Subgroups — C₂×#20243539 #26216656

Order 12032.

1470 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₇⋊C₈	C₂	1	4	yes	maximal
3	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243539	C₂	1	8	yes	maximal, direct product factor
5	C₂²×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
10	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
13	C₂×Dic₄₇⋊C₈	—	2	16	no	—
14	C₂×Dic₄₇⋊C₈	C₂²	1	16	yes	—
15	C₄²×Dic₄₇	C₄	1	4	yes	—
16	C₄²×Dic₄₇	C₂²	1	4	yes	—
17	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₄₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
22	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄₇⋊C₈	—	2	8	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄×Dic₄₇	—	2	4	no	—
27	C₂×C₄×Dic₄₇	—	2	8	no	—
28	C₂×C₄×Dic₄₇	C₈	1	8	yes	complemented
29	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₄₇	D₄	1	8	yes	—
34	C₂×C₄×Dic₄₇	Q₈	1	8	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
36	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
37	Dic₄₇⋊C₈	—	2	32	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	8	no	—
42	C₂²×C₈×C₄₇	—	2	4	no	—
43	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	4	yes	—
46	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
48	C₄⋊C₈×C₄₇	—	2	8	no	—
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	2	4	no	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
66	C₂²×Dic₄₇	C₄⋊C₄	1	8	yes	—
67	C₂²×Dic₄₇	C₂×C₈	1	8	yes	—
68	C₂²×Dic₄₇	M₁₆	1	8	yes	—
69	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
70	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	8	no	—
73	C₂×C₈×C₄₇	—	4	8	no	—
74	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₄₇	C₂×D₄	1	4	yes	—
76	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
77	C₄×Dic₄₇	—	2	8	no	—
78	C₄×Dic₄₇	—	2	16	no	—
79	C₄×Dic₄₇	C₄⋊C₄	1	16	yes	—
80	C₄×Dic₄₇	C₂×C₈	1	16	yes	complemented
81	C₄×Dic₄₇	C₂×D₄	1	8	yes	—
82	C₄×Dic₄₇	C₂×Q₈	1	8	yes	—
83	C₂×C₄₇⋊C₈	—	2	8	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	4	8	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
89	C₂³×C₄₇	C₂×M₁₆	1	4	yes	—
90	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
91	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	C₄⋊C₈	1	16	yes	complemented
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₇	#127	1	4	yes	—
100	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₄⋊Q₈	1	8	yes	—
102	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
107	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	16	no	—
111	C₂×Dic₄₇	C₄⋊C₈	1	32	yes	complemented
112	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×Dic₄₇	C₂²×C₈	1	8	yes	—
115	C₂×Dic₄₇	C₂×M₁₆	1	8	yes	—
116	C₈×C₄₇	—	4	4	no	—
117	C₄₇⋊C₈	—	4	4	no	—
118	C₂×#1842	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	#384	1	4	yes	—
120	C₂²×C₄₇	C₂×C₄⋊C₈	1	8	yes	—
121	C₂²×C₄₇	C₂×C₄⋊C₈	1	4	yes	—
122	C₂²×C₄₇	#423	1	8	yes	—
123	C₂²×C₄₇	#423	1	4	yes	—
124	C₂²×C₄₇	C₂²×C₄⋊C₄	1	2	yes	—
125	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
127	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
128	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	#384	1	4	yes	—
131	C₄×C₄₇	C₂×C₄⋊C₈	1	8	yes	complemented
132	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
133	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₄₇	C₂×C₄⋊Q₈	1	2	yes	—
135	Dic₄₇	—	2	8	no	—
136	Dic₄₇	C₂×C₄⋊C₈	1	16	yes	complemented
137	C₂²×C₄⋊C₈	—	47	1	no	—
138	C₂²×C₄⋊C₈	—	47	4	no	—
139	C₂²×C₄⋊C₈	D₄₇	1	1	yes	characteristic
140	C₂×C₄³	—	47	1	no	—
141	#1842	—	47	8	no	—
142	C₂×C₄₇	#1842	1	8	yes	complemented
143	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
144	C₂×C₄₇	C₂²×C₄⋊C₈	1	2	yes	—
145	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#423	1	2	yes	—
147	C₂×C₄₇	C₂×#423	1	1	yes	characteristic
148	C₂³×C₈	—	47	2	no	—
149	C₂³×C₈	C₂×D₄₇	1	2	yes	—
150	C₂²×C₄²	—	47	1	no	—
151	C₂²×C₄²	—	47	2	no	—
152	C₂²×C₄²	—	47	2	no	—
153	C₂²×C₄²	—	47	1	no	—
154	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
155	C₂×C₄⋊C₈	—	2	4	no	—
156	C₂×C₄⋊C₈	—	47	16	no	—
157	C₂×C₄⋊C₈	—	47	4	no	—
158	C₂×C₄⋊C₈	—	94	16	no	—
159	C₂×C₄⋊C₈	—	94	4	no	—
160	C₂×C₄⋊C₈	C₂×D₄₇	1	4	yes	—
161	C₄³	—	47	4	no	—
162	C₄³	—	47	4	no	—
163	C₄₇	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	—	47	2	no	—
165	C₂³×C₄	—	47	2	no	—
