Subgroups — C₂×#20243514 #26216460

Order 12032.

1182 conjugacy classes of subgroups.

The subgroup lattice has 290 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	C₂×#15688992	C₂	1	2	yes	maximal
3	#20243514	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688885	C₂	1	2	yes	maximal
6	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
10	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₄	1	8	yes	complemented
13	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
14	#15688992	—	2	4	no	—
15	#15688992	C₂²	1	8	yes	—
16	#15688886	C₂²	1	4	yes	—
17	#15688885	C₂²	1	8	yes	—
18	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₄₇×#340	C₂²	1	4	yes	—
21	C₂³×Dic₄₇	—	2	2	no	—
22	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
23	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	—	2	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×Dic₄₇⋊C₄	—	2	8	no	—
32	C₂×Dic₄₇⋊C₄	—	2	8	no	—
33	C₂×Dic₄₇⋊C₄	—	4	8	no	—
34	C₂×Dic₄₇⋊C₄	—	4	8	no	—
35	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
38	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
39	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
40	C₂³×C₄×C₄₇	—	2	2	no	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₇	—	2	4	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
49	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	4	4	no	—
52	C₂²×C₄×C₄₇	—	4	4	no	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	4	yes	—
58	C₂²×C₄×C₄₇	#41	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	4	4	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
65	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
66	C₂²×Dic₄₇	D₈	1	4	yes	—
67	C₂²×Dic₄₇	SD₁₆	1	8	yes	—
68	C₂²×Dic₄₇	Q₁₆	1	4	yes	—
69	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
70	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₄₇	—	2	8	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
74	C₄⋊C₄×C₄₇	—	2	4	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄⋊Dic₄₇	—	2	4	no	—
78	C₄⋊Dic₄₇	—	2	4	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	Dic₄₇⋊C₄	—	4	16	no	—
81	Dic₄₇⋊C₄	—	4	16	no	—
82	C₄×Dic₄₇	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
84	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
85	C₂×C₄₇⋊C₈	—	2	8	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₇	#124	1	4	yes	—
90	C₂³×C₄₇	C₂×D₈	1	2	yes	—
91	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
92	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	4	4	no	—
95	C₂×C₄×C₄₇	—	4	4	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	—	8	8	no	—
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₇	#117	1	4	yes	—
100	C₂×C₄×C₄₇	#117	1	4	yes	—
101	C₂×C₄×C₄₇	#124	1	4	yes	—
102	C₂×C₄×C₄₇	#125	1	8	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	4	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	8	8	no	—
111	C₂×Dic₄₇	#106	1	16	yes	—
112	C₂×Dic₄₇	#107	1	16	yes	—
113	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×Dic₄₇	C₂×D₈	1	4	yes	—
116	C₂×Dic₄₇	C₂×SD₁₆	1	8	yes	—
117	C₂×Dic₄₇	C₂×Q₁₆	1	4	yes	—
118	C₈×C₄₇	—	2	4	no	—
119	C₄₇⋊C₈	—	2	4	no	—
120	C₂×#1834	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	#382	1	4	yes	—
122	C₂²×C₄₇	C₂×#106	1	4	yes	—
123	C₂²×C₄₇	C₂×#107	1	4	yes	—
