Subgroups — C₄₇×#20689 #26060387

Order 12032.

1602 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#518	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2308	C₂	1	2	yes	maximal
3	C₄₇×#1820	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1819	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₇×#342	C₂	1	4	yes	maximal, complemented
6	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#518	—	2	8	no	—
8	C₂×C₄×D₄×C₄₇	C₄	1	8	yes	complemented
9	C₂×#117×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
11	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
13	C₄₇×#386	—	2	8	no	—
14	C₄₇×#381	C₂²	1	4	yes	—
15	C₄₇×#379	C₂²	1	4	yes	—
16	C₄₇×#342	C₂²	1	16	yes	—
17	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
18	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×D₄×C₄₇	—	2	8	no	—
20	C₂²×D₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂²×D₄×C₄₇	D₄	1	8	yes	—
22	C₂³×C₄×C₄₇	—	2	4	no	—
23	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	complemented
25	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₄₇	—	2	32	no	—
27	#117×C₄₇	—	2	8	no	—
28	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
29	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
30	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
31	C₂×C₂²⋊C₄×C₄₇	—	2	16	no	—
32	C₂×C₂²⋊C₄×C₄₇	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₄₇	—	2	8	no	—
34	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	complemented
36	C₂×C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
37	C₂×C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
39	#95×C₄₇	—	2	8	no	—
40	C₂⁴×C₄₇	—	2	4	no	—
41	C₂⁴×C₄₇	—	8	2	no	—
42	C₂⁴×C₄₇	—	8	2	no	—
43	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
44	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
46	C₂⁴×C₄₇	C₂×D₄	1	4	yes	—
47	C₂⁴×C₄₇	#41	1	8	yes	—
48	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
49	C₂×D₄×C₄₇	—	2	16	no	—
50	C₂×D₄×C₄₇	—	2	16	no	—
51	C₂×D₄×C₄₇	—	2	16	no	—
52	C₂×D₄×C₄₇	C₂²⋊C₄	1	16	yes	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	8	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	8	no	—
57	C₂²×C₄×C₄₇	—	4	8	no	—
58	C₂²×C₄×C₄₇	—	8	8	no	—
59	C₂²×C₄×C₄₇	—	8	4	no	—
60	C₂²×C₄×C₄₇	—	8	4	no	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₄₇	—	2	8	no	—
65	C₂²⋊C₄×C₄₇	—	2	8	no	—
66	C₂²⋊C₄×C₄₇	—	2	8	no	—
67	C₂²⋊C₄×C₄₇	—	2	8	no	—
68	C₂²⋊C₄×C₄₇	—	2	16	no	—
69	C₂²⋊C₄×C₄₇	—	4	8	no	—
70	C₂²⋊C₄×C₄₇	—	8	8	no	—
71	C₄²×C₄₇	—	2	8	no	—
72	C₂³×C₄₇	—	2	4	no	—
73	C₂³×C₄₇	—	2	8	no	—
74	C₂³×C₄₇	—	4	4	no	—
75	C₂³×C₄₇	—	4	4	no	—
76	C₂³×C₄₇	—	4	8	no	—
77	C₂³×C₄₇	—	4	2	no	—
78	C₂³×C₄₇	—	4	8	no	—
79	C₂³×C₄₇	—	8	8	no	—
80	C₂³×C₄₇	—	8	8	no	—
81	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂³×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
83	C₂³×C₄₇	#117	1	8	yes	—
84	C₂³×C₄₇	#123	1	8	yes	—
85	C₂³×C₄₇	#123	1	8	yes	—
86	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
