Subgroups — C₂×#20245002 #26368951

Order 12032.

1190 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245002	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
5	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093964	—	2	4	no	—
7	C₂×#13093964	C₂²	1	4	yes	complemented
8	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	complemented
12	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	16	yes	complemented
13	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
14	C₂×#131×C₄₇	—	2	4	no	—
15	C₂×#131×C₄₇	—	2	4	no	—
16	C₂×#131×C₄₇	C₄	1	4	yes	—
17	C₂×#131×C₄₇	C₂²	1	4	yes	—
18	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
19	C₂²×M₁₆×C₄₇	C₄	1	2	yes	—
20	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#13093964	—	2	8	no	—
23	C₂×C₄₇⋊C₁₆	—	2	4	no	—
24	C₂×C₄₇⋊C₁₆	—	2	8	no	—
25	C₂×C₄₇⋊C₁₆	—	2	4	no	—
26	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
27	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
28	C₂×#41×C₄₇	—	2	4	no	—
29	C₂×#41×C₄₇	—	2	4	no	—
30	C₂×#41×C₄₇	C₈	1	4	yes	—
31	C₂×#41×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
33	C₂²×D₄×C₄₇	C₈	1	2	yes	—
34	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	#131×C₄₇	—	2	16	no	—
38	#131×C₄₇	—	2	8	no	—
39	#131×C₄₇	—	2	8	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	—	2	8	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	—	2	4	no	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₄₇	D₄	1	4	yes	—
51	C₂²×C₈×C₄₇	D₄	1	4	yes	—
52	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂⁴×C₄₇	C₂×C₈	1	2	yes	—
56	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
57	#41×C₄₇	—	2	16	no	—
58	#41×C₄₇	—	2	8	no	—
59	#41×C₄₇	—	2	8	no	—
60	C₂×Q₈×C₄₇	—	2	4	no	—
61	C₂×Q₈×C₄₇	C₂×C₈	1	4	yes	—
62	C₂×D₄×C₄₇	—	2	8	no	—
63	C₂×D₄×C₄₇	—	2	4	no	—
64	C₂×D₄×C₄₇	C₂×C₈	1	8	yes	—
65	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
77	M₁₆×C₄₇	—	2	8	no	—
78	M₁₆×C₄₇	—	2	16	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	8	no	—
81	C₂×C₈×C₄₇	—	2	4	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄₇⋊C₁₆	—	2	4	no	—
94	C₄₇⋊C₁₆	—	2	4	no	—
95	C₂³×C₄₇	—	2	4	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	2	8	no	—
98	C₂³×C₄₇	C₂²⋊C₈	1	4	yes	—
99	C₂³×C₄₇	C₂²×C₈	1	4	yes	—
100	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
101	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
102	Q₈×C₄₇	—	2	8	no	—
103	D₄×C₄₇	—	2	16	no	—
104	D₄×C₄₇	—	2	8	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	8	no	—
109	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
110	C₂×C₄×C₄₇	C₂²⋊C₈	1	8	yes	—
111	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
118	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₄₇	—	2	4	no	—
120	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
121	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
122	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×#2113	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	8	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	—	4	8	no	—
130	C₂²×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
131	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
132	C₂²×C₄₇	#504	1	4	yes	—
133	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
136	C₄×C₄₇	C₂×C₂²⋊C₈	1	4	yes	—
137	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
139	C₂²×#131	D₄₇	1	1	yes	characteristic
140	C₂²×M₃₂	—	47	1	no	—
141	C₂³×C₁₆	—	47	1	no	—
142	#2113	—	47	8	no	—
143	C₂×C₂²⋊C₁₆	—	47	4	no	—
144	C₂×C₄₇	—	2	4	no	—
145	C₂×C₄₇	—	4	4	no	—
146	C₂×C₄₇	#2113	1	4	yes	complemented
147	C₂×C₄₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
148	C₂×C₄₇	C₂×#504	1	2	yes	—
149	C₂²×#41	Dic₄₇	1	1	yes	characteristic
150	C₂×#131	—	2	4	no	—
151	C₂×#131	—	2	4	no	—
152	C₂×#131	Dic₄₇	1	4	yes	—
153	C₂×#131	C₂×D₄₇	1	4	yes	—
154	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
155	C₂²×M₁₆	Dic₄₇	1	2	yes	—
156	C₂³×C₈	C₂×D₄₇	1	2	yes	—
157	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₂×M₃₂	—	47	4	no	—
159	C₂×M₃₂	—	94	4	no	—
160	C₂²×C₁₆	—	47	4	no	—
161	C₂²×C₁₆	—	47	2	no	—
162	C₂²×C₁₆	—	47	2	no	—
163	C₂²×C₁₆	—	94	4	no	—
164	C₂²⋊C₁₆	—	47	16	no	—
165	C₄₇	C₂×#2113	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	2	4	no	—
