Subgroups — C₄×C₄₇×#446 #26060215

Order 12032.

958 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#446	C₂	1	2	yes	maximal
2	C₄²×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1771	C₂	1	4	yes	maximal
4	C₄₇×#1770	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₄₇	C₂	1	2	yes	maximal
6	C₄×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₄×#97×C₄₇	C₂	1	2	yes	maximal
8	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₄₇	C₄	1	2	yes	—
10	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#446	C₄	1	16	yes	direct product factor
12	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
14	C₂×#97×C₄₇	C₂²	1	4	yes	—
15	C₂×#97×C₄₇	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
25	C₄³×C₄₇	C₂²	1	1	yes	characteristic
26	C₄³×C₄₇	C₂²	1	2	yes	—
27	C₄₇×#336	C₂²	1	4	yes	—
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
32	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
33	C₄×Q₈×C₄₇	—	2	8	no	—
34	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₇	C₂×C₄	1	8	yes	—
49	C₄⋊C₈×C₄₇	C₂×C₄	1	16	yes	—
50	#97×C₄₇	—	2	8	no	—
51	#97×C₄₇	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₄₇	—	2	4	no	—
53	C₄×C₈×C₄₇	—	2	2	no	—
54	C₄×C₈×C₄₇	—	2	4	no	—
55	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₄₇	Q₈	1	4	yes	complemented
57	#95×C₄₇	C₂×C₄	1	2	yes	—
58	#95×C₄₇	C₂×C₄	1	4	yes	—
59	C₂×Q₈×C₄₇	—	2	4	no	—
60	C₂×Q₈×C₄₇	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₄₇	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₄₇	#41	1	4	yes	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	4	no	—
80	C₄⋊C₄×C₄₇	C₄²	1	8	yes	—
81	C₄⋊C₄×C₄₇	C₄²	1	4	yes	—
82	C₄²×C₄₇	—	2	8	no	—
83	C₄²×C₄₇	—	2	4	no	—
84	C₄²×C₄₇	—	2	8	no	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	M₁₆	1	8	yes	—
88	C₄²×C₄₇	C₂²×C₄	1	4	yes	—
89	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
90	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
91	C₄²×C₄₇	#41	1	2	yes	—
92	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
93	Q₈×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
110	C₂×C₄×C₄₇	#131	1	4	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
115	C₈×C₄₇	—	2	4	no	—
116	C₈×C₄₇	—	2	8	no	—
117	C₈×C₄₇	C₄×Q₈	1	8	yes	complemented
118	C₄×#446	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
122	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
123	C₂²×C₄₇	C₂×#131	1	2	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	—	2	2	no	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	4	4	no	—
129	C₄×C₄₇	C₄×M₁₆	1	4	yes	—
130	C₄×C₄₇	#446	1	8	yes	direct product factor
131	C₄×C₄₇	C₂×C₄×Q₈	1	2	yes	—
132	C₄×C₄₇	C₄×#41	1	2	yes	—
133	C₂×#446	C₂×C₄₇	1	2	yes	—
134	C₄²×Q₈	C₂×C₄₇	1	1	yes	characteristic
135	#1771	C₂×C₄₇	1	4	yes	—
136	#1770	C₂×C₄₇	1	2	yes	—
137	C₄×C₄⋊C₈	C₂×C₄₇	1	2	yes	—
138	C₄×C₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic
139	C₄×#97	C₂×C₄₇	1	2	yes	—
140	C₄²×C₈	C₂×C₄₇	1	1	yes	characteristic
141	C₂×C₄₇	C₄²×Q₈	1	1	yes	characteristic
142	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
143	C₂×C₄₇	C₄×#131	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#446	1	2	yes	—
145	C₂×C₄₇	#2965	1	2	yes	—
146	C₂×C₄×Q₈	C₄×C₄₇	1	2	yes	—
147	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
148	#446	C₄×C₄₇	1	16	yes	direct product factor
149	C₂×C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
151	C₂×#97	C₂²×C₄₇	1	4	yes	—
152	C₂×#97	C₂²×C₄₇	1	4	yes	—
153	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
154	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
156	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
157	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