87	C₂³×C₄₇	C₂²×D₄	1	2	yes	—
88	C₂³×C₄₇	C₂×#41	1	4	yes	—
89	D₄×C₄₇	—	2	32	no	—
90	C₂×C₄×C₄₇	—	2	4	no	—
91	C₂×C₄×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	8	no	—
93	C₂×C₄×C₄₇	—	4	8	no	—
94	C₂×C₄×C₄₇	—	4	16	no	—
95	C₂×C₄×C₄₇	—	4	8	no	—
96	C₂×C₄×C₄₇	—	8	8	no	—
97	C₂×C₄×C₄₇	—	8	8	no	—
98	C₂×C₄×C₄₇	—	8	16	no	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
100	#20689	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₄₇	—	2	2	no	—
102	C₂²×C₄₇	—	2	4	no	—
103	C₂²×C₄₇	—	2	8	no	—
104	C₂²×C₄₇	—	2	8	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	4	8	no	—
107	C₂²×C₄₇	—	4	4	no	—
108	C₂²×C₄₇	—	4	8	no	—
109	C₂²×C₄₇	—	4	8	no	—
110	C₂²×C₄₇	—	4	8	no	—
111	C₂²×C₄₇	—	8	8	no	—
112	C₂²×C₄₇	#381	1	8	yes	—
113	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₇	C₂×#117	1	2	yes	—
115	C₂²×C₄₇	C₂×#123	1	4	yes	—
116	C₄×C₄₇	—	2	4	no	—
117	C₄×C₄₇	—	4	8	no	—
118	C₄×C₄₇	—	8	8	no	—
119	C₂×#518	C₂×C₄₇	1	1	yes	characteristic
120	#2308	C₂×C₄₇	1	2	yes	—
121	#1820	C₂×C₄₇	1	4	yes	complemented
122	#1819	C₂×C₄₇	1	4	yes	complemented
123	C₂×#342	C₂×C₄₇	1	4	yes	complemented
124	C₂×C₄₇	—	2	4	no	—
125	C₂×C₄₇	—	2	2	no	—
126	C₂×C₄₇	—	2	8	no	—
127	C₂×C₄₇	—	4	4	no	—
128	C₂×C₄₇	C₂×#381	1	1	yes	characteristic
129	C₂×C₄₇	#2883	1	2	yes	—
130	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
131	#518	—	2	8	no	—
132	C₂×C₄×D₄	C₄×C₄₇	1	8	yes	complemented
133	C₂×#117	C₂²×C₄₇	1	2	yes	—
134	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
135	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₄₇	1	2	yes	—
137	#386	—	2	8	no	—
138	#381	C₂²×C₄₇	1	4	yes	—
139	#379	C₂²×C₄₇	1	4	yes	—
140	#342	C₂²×C₄₇	1	16	yes	—
141	C₄₇	#20689	1	1	yes	characteristic, direct product factor, Sylow
142	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
143	C₂⁵	C₂³×C₄₇	1	1	yes	characteristic
144	C₂²×D₄	—	2	8	no	—
145	C₂²×D₄	C₂×C₄×C₄₇	1	4	yes	—
146	C₂²×D₄	D₄×C₄₇	1	8	yes	—
147	C₂³×C₄	—	2	4	no	—
148	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
149	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	complemented
150	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
151	C₄×D₄	—	2	32	no	—
152	#117	—	2	8	no	—
153	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	complemented
154	C₂×C₂²⋊C₄	—	2	8	no	—
155	C₂×C₂²⋊C₄	—	2	8	no	—
156	C₂×C₂²⋊C₄	—	2	16	no	—
157	C₂×C₂²⋊C₄	—	2	4	no	—
158	C₂×C₂²⋊C₄	—	2	8	no	—
159	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
160	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	complemented
161	C₂×C₂²⋊C₄	C₂³×C₄₇	1	8	yes	—
162	C₂×C₂²⋊C₄	C₂³×C₄₇	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	complemented
164	#95	—	2	8	no	—
165	C₂⁴	—	2	4	no	—
166	C₂⁴	—	8	2	no	—
167	C₂⁴	—	8	2	no	—
168	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
169	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
170	C₂⁴	C₂×D₄×C₄₇	1	4	yes	—
171	C₂⁴	C₂×D₄×C₄₇	1	4	yes	—
172	C₂⁴	#41×C₄₇	1	8	yes	—
