Subgroups — C₂×#20245145 #26369259

Order 12032.

734 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093963	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
4	#20245145	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625324	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	C₂×#13093981	C₄	1	1	yes	characteristic
12	C₂×#13093963	—	2	4	no	—
13	C₂×#13093963	C₂²	1	4	yes	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₄₇	—	2	4	no	—
17	C₂×C₁₆×D₄₇	C₂²	1	4	yes	—
18	#15689077	C₄	1	4	yes	—
19	#15689077	C₂²	1	4	yes	—
20	C₂×#13093935	C₄	1	1	yes	characteristic
21	C₂×#13093934	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
24	Dic₄₇⋊C₁₆	C₂²	1	4	yes	complemented
25	Dic₄₇⋊C₁₆	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×C₄⋊C₁₆	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625324	—	2	4	no	—
32	C₂×#10625324	C₂×C₄	1	4	yes	—
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₄₇	D₄	1	4	yes	—
37	C₂×C₈×D₄₇	Q₈	1	4	yes	—
38	C₂×C₈×D₄₇	C₂³	1	2	yes	—
39	#13093981	C₈	1	4	yes	—
40	#13093981	C₂×C₄	1	4	yes	—
41	C₂×Dic₄₇⋊C₄	C₈	1	1	yes	characteristic
42	C₂×Dic₄₇⋊C₄	C₈	1	1	yes	characteristic
43	C₂×(C₂×D₄₇)⋊C₄	C₈	1	1	yes	characteristic
44	C₂×(C₂×D₄₇)⋊C₄	C₈	1	1	yes	characteristic
45	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
46	#13093963	—	2	8	no	—
47	C₂×C₄₇⋊C₁₆	—	2	2	no	—
48	C₂×C₄₇⋊C₁₆	—	2	2	no	—
49	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	C₁₆×D₄₇	—	2	8	no	—
52	#13093935	C₂×C₄	1	4	yes	—
53	#13093934	C₂×C₄	1	4	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₄₇	—	2	2	no	—
60	C₂×C₁₆×C₄₇	—	2	2	no	—
61	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
62	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₄₇	C₂³	1	2	yes	—
65	C₂³×D₄₇	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
73	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₄₇	M₁₆	1	4	yes	—
75	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
82	C₄²×C₄₇	C₂×C₈	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
85	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
86	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
87	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
88	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
89	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
94	#10625324	—	2	8	no	—
95	C₈×D₄₇	C₄⋊C₄	1	8	yes	—
96	C₈×D₄₇	C₂×D₄	1	4	yes	—
97	C₈×D₄₇	C₂×Q₈	1	4	yes	—
98	C₁₆×C₄₇	—	2	2	no	—
99	C₁₆×C₄₇	—	2	2	no	—
100	C₄₇⋊C₁₆	—	2	2	no	—
101	C₄₇⋊C₁₆	—	2	2	no	—
102	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₄₇	—	2	2	no	—
104	C₂²×D₄₇	—	2	2	no	—
105	C₂²×D₄₇	C₄⋊C₈	1	4	yes	—
106	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₄₇	—	2	2	no	—
116	C₂×Dic₄₇	—	2	2	no	—
117	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
118	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
119	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
120	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
121	C₄×D₄₇	C₄⋊C₈	1	8	yes	—
122	C₄×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
123	C₄×D₄₇	C₂×M₁₆	1	4	yes	—
124	C₈×C₄₇	—	2	2	no	—
125	C₈×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₄₇⋊C₈	—	2	2	no	—
129	C₄₇⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×#2151	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
132	C₂²×C₄₇	#504	1	2	yes	—
133	C₂²×C₄₇	#504	1	2	yes	—
134	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₄₇	—	2	2	no	—
136	C₂×D₄₇	—	2	2	no	—
137	C₂×D₄₇	—	2	8	no	—
138	C₂×D₄₇	C₂×C₄⋊C₈	1	4	yes	—
139	C₄×C₄₇	—	2	2	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
142	C₄×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
144	Dic₄₇	—	2	2	no	—
145	Dic₄₇	—	2	2	no	—
146	Dic₄₇	C₂×C₄⋊C₈	1	4	yes	—
147	C₂²×M₃₂	—	47	1	no	—
148	C₂³×C₁₆	—	47	1	no	—
149	C₂×#405	—	47	1	no	—
150	#2151	—	47	8	no	—
151	C₂×C₄⋊C₁₆	—	47	1	no	—
152	C₂×C₄⋊C₁₆	—	47	1	no	—
153	C₂×C₄⋊C₁₆	—	47	1	no	—
154	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
155	C₂×C₄₇	#2151	1	4	yes	complemented
156	C₂×C₄₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
158	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
159	D₄₇	—	2	4	no	—
160	C₂²×M₁₆	—	47	1	no	—
161	C₂³×C₈	—	47	1	no	—
162	C₂×#117	—	47	1	no	—
163	C₂×M₃₂	—	47	4	no	—
164	C₂×M₃₂	—	94	4	no	—
165	C₂²×C₁₆	—	47	1	no	—
166	C₂²×C₁₆	—	47	1	no	—
167	C₂²×C₁₆	—	47	4	no	—
168	C₂²×C₁₆	—	94	4	no	—
169	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