158	C₄×C₄⋊C₄	C₄×C₄₇	1	4	yes	—
159	C₄×C₄⋊C₄	C₄×C₄₇	1	2	yes	—
160	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
161	C₄×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
162	C₄³	C₂²×C₄₇	1	1	yes	characteristic
163	C₄³	C₂²×C₄₇	1	2	yes	—
164	#336	C₂²×C₄₇	1	4	yes	—
165	#336	C₂²×C₄₇	1	2	yes	—
166	C₄₇	C₄×#446	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
169	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
170	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
171	C₄×Q₈	—	2	8	no	—
172	C₄×Q₈	C₂×C₄×C₄₇	1	8	yes	—
173	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
176	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
179	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
181	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
183	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
185	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₄₇	1	8	yes	—
187	C₄⋊C₈	C₂×C₄×C₄₇	1	16	yes	—
188	#97	—	2	8	no	—
189	#97	C₂×C₄×C₄₇	1	16	yes	—
190	C₄×C₈	—	2	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	C₂×C₄×C₄₇	1	8	yes	—
194	C₄×C₈	Q₈×C₄₇	1	4	yes	complemented
195	#95	C₂×C₄×C₄₇	1	2	yes	—
196	#95	C₂×C₄×C₄₇	1	4	yes	—
197	C₂×Q₈	—	2	4	no	—
198	C₂×Q₈	C₄²×C₄₇	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
200	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
202	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₄₇	1	8	yes	—
213	C₂×C₈	C₂²×C₄×C₄₇	1	4	yes	—
214	C₂×C₈	C₂×Q₈×C₄₇	1	4	yes	—
215	C₂×C₈	#41×C₄₇	1	4	yes	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	C₄²×C₄₇	1	8	yes	—
219	C₄⋊C₄	C₄²×C₄₇	1	4	yes	—
220	C₄²	—	2	8	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	8	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	4	no	—
225	C₄²	M₁₆×C₄₇	1	8	yes	—
226	C₄²	C₂²×C₄×C₄₇	1	4	yes	—
227	C₄²	C₂²×C₄×C₄₇	1	8	yes	—
228	C₄²	C₂×Q₈×C₄₇	1	2	yes	—
229	C₄²	#41×C₄₇	1	2	yes	—
230	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×C₄²×C₄₇	1	2	yes	—
241	C₂×C₄	C₂×C₄²×C₄₇	1	4	yes	—
242	C₂×C₄	C₂×C₄²×C₄₇	1	4	yes	—
243	C₂×C₄	C₂×C₄²×C₄₇	1	2	yes	—
244	C₂×C₄	C₄×Q₈×C₄₇	1	4	yes	—
245	C₂×C₄	C₂×M₁₆×C₄₇	1	4	yes	—
246	C₂×C₄	C₂×M₁₆×C₄₇	1	4	yes	—
247	C₂×C₄	C₂×M₁₆×C₄₇	1	4	yes	—
248	C₂×C₄	#131×C₄₇	1	4	yes	—
249	C₂×C₄	C₂³×C₄×C₄₇	1	2	yes	—
250	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
252	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₄×Q₈×C₄₇	1	8	yes	complemented
256	C₂²	C₂²×C₄²×C₄₇	1	1	yes	characteristic
257	C₂²	C₂×C₄×Q₈×C₄₇	1	1	yes	characteristic
258	C₂²	C₄×#41×C₄₇	1	1	yes	characteristic
259	C₂²	C₂²×M₁₆×C₄₇	1	2	yes	—
260	C₂²	C₂×#131×C₄₇	1	2	yes	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	4	no	—
264	C₄	—	2	4	no	—
265	C₄	—	4	4	no	—
266	C₄	C₄×M₁₆×C₄₇	1	4	yes	—
267	C₄	C₄₇×#446	1	8	yes	direct product factor
268	C₄	C₂×C₄×Q₈×C₄₇	1	2	yes	—
269	C₄	C₄×#41×C₄₇	1	2	yes	—
270	C₂	C₄²×Q₈×C₄₇	1	1	yes	characteristic
271	C₂	C₂×C₄×M₁₆×C₄₇	1	1	yes	characteristic
272	C₂	C₄×#131×C₄₇	1	1	yes	characteristic
273	C₂	C₂×C₄₇×#446	1	2	yes	—
274	C₂	C₄₇×#2965	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₄₇×#446	C₄	direct (N × H)
56. C₄×C₈×C₄₇	Q₈	semidirect
117. C₈×C₄₇	C₄×Q₈	semidirect
118. C₄×#446	C₄₇	direct (N × H)
130. C₄×C₄₇	#446	direct (N × H)
148. #446	C₄×C₄₇	direct (N × H)
166. C₄₇	C₄×#446	direct (N × H)
194. C₄×C₈	Q₈×C₄₇	semidirect
255. C₈	C₄×Q₈×C₄₇	semidirect
267. C₄	C₄₇×#446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#446	2
C₄²×Q₈×C₄₇	2
C₄₇×#1771	2
C₄₇×#1770	2
C₄×C₄⋊C₈×C₄₇	2
C₄×#97×C₄₇	2
C₄²×C₈×C₄₇	2
C₄×#446	47

Subgroups — C4×C47×#446 #26060215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₄₇×#446 #26060215