173	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
174	C₂×D₄	—	2	16	no	—
175	C₂×D₄	—	2	16	no	—
176	C₂×D₄	—	2	16	no	—
177	C₂×D₄	C₂²⋊C₄×C₄₇	1	16	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	4	8	no	—
183	C₂²×C₄	—	8	8	no	—
184	C₂²×C₄	—	8	4	no	—
185	C₂²×C₄	—	8	4	no	—
186	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
188	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
189	C₄⋊C₄	—	2	8	no	—
190	C₂²⋊C₄	—	2	8	no	—
191	C₂²⋊C₄	—	2	8	no	—
192	C₂²⋊C₄	—	2	8	no	—
193	C₂²⋊C₄	—	2	16	no	—
194	C₂²⋊C₄	—	4	8	no	—
195	C₂²⋊C₄	—	8	8	no	—
196	C₄²	—	2	8	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	4	no	—
200	C₂³	—	4	4	no	—
201	C₂³	—	4	8	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	8	no	—
204	C₂³	—	8	8	no	—
205	C₂³	—	8	8	no	—
206	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
208	C₂³	#117×C₄₇	1	8	yes	—
209	C₂³	#123×C₄₇	1	8	yes	—
210	C₂³	#123×C₄₇	1	8	yes	—
211	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
212	C₂³	C₂²×D₄×C₄₇	1	2	yes	—
213	C₂³	C₂×#41×C₄₇	1	4	yes	—
214	D₄	—	2	32	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	8	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	4	8	no	—
219	C₂×C₄	—	4	16	no	—
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	8	8	no	—
222	C₂×C₄	—	8	8	no	—
223	C₂×C₄	—	8	16	no	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	8	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	8	no	—
230	C₂²	—	4	8	no	—
231	C₂²	—	4	4	no	—
232	C₂²	—	4	8	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	8	no	—
235	C₂²	—	8	8	no	—
236	C₂²	C₄₇×#381	1	8	yes	—
237	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
238	C₂²	C₂×#117×C₄₇	1	2	yes	—
239	C₂²	C₂×#123×C₄₇	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	8	no	—
242	C₄	—	8	8	no	—
243	C₂	—	2	4	no	—
244	C₂	—	2	2	no	—
245	C₂	—	2	8	no	—
246	C₂	—	4	4	no	—
247	C₂	C₂×C₄₇×#381	1	1	yes	characteristic
248	C₂	C₄₇×#2883	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₄₇
Frattini, cosocle	C₂⁴	C₂⁴×C₄₇
prime radical	C₂³×D₄×C₄₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂³×D₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#1820	C₂	semidirect
4. C₄₇×#1819	C₂	semidirect
5. C₂×C₄₇×#342	C₂	semidirect
8. C₂×C₄×D₄×C₄₇	C₄	semidirect
24. C₂³×C₄×C₄₇	C₂×C₄	semidirect
28. C₂×C₄⋊C₄×C₄₇	C₂×C₄	semidirect
35. C₂×C₂²⋊C₄×C₄₇	C₂×C₄	semidirect
38. C₂×C₄²×C₄₇	C₂×C₄	semidirect
100. #20689	C₄₇	direct (N × H)
121. #1820	C₂×C₄₇	semidirect
122. #1819	C₂×C₄₇	semidirect
123. C₂×#342	C₂×C₄₇	semidirect
132. C₂×C₄×D₄	C₄×C₄₇	semidirect
141. C₄₇	#20689	direct (N × H)
149. C₂³×C₄	C₂×C₄×C₄₇	semidirect
153. C₂×C₄⋊C₄	C₂×C₄×C₄₇	semidirect
160. C₂×C₂²⋊C₄	C₂×C₄×C₄₇	semidirect
163. C₂×C₄²	C₂×C₄×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×#518	2
C₄₇×#2308	2
C₄₇×#1820	2
C₄₇×#1819	2
C₂×C₄₇×#342	2
#20689	47

Subgroups — C47×#20689 #26060387

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#20689 #26060387