Subgroups — C₂×#20245188 #26369345

Order 12032.

686 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20245188	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₄₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688898	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₄	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
12	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	Dic₄₇⋊C₁₆	C₂²	1	8	yes	complemented
17	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	—
18	#15688898	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
21	C₄₇×C₂²⋊C₁₆	C₂²	1	4	yes	—
22	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
23	C₂³×Dic₄₇	C₈	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	C₈	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄₇⋊C₁₆	—	2	4	no	—
35	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
36	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
37	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
39	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
40	C₈×Dic₄₇	C₂³	1	2	yes	—
41	C₈×Dic₄₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₄₇	—	2	4	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₄₇	—	2	4	no	—
49	C₂×C₁₆×C₄₇	C₂³	1	4	yes	—
50	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	4	no	—
58	C₂²×Dic₄₇	—	2	4	no	—
59	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
60	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
64	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	#41	1	4	yes	—
71	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
72	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
73	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
74	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
75	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
76	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
77	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	2	8	no	—
80	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
81	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
83	C₁₆×C₄₇	—	2	4	no	—
84	C₄₇⋊C₁₆	—	2	4	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	M₃₂	1	4	yes	—
88	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
89	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	#117	1	2	yes	—
95	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	8	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	8	no	—
104	C₂×Dic₄₇	M₃₂	1	4	yes	—
105	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
106	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
107	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
108	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
109	C₈×C₄₇	—	2	2	no	—
110	C₈×C₄₇	#117	1	2	yes	—
111	C₈×C₄₇	C₂×#41	1	2	yes	—
112	C₄₇⋊C₈	—	2	2	no	—
113	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
114	C₂×#2163	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	2	no	—
117	C₂²×C₄₇	—	2	8	no	—
118	C₂²×C₄₇	#431	1	2	yes	—
119	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
120	C₂²×C₄₇	C₂×M₃₂	1	4	yes	—
121	C₂²×C₄₇	C₂×M₃₂	1	2	yes	—
122	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	#431	1	2	yes	—
125	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
126	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₄₇	—	2	2	no	—
128	Dic₄₇	—	2	2	no	—
129	Dic₄₇	—	2	2	no	—
130	Dic₄₇	C₂×M₃₂	1	4	yes	—
131	C₂²×C₄×C₈	—	47	1	no	—
132	#2163	—	47	8	no	—
133	C₂×C₄⋊C₁₆	—	47	2	no	—
134	C₂×C₂²⋊C₁₆	—	47	1	no	—
135	C₂×C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic
136	C₂×#346	—	47	2	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	#2163	1	4	yes	complemented
139	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
140	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
141	C₂×C₄₇	C₂²×M₃₂	1	1	yes	characteristic
142	C₂³×C₈	—	47	1	no	—
143	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂²×C₄²	—	47	1	no	—
