Subgroups — C₂×C₄₇×#1296 #26047735

Order 12032.

878 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₄₇	C₂	1	2	yes	maximal
2	C₂×C₄₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1296	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₄₇	—	2	2	no	—
5	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
6	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
9	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
10	C₂×C₄⋊C₈×C₄₇	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
12	C₄₇×#384	—	2	2	no	—
13	C₄₇×#384	C₄	1	2	yes	—
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₂³×C₄×C₄₇	—	2	2	no	—
16	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₄₇	D₄	1	2	yes	—
18	C₂²×C₈×C₄₇	—	2	8	no	—
19	C₂²×C₈×C₄₇	—	2	4	no	—
20	C₂²×C₈×C₄₇	—	4	4	no	—
21	C₂²×C₈×C₄₇	—	4	4	no	—
22	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
26	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
27	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
28	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₄₇	D₄	1	2	yes	—
32	C₂×C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₇	—	2	8	no	—
36	C₄⋊C₈×C₄₇	—	2	4	no	—
37	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	complemented
38	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	—	2	2	no	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	—	2	4	no	—
42	C₂²×C₄×C₄₇	—	4	4	no	—
43	C₂²×C₄×C₄₇	—	4	8	no	—
44	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
48	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₄₇	D₈	1	4	yes	—
51	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
52	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
53	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₇	—	2	8	no	—
57	C₂×C₈×C₄₇	—	4	16	no	—
58	C₂×C₈×C₄₇	—	4	4	no	—
59	C₂×C₈×C₄₇	—	4	4	no	—
60	C₄⋊C₄×C₄₇	—	4	4	no	—
61	C₄⋊C₄×C₄₇	—	4	4	no	—
62	C₄⋊C₄×C₄₇	—	4	8	no	—
63	C₄⋊C₄×C₄₇	—	4	4	no	—
64	C₄⋊C₄×C₄₇	—	4	4	no	—
65	C₄²×C₄₇	C₄²	1	2	yes	—
66	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	4	yes	—
68	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂³×C₄₇	#95	1	2	yes	—
73	C₂³×C₄₇	#100	1	2	yes	—
74	C₂³×C₄₇	#101	1	2	yes	—
75	C₂³×C₄₇	#102	1	4	yes	—
76	C₂³×C₄₇	#103	1	4	yes	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	—	2	4	no	—
79	C₂×C₄×C₄₇	—	2	4	no	—
80	C₂×C₄×C₄₇	—	2	1	no	—
81	C₂×C₄×C₄₇	—	2	1	no	—
82	C₂×C₄×C₄₇	—	2	1	no	—
83	C₂×C₄×C₄₇	—	2	1	no	—
84	C₂×C₄×C₄₇	—	4	4	no	—
85	C₂×C₄×C₄₇	—	4	8	no	—
86	C₂×C₄×C₄₇	—	8	8	no	—
87	C₂×C₄×C₄₇	#95	1	2	yes	—
88	C₂×C₄×C₄₇	#95	1	4	yes	—
89	C₂×C₄×C₄₇	#102	1	4	yes	—
90	C₂×C₄×C₄₇	#102	1	8	yes	—
91	C₂×C₄×C₄₇	#103	1	4	yes	—
92	C₂×C₄×C₄₇	#103	1	8	yes	—
93	C₂×C₄×C₄₇	#106	1	8	yes	—
94	C₂×C₄×C₄₇	#107	1	8	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
102	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
103	C₈×C₄₇	—	4	8	no	—
104	C₂×#1296	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₄₇	#331	1	4	yes	—
106	C₂²×C₄₇	#332	1	8	yes	—
107	C₂²×C₄₇	#333	1	4	yes	—
108	C₂²×C₄₇	#340	1	8	yes	—
109	C₂²×C₄₇	#343	1	4	yes	—
110	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
111	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
112	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
113	C₂²×C₄₇	C₂×#102	1	2	yes	—
114	C₂²×C₄₇	C₂×#103	1	2	yes	—
115	C₄×C₄₇	—	2	2	no	—
116	C₄×C₄₇	—	2	1	no	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	8	4	no	—
119	C₄×C₄₇	#340	1	8	yes	—
120	C₄×C₄₇	C₂×#102	1	2	yes	—
121	C₄×C₄₇	C₂×#103	1	2	yes	—
122	C₄×C₄₇	C₂×#106	1	2	yes	—
123	C₄×C₄₇	C₂×#107	1	2	yes	—
124	C₂²×C₄⋊C₈	C₂×C₄₇	1	2	yes	—
125	C₂×#384	C₂×C₄₇	1	1	yes	characteristic
126	#1296	C₂×C₄₇	1	4	yes	direct product factor
127	C₂×C₄₇	#1296	1	8	yes	direct product factor
128	C₂×C₄₇	C₂×#331	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#332	1	2	yes	—
130	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
131	C₂×C₄₇	C₂×#340	1	2	yes	—
132	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
133	C₂³×C₈	—	2	2	no	—
134	C₂²×C₄⋊C₄	—	2	2	no	—
135	C₂²×C₄⋊C₄	C₄×C₄₇	1	2	yes	—
136	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	2	2	no	—
138	C₂×C₄⋊C₈	—	2	4	no	—
139	C₂×C₄⋊C₈	C₄×C₄₇	1	8	yes	complemented