166	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×C₈	—	2	8	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	47	4	no	—
172	C₂²×C₈	—	94	8	no	—
173	C₂²×C₈	—	94	4	no	—
174	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
175	C₂×C₄²	—	47	8	no	—
176	C₂×C₄²	—	47	4	no	—
177	C₂×C₄²	—	47	4	no	—
178	C₂×C₄²	—	47	8	no	—
179	C₂×C₄²	—	47	8	no	—
180	C₂×C₄²	—	47	4	no	—
181	C₂×C₄²	—	47	8	no	—
182	C₂×C₄²	—	47	4	no	—
183	C₂×C₄²	—	94	8	no	—
184	C₂×C₄²	—	94	4	no	—
185	C₂×C₄²	Dic₉₄	1	4	yes	—
186	C₂×C₄²	C₄×D₄₇	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
188	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
189	C₄⋊C₈	—	2	8	no	—
190	C₄⋊C₈	—	94	32	no	—
191	C₄⋊C₈	—	94	8	no	—
192	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	47	8	no	—
195	C₂²×C₄	—	47	8	no	—
196	C₂²×C₄	—	47	8	no	—
197	C₂²×C₄	—	47	4	no	—
198	C₂²×C₄	—	47	8	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	—	94	4	no	—
202	C₂²×C₄	C₈×D₄₇	1	4	yes	—
203	C₂²×C₄	#10625324	1	4	yes	—
204	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
205	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
206	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
210	C₂²×C₄	D₄×D₄₇	1	4	yes	—
211	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
212	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
213	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
214	C₂×C₈	—	2	8	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	4	8	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	188	8	no	—
220	C₄²	—	47	16	no	—
221	C₄²	—	47	16	no	—
222	C₄²	—	47	8	no	—
223	C₄²	—	47	8	no	—
224	C₄²	—	94	16	no	—
225	C₄²	—	94	8	no	—
226	C₄²	Dic₄₇⋊C₄	1	8	yes	—
227	C₄²	C₂×Dic₉₄	1	4	yes	—
228	C₄²	C₂×C₂²⋊D₄₇	1	4	yes	—
229	C₂³	C₄⋊C₄×D₄₇	1	4	yes	—
230	C₂³	C₂×Dic₄₇⋊C₄	1	2	yes	—
231	C₂³	C₂×C₈×D₄₇	1	2	yes	—
232	C₂³	M₁₆×D₄₇	1	4	yes	—
233	C₂³	C₂×#10625324	1	2	yes	—
234	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	47	8	no	—
238	C₂×C₄	—	47	8	no	—
239	C₂×C₄	—	47	8	no	—
240	C₂×C₄	—	47	32	no	—
241	C₂×C₄	—	47	8	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	94	16	no	—
245	C₂×C₄	Dic₄₇⋊C₈	1	16	yes	complemented
246	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
247	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
248	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
249	C₂×C₄	C₂×Dic₄₇⋊C₄	1	4	yes	—
250	C₂×C₄	#13094020	1	4	yes	—
251	C₂×C₄	#13094022	1	4	yes	—
252	C₂×C₄	#13094023	1	8	yes	—
253	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
254	C₂×C₄	C₂×#10625324	1	4	yes	—
255	C₂×C₄	C₂×D₄×D₄₇	1	2	yes	—
256	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
258	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
259	C₈	—	4	4	no	—
260	C₈	—	188	4	no	—
261	C₂²	#15688992	1	4	yes	—
262	C₂²	C₄⋊C₈×D₄₇	1	8	yes	—
263	C₂²	C₂×Dic₄₇⋊C₈	1	4	yes	—
264	C₂²	#15689145	1	8	yes	—
265	C₂²	#15689146	1	4	yes	—
266	C₂²	C₂×C₄⋊C₄×D₄₇	1	2	yes	—
267	C₂²	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
268	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
269	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
270	C₂²	C₂²×#10625324	1	1	yes	characteristic
271	C₄	—	2	4	no	—
272	C₄	—	47	16	no	—
273	C₄	—	94	8	no	—
274	C₄	#15688992	1	4	yes	—
275	C₄	C₂×Dic₄₇⋊C₈	1	8	yes	complemented
276	C₄	C₂×#13094020	1	1	yes	characteristic
277	C₄	C₂×#13094022	1	1	yes	characteristic
278	C₄	C₂×#13094023	1	2	yes	—
279	C₂	#20243539	1	8	yes	direct product factor
280	C₂	C₂×#15688992	1	1	yes	characteristic
281	C₂	C₂×C₄⋊C₈×D₄₇	1	2	yes	—
282	C₂	C₂²×Dic₄₇⋊C₈	1	1	yes	characteristic
283	C₂	C₂×#15689145	1	2	yes	—
284	C₂	C₂×#15689146	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂²×C₄⋊C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₄₇
hypocenter	C₄₇	C₂×#1842
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #20243539	C₂	direct (N × H)
28. C₂×C₄×Dic₄₇	C₈	semidirect
80. C₄×Dic₄₇	C₂×C₈	semidirect
94. C₂×C₄×C₄₇	C₄⋊C₈	semidirect
111. C₂×Dic₄₇	C₄⋊C₈	semidirect
131. C₄×C₄₇	C₂×C₄⋊C₈	semidirect
136. Dic₄₇	C₂×C₄⋊C₈	semidirect
142. C₂×C₄₇	#1842	semidirect
163. C₄₇	C₂×#1842	semidirect
245. C₂×C₄	Dic₄₇⋊C₈	semidirect
275. C₄	C₂×Dic₄₇⋊C₈	semidirect
279. C₂	#20243539	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⋊(C₄₇⋊C₈)	2
C₂²×Dic₄₇⋊C₈	2
C₂×C₄²×Dic₄₇	2
#20243539	2
C₂²×C₄⋊C₈×C₄₇	2
C₂×#1842	47

Subgroups — C2×#20243539 #26216656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243539 #26216656