1	2	yes	—
113	#2253	C₂×C₄₇	1	1	yes	characteristic, complemented
114	#2247	C₂×C₄₇	1	1	yes	characteristic, complemented
115	#2245	C₂×C₄₇	1	2	yes	complemented
116	#2219	C₂×C₄₇	1	2	yes	—
117	#2216	C₂×C₄₇	1	2	yes	—
118	#2176	C₂×C₄₇	1	2	yes	complemented
119	C₄×Q₃₂	C₂×C₄₇	1	2	yes	complemented
120	#2108	C₂×C₄₇	1	1	yes	characteristic
121	C₂×C₄₇	—	4	1	no	—
122	C₂×C₄₇	#3169	1	1	yes	characteristic
123	C₂×C₄₇	#3417	1	2	yes	—
124	#541	—	2	2	no	—
125	C₂×Q₃₂	C₂²×C₄₇	1	2	yes	—
126	C₂×Q₃₂	C₂²×C₄₇	1	2	yes	—
127	C₂×M₃₂	C₂²×C₄₇	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
129	#499	C₂²×C₄₇	1	2	yes	—
130	#494	C₂²×C₄₇	1	2	yes	—
131	#491	—	2	2	no	—
132	#470	—	2	2	no	—
133	#467	C₂²×C₄₇	1	4	yes	—
134	#441	—	2	2	no	—
135	C₄×Q₁₆	C₂²×C₄₇	1	4	yes	—
136	#405	C₂²×C₄₇	1	1	yes	characteristic
137	#367	C₂²×C₄₇	1	2	yes	—
138	#366	C₂²×C₄₇	1	2	yes	—
139	#358	C₂²×C₄₇	1	4	yes	—
140	#358	C₂²×C₄₇	1	4	yes	—
141	#346	C₂²×C₄₇	1	1	yes	characteristic
142	#346	C₂²×C₄₇	1	1	yes	characteristic
143	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
144	C₄×C₁₆	C₂²×C₄₇	1	1	yes	characteristic
145	C₄₇	#33741	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	C₂³×C₄₇	1	4	yes	—
148	C₂×M₁₆	D₄×C₄₇	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	2	no	—
151	#126	—	4	4	no	—
152	#125	—	2	2	no	—
153	#125	—	4	4	no	—
154	C₂²⋊Q₈	—	2	4	no	—
155	C₄×Q₈	—	2	4	no	—
156	#117	D₄×C₄₇	1	1	yes	characteristic
157	Q₃₂	—	2	4	no	—
158	Q₃₂	—	2	4	no	—
159	M₃₂	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
164	C₂×C₁₆	C₂³×C₄₇	1	1	yes	characteristic
165	#107	C₂³×C₄₇	1	4	yes	—
166	#106	—	2	2	no	—
167	#103	—	2	4	no	—
168	#103	—	2	4	no	—
169	#97	D₄×C₄₇	1	1	yes	characteristic
170	#97	D₄×C₄₇	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
172	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
173	C₂×Q₈	—	2	4	no	—
174	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	4	4	no	—
178	M₁₆	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
184	C₂×C₈	C₂⁴×C₄₇	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	D₈×C₄₇	1	2	yes	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₂²⋊C₄	D₈×C₄₇	1	2	yes	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
196	C₁₆	—	2	1	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₄₇	1	2	yes	—
199	C₁₆	#41×C₄₇	1	2	yes	—
200	C₂³	C₂×D₈×C₄₇	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
208	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	#135×C₄₇	1	2	yes	—
214	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
215	C₈	C₂×#41×C₄₇	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂²×D₈×C₄₇	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	—	8	4	no	—
224	C₄	C₄₇×#532	1	1	yes	characteristic
225	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
226	C₂	—	4	1	no	—
227	C₂	C₄₇×#3169	1	1	yes	characteristic
228	C₂	C₄₇×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₇
prime radical	C₂³×C₄₇	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ #117×C₄₇ ◁ C₄₇×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2253	C₂	semidirect
3. C₄₇×#2247	C₂	semidirect
4. C₄₇×#2245	C₂	semidirect
7. C₄₇×#2176	C₂	semidirect
8. C₄×Q₃₂×C₄₇	C₂	semidirect
101. #33741	C₄₇	direct (N × H)
113. #2253	C₂×C₄₇	semidirect
114. #2247	C₂×C₄₇	semidirect
115. #2245	C₂×C₄₇	semidirect
118. #2176	C₂×C₄₇	semidirect
119. C₄×Q₃₂	C₂×C₄₇	semidirect
145. C₄₇	#33741	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#3177	2
C₄₇×#2253	2
C₄₇×#2247	2
C₄₇×#2245	2
C₄₇×#2219	2
C₄₇×#2216	2
C₄₇×#2176	2
C₄×Q₃₂×C₄₇	2
C₄₇×#2108	2
#33741	47

Subgroups — C47×#33741 #26073439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#33741 #26073439