145	C₂²×C₁₆	—	47	2	no	—
146	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
147	C₂×C₄×C₈	—	47	1	no	—
148	C₂×C₄×C₈	—	47	4	no	—
149	C₂×C₄×C₈	—	47	1	no	—
150	C₂×C₄×C₈	—	47	4	no	—
151	C₂×C₄×C₈	—	47	1	no	—
152	C₂×C₄×C₈	—	47	1	no	—
153	C₄⋊C₁₆	—	47	8	no	—
154	C₂²⋊C₁₆	—	47	4	no	—
155	C₂²⋊C₁₆	C₂×D₄₇	1	4	yes	—
156	#346	—	47	8	no	—
157	C₄₇	C₂×#2163	1	1	yes	characteristic, complemented, Sylow
158	C₂³×C₄	—	47	1	no	—
159	C₂³×C₄	—	47	1	no	—
160	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
161	C₂²×C₈	—	2	4	no	—
162	C₂²×C₈	—	47	1	no	—
163	C₂²×C₈	—	47	4	no	—
164	C₂²×C₈	—	47	1	no	—
165	C₂²×C₈	—	94	4	no	—
166	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₄×D₄₇	1	2	yes	—
168	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
169	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×C₄²	—	47	1	no	—
171	C₂×C₄²	—	47	4	no	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	47	4	no	—
175	C₂×C₄²	—	47	1	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	47	4	no	—
178	C₂×C₁₆	—	94	4	no	—
179	C₂×C₁₆	C₂²×D₄₇	1	4	yes	—
180	C₄×C₈	—	47	4	no	—
181	C₄×C₈	—	47	2	no	—
182	C₄×C₈	—	47	4	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	47	2	no	—
185	C₄×C₈	—	47	2	no	—
186	C₂⁴	C₈×D₄₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	47	1	no	—
190	C₂²×C₄	—	47	4	no	—
191	C₂²×C₄	—	47	1	no	—
192	C₂²×C₄	—	47	4	no	—
193	C₂²×C₄	—	47	1	no	—
194	C₂²×C₄	—	47	1	no	—
195	C₂²×C₄	—	94	4	no	—
196	C₂²×C₄	—	94	4	no	—
197	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	C₈×D₄₇	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	47	4	no	—
205	C₂×C₈	—	47	2	no	—
206	C₂×C₈	—	47	2	no	—
207	C₂×C₈	—	94	8	no	—
208	C₂×C₈	—	94	2	no	—
209	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂×C₈	#10625352	1	4	yes	—
212	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
213	C₄²	—	47	4	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	47	4	no	—
216	C₄²	—	47	2	no	—
217	C₄²	—	47	2	no	—
218	C₄²	—	47	2	no	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	94	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	#13093963	1	4	yes	—
224	C₂³	C₂×C₈×D₄₇	1	2	yes	—
225	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	47	4	no	—
231	C₂×C₄	—	47	2	no	—
232	C₂×C₄	—	47	2	no	—
233	C₂×C₄	—	47	2	no	—
234	C₂×C₄	—	47	2	no	—
235	C₂×C₄	—	94	8	no	—
236	C₂×C₄	—	94	2	no	—
237	C₂×C₄	—	94	8	no	—
238	C₂×C₄	—	94	2	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	#13093980	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	47	4	no	—
247	C₈	—	94	2	no	—
248	C₈	#13093980	1	2	yes	—
249	C₈	C₂×#10625352	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	#15689173	1	2	yes	—
254	C₂²	M₃₂×D₄₇	1	2	yes	—
255	C₂²	C₂×#13093963	1	4	yes	—
256	C₂²	C₂×#13093963	1	2	yes	—
257	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	47	4	no	—
260	C₄	—	94	2	no	—
261	C₄	—	94	2	no	—
262	C₄	—	94	2	no	—
263	C₄	#15689173	1	2	yes	—
264	C₄	C₂×#13093980	1	1	yes	characteristic
265	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
266	C₂	—	2	4	no	—
267	C₂	#20245188	1	4	yes	direct product factor
268	C₂	C₂×#15689173	1	1	yes	characteristic
269	C₂	C₂×M₃₂×D₄₇	1	1	yes	characteristic
270	C₂	C₂²×#13093963	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×C₂²⋊C₁₆	C₂
socle	C₂³×C₄₇	C₂²×C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	C₂×#2163
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20245188	C₂	direct (N × H)
3. C₂×Dic₄₇⋊C₁₆	C₂	semidirect
5. C₂×#15688898	C₂	semidirect
16. Dic₄₇⋊C₁₆	C₂²	semidirect
18. #15688898	C₂²	semidirect
138. C₂×C₄₇	#2163	semidirect
157. C₄₇	C₂×#2163	semidirect
267. C₂	#20245188	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×C₈×Dic₄₇	2
#20245188	2
C₂×Dic₄₇⋊C₁₆	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂×#15688898	2
C₂×C₄₇×C₂²⋊C₁₆	2
C₂×#2163	47

Subgroups — C2×#20245188 #26369345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245188 #26369345