140	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
141	#384	—	2	2	no	—
142	#384	C₄×C₄₇	1	2	yes	—
143	#384	C₂²×C₄₇	1	2	yes	—
144	C₄₇	C₂×#1296	1	1	yes	characteristic, direct product factor, Sylow
145	C₂³×C₄	—	2	2	no	—
146	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
147	C₂³×C₄	D₄×C₄₇	1	2	yes	—
148	C₂²×C₈	—	2	8	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	4	4	no	—
151	C₂²×C₈	—	4	4	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	4	4	no	—
156	C₂×C₄⋊C₄	—	4	4	no	—
157	C₂×C₄⋊C₄	—	4	4	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
159	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
160	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
161	C₂×C₄²	D₄×C₄₇	1	2	yes	—
162	C₂×C₄²	D₄×C₄₇	1	2	yes	—
163	C₂×C₄²	Q₈×C₄₇	1	2	yes	—
164	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
165	C₄⋊C₈	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	C₂×C₄×C₄₇	1	8	yes	complemented
168	C₂⁴	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	—	4	8	no	—
174	C₂²×C₄	C₄²×C₄₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
176	C₂²×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
177	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
178	C₂²×C₄	C₄⋊C₄×C₄₇	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₄₇	1	4	yes	—
180	C₂²×C₄	D₈×C₄₇	1	4	yes	—
181	C₂²×C₄	SD₁₆×C₄₇	1	4	yes	—
182	C₂²×C₄	SD₁₆×C₄₇	1	4	yes	—
183	C₂²×C₄	Q₁₆×C₄₇	1	4	yes	—
184	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	4	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	—	4	16	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	4	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	8	no	—
195	C₄²	C₄²×C₄₇	1	2	yes	—
196	C₄²	C₂²⋊C₄×C₄₇	1	4	yes	—
197	C₄²	C₄⋊C₄×C₄₇	1	2	yes	—
198	C₄²	C₄⋊C₄×C₄₇	1	4	yes	—
199	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
200	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
201	C₄²	C₂×Q₈×C₄₇	1	1	yes	characteristic
202	C₂³	#95×C₄₇	1	2	yes	—
203	C₂³	#100×C₄₇	1	2	yes	—
204	C₂³	#101×C₄₇	1	2	yes	—
205	C₂³	#102×C₄₇	1	4	yes	—
206	C₂³	#103×C₄₇	1	4	yes	—
207	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	#95×C₄₇	1	2	yes	—
218	C₂×C₄	#95×C₄₇	1	4	yes	—
219	C₂×C₄	#102×C₄₇	1	4	yes	—
220	C₂×C₄	#102×C₄₇	1	8	yes	—
221	C₂×C₄	#103×C₄₇	1	4	yes	—
222	C₂×C₄	#103×C₄₇	1	8	yes	—
223	C₂×C₄	#106×C₄₇	1	8	yes	—
224	C₂×C₄	#107×C₄₇	1	8	yes	—
225	C₂×C₄	C₂×C₄²×C₄₇	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
227	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₄₇	1	2	yes	—
230	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
231	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
232	C₂×C₄	C₂×Q₁₆×C₄₇	1	2	yes	—
233	C₈	—	4	8	no	—
234	C₂²	C₄₇×#331	1	4	yes	—
235	C₂²	C₄₇×#332	1	8	yes	—
236	C₂²	C₄₇×#333	1	4	yes	—
237	C₂²	C₄₇×#340	1	8	yes	—
238	C₂²	C₄₇×#343	1	4	yes	—
239	C₂²	C₂×#95×C₄₇	1	1	yes	characteristic
240	C₂²	C₂×#100×C₄₇	1	1	yes	characteristic
241	C₂²	C₂×#101×C₄₇	1	1	yes	characteristic
242	C₂²	C₂×#102×C₄₇	1	2	yes	—
243	C₂²	C₂×#103×C₄₇	1	2	yes	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	2	no	—
247	C₄	—	8	4	no	—
248	C₄	C₄₇×#340	1	8	yes	—
249	C₄	C₂×#102×C₄₇	1	2	yes	—
250	C₄	C₂×#103×C₄₇	1	2	yes	—
251	C₄	C₂×#106×C₄₇	1	2	yes	—
252	C₄	C₂×#107×C₄₇	1	2	yes	—
253	C₂	C₄₇×#1296	1	8	yes	direct product factor
254	C₂	C₂×C₄₇×#331	1	1	yes	characteristic
255	C₂	C₂×C₄₇×#332	1	2	yes	—
256	C₂	C₂×C₄₇×#333	1	1	yes	characteristic
257	C₂	C₂×C₄₇×#340	1	2	yes	—
258	C₂	C₂×C₄₇×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₇	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₄₇
Frattini, cosocle	C₂×C₄²	C₂³×C₄₇
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#1296	C₂	direct (N × H)
10. C₂×C₄⋊C₈×C₄₇	C₄	semidirect
37. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
104. C₂×#1296	C₄₇	direct (N × H)
126. #1296	C₂×C₄₇	direct (N × H)
127. C₂×C₄₇	#1296	direct (N × H)
139. C₂×C₄⋊C₈	C₄×C₄₇	semidirect
144. C₄₇	C₂×#1296	direct (N × H)
167. C₄⋊C₈	C₂×C₄×C₄₇	semidirect
253. C₂	C₄₇×#1296	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₄₇	2
C₂×C₄₇×#384	2
C₄₇×#1296	2
C₂×#1296	47

Subgroups — C2×C47×#1296 #26047735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₇×#1296 #26047735