167	C₂×#41	—	2	4	no	—
168	C₂×#41	C₄₇⋊C₈	1	4	yes	—
169	C₂×#41	C₂×Dic₄₇	1	4	yes	—
170	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
171	C₂²×D₄	C₄₇⋊C₈	1	2	yes	—
172	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
173	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
174	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
175	#131	—	2	16	no	—
176	#131	—	2	8	no	—
177	#131	—	2	8	no	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	8	no	—
180	C₂×M₁₆	C₂×Dic₄₇	1	4	yes	—
181	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
185	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
186	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
189	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
190	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
193	M₃₂	—	94	8	no	—
194	C₂×C₁₆	—	47	4	no	—
195	C₂×C₁₆	—	47	4	no	—
196	C₂×C₁₆	—	94	4	no	—
197	C₂×C₁₆	—	94	8	no	—
198	C₂×C₁₆	—	94	4	no	—
199	C₂⁴	C₂×C₄₇⋊C₈	1	2	yes	—
200	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
201	#41	—	2	16	no	—
202	#41	—	2	8	no	—
203	#41	—	2	8	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	C₂×C₄₇⋊C₈	1	4	yes	—
206	C₂×D₄	—	2	8	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×C₄₇⋊C₈	1	8	yes	—
209	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
213	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
214	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
216	C₂²×C₄	#10625329	1	4	yes	—
217	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
219	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
221	M₁₆	—	2	8	no	—
222	M₁₆	—	2	16	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
228	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
229	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
230	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
234	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
237	C₁₆	—	94	4	no	—
238	C₁₆	—	94	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	4	no	—
242	C₂³	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
243	C₂³	C₂²×C₄₇⋊C₈	1	4	yes	—
244	C₂³	C₂²×C₄₇⋊C₈	1	2	yes	—
245	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
246	Q₈	—	2	8	no	—
247	D₄	—	2	16	no	—
248	D₄	—	2	8	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
254	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	8	yes	—
255	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
257	C₂×C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
258	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
260	C₂×C₄	C₂×#10625329	1	2	yes	—
261	C₂×C₄	C₂×#10625329	1	4	yes	—
262	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
263	C₈	—	2	4	no	—
264	C₈	C₂×C₂²⋊Dic₄₇	1	2	yes	—
265	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
266	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
268	C₂²	—	2	8	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	8	no	—
273	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
274	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
275	C₂²	#15689540	1	4	yes	—
276	C₂²	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
279	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
280	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₄	C₂²×#10625329	1	1	yes	characteristic
282	C₂	—	2	4	no	—
283	C₂	—	4	4	no	—
284	C₂	#20245002	1	4	yes	direct product factor
285	C₂	C₂²×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
286	C₂	C₂×#15689540	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	C₂×#2113
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. C₂³×C₄₇⋊C₁₆	C₂	semidirect
3. #20245002	C₂	direct (N × H)
4. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
7. C₂×#13093964	C₂²	semidirect
11. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
12. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
146. C₂×C₄₇	#2113	semidirect
165. C₄₇	C₂×#2113	semidirect
284. C₂	#20245002	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×#13093964	2
C₂³×C₄₇⋊C₁₆	2
#20245002	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂²×#131×C₄₇	2
C₂×#2113	47

Subgroups — C2×#20245002 #26368951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245002 #26368951