170	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
171	#405	—	47	4	no	—
172	#405	—	47	4	no	—
173	C₂×#97	—	47	1	no	—
174	C₂×#97	—	47	1	no	—
175	C₂×C₄×C₈	—	47	1	no	—
176	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
177	C₄⋊C₁₆	—	47	4	no	—
178	C₄⋊C₁₆	—	47	4	no	—
179	C₄⋊C₁₆	—	47	4	no	—
180	C₄⋊C₁₆	C₂×D₄₇	1	4	yes	complemented
181	C₄₇	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	1	no	—
183	C₂×M₁₆	—	47	4	no	—
184	C₂×M₁₆	—	94	4	no	—
185	C₂²×C₈	—	47	1	no	—
186	C₂²×C₈	—	47	2	no	—
187	C₂²×C₈	—	47	4	no	—
188	C₂²×C₈	—	47	4	no	—
189	C₂²×C₈	—	47	2	no	—
190	C₂²×C₈	—	47	1	no	—
191	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	#117	—	47	4	no	—
194	#117	—	47	4	no	—
195	C₂×C₄⋊C₄	—	47	1	no	—
196	C₂×C₄⋊C₄	—	47	1	no	—
197	C₂×C₂²⋊C₄	—	47	1	no	—
198	C₂×C₂²⋊C₄	—	47	1	no	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
201	M₃₂	—	94	8	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	47	2	no	—
205	C₂×C₁₆	—	47	2	no	—
206	C₂×C₁₆	—	94	8	no	—
207	C₂×C₁₆	—	94	2	no	—
208	C₂×C₁₆	—	94	2	no	—
209	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
210	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
211	#97	—	47	4	no	—
212	#97	—	47	4	no	—
213	C₄×C₈	—	47	2	no	—
214	C₄×C₈	—	47	2	no	—
215	C₄×C₈	C₄×D₄₇	1	2	yes	—
216	C₄×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂⁴	—	47	1	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	4	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	4	no	—
222	C₂²×C₄	—	47	2	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	1	no	—
225	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
228	M₁₆	—	94	8	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	47	4	no	—
231	C₂×C₈	—	47	2	no	—
232	C₂×C₈	—	47	8	no	—
233	C₂×C₈	—	47	4	no	—
234	C₂×C₈	—	47	4	no	—
235	C₂×C₈	—	47	2	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
239	C₂×C₈	D₄×D₄₇	1	2	yes	—
240	C₂×C₈	Q₈×D₄₇	1	2	yes	—
241	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
242	C₄⋊C₄	—	47	4	no	—
243	C₄⋊C₄	—	47	4	no	—
244	C₂²⋊C₄	—	47	4	no	—
245	C₂²⋊C₄	—	47	4	no	—
246	C₄²	—	47	2	no	—
247	C₄²	—	47	2	no	—
248	C₄²	C₈×D₄₇	1	2	yes	—
249	C₄²	C₂×C₄×D₄₇	1	2	yes	—
250	C₁₆	—	2	2	no	—
251	C₁₆	—	2	2	no	—
252	C₁₆	—	94	2	no	—
253	C₁₆	—	94	2	no	—
254	C₂³	—	47	2	no	—
255	C₂³	—	47	4	no	—
256	C₂³	—	94	2	no	—
257	C₂³	—	94	2	no	—
258	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	47	2	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	47	4	no	—
264	C₂×C₄	—	47	4	no	—
265	C₂×C₄	—	47	8	no	—
266	C₂×C₄	—	47	2	no	—
267	C₂×C₄	—	47	2	no	—
268	C₂×C₄	—	94	2	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
274	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	47	4	no	—
278	C₈	—	94	2	no	—
279	C₈	C₄⋊C₄×D₄₇	1	2	yes	—
280	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
281	C₈	C₂×Q₈×D₄₇	1	1	yes	characteristic
282	C₂²	—	47	4	no	—
283	C₂²	—	94	8	no	—
284	C₂²	—	94	2	no	—
285	C₂²	—	94	2	no	—
286	C₂²	C₄⋊C₈×D₄₇	1	2	yes	—
287	C₂²	#15689539	1	2	yes	—
288	C₂²	#15689541	1	2	yes	—
289	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	47	4	no	—
293	C₄	—	94	2	no	—
294	C₄	—	94	2	no	—
295	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
296	C₄	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
297	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
298	C₂	—	94	4	no	—
299	C₂	#20245145	1	4	yes	direct product factor
300	C₂	C₂×C₄⋊C₈×D₄₇	1	1	yes	characteristic
301	C₂	C₂×#15689539	1	1	yes	characteristic
302	C₂	C₂×#15689541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂³×D₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2151
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #20245145	C₂	direct (N × H)
5. C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	semidirect
6. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
7. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
8. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
23. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
24. Dic₄₇⋊C₁₆	C₂²	semidirect
25. Dic₄₇⋊C₁₆	C₂²	semidirect
29. C₄₇×C₄⋊C₁₆	C₂²	semidirect
154. C₂×C₄⋊C₁₆	D₄₇	semidirect
155. C₂×C₄₇	#2151	semidirect
180. C₄⋊C₁₆	C₂×D₄₇	semidirect
181. C₄₇	C₂×#2151	semidirect
299. C₂	#20245145	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×#13093963	2
C₂²×C₁₆×D₄₇	2
C₂×#15689077	2
#20245145	2
C₂×C₄⋊(C₄₇⋊C₁₆)	2
C₂×Dic₄₇⋊C₁₆	2
C₂×C₄₇×C₄⋊C₁₆	2
C₂×#2151	47

Subgroups — C2×#20245145 #26369259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245145 #26369259