124	C₂²×C₄₇	#486	1	8	yes	—
125	C₂²×C₄₇	#487	1	8	yes	—
126	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₇	C₂×#124	1	2	yes	—
128	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
129	C₂²×C₄₇	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄×C₄₇	—	8	4	no	—
132	C₄×C₄₇	#382	1	4	yes	—
133	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₇	C₂×#125	1	2	yes	—
136	Dic₄₇	—	8	4	no	—
137	Dic₄₇	C₂×#106	1	8	yes	—
138	Dic₄₇	C₂×#107	1	8	yes	—
139	C₂²×C₄×C₈	—	47	1	no	—
140	C₂×#384	—	47	2	no	—
141	#1834	—	47	8	no	—
142	C₂×#340	—	47	1	no	—
143	C₂×#340	—	47	2	no	—
144	C₂×#340	D₄₇	1	1	yes	characteristic
145	C₂×C₄₇	#1834	1	8	yes	complemented
146	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
147	C₂×C₄₇	C₂²×#106	1	1	yes	characteristic
148	C₂×C₄₇	C₂²×#107	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#486	1	2	yes	—
150	C₂×C₄₇	C₂×#487	1	2	yes	—
151	C₂³×C₈	—	47	1	no	—
152	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	47	2	no	—
154	C₂²×C₄⋊C₄	—	94	2	no	—
155	C₂²×C₄⋊C₄	—	94	2	no	—
156	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
157	C₂²×C₄²	—	47	1	no	—
158	C₂×C₄×C₈	—	47	8	no	—
159	C₂×C₄×C₈	—	47	4	no	—
160	#384	—	47	8	no	—
161	#384	—	94	4	no	—
162	#340	—	47	4	no	—
163	#340	—	47	8	no	—
164	#340	C₂×D₄₇	1	4	yes	—
165	C₄₇	C₂×#1834	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	—	47	1	no	—
168	C₂³×C₄	—	47	1	no	—
169	C₂³×C₄	—	94	2	no	—
170	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	47	4	no	—
173	C₂²×C₈	—	47	2	no	—
174	C₂²×C₈	—	94	4	no	—
175	C₂²×C₈	C₄×D₄₇	1	4	yes	—
176	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	47	4	no	—
181	C₂×C₄⋊C₄	—	94	4	no	—
182	C₂×C₄⋊C₄	—	94	4	no	—
183	C₂×C₄⋊C₄	—	94	8	no	—
184	C₂×C₄⋊C₄	—	94	8	no	—
185	C₂×C₄⋊C₄	—	94	2	no	—
186	C₂×C₄⋊C₄	—	188	8	no	—
187	C₂×C₄⋊C₄	—	188	8	no	—
188	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
189	C₂×C₄²	—	47	4	no	—
190	C₂×C₄²	—	47	2	no	—
191	C₂×C₄²	—	47	4	no	—
192	C₂×C₄²	—	47	2	no	—
193	C₄×C₈	—	47	16	no	—
194	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	4	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	47	4	no	—
200	C₂²×C₄	—	47	4	no	—
201	C₂²×C₄	—	47	4	no	—
202	C₂²×C₄	—	47	8	no	—
203	C₂²×C₄	—	47	4	no	—
204	C₂²×C₄	—	47	2	no	—
205	C₂²×C₄	—	47	2	no	—
206	C₂²×C₄	—	94	4	no	—
207	C₂²×C₄	—	188	4	no	—
208	C₂²×C₄	—	188	4	no	—
209	C₂²×C₄	—	188	4	no	—
210	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
211	C₂²×C₄	#10625350	1	4	yes	—
212	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
213	C₂²×C₄	#10625352	1	4	yes	—
214	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	47	4	no	—
219	C₂×C₈	—	94	8	no	—
220	C₂×C₈	—	94	4	no	—
221	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	—	94	4	no	—
227	C₄⋊C₄	—	188	16	no	—
228	C₄⋊C₄	—	188	16	no	—
229	C₄⋊C₄	—	188	4	no	—
230	C₄²	—	47	8	no	—
231	C₄²	—	47	4	no	—
232	C₄²	—	47	4	no	—
233	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
234	C₂³	#13094010	1	4	yes	—
235	C₂³	D₈×D₄₇	1	2	yes	—
236	C₂³	SD₁₆×D₄₇	1	4	yes	—
237	C₂³	Q₁₆×D₄₇	1	2	yes	—
238	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	8	no	—
243	C₂×C₄	—	47	4	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	4	no	—
246	C₂×C₄	—	47	8	no	—
247	C₂×C₄	—	47	4	no	—
248	C₂×C₄	—	47	16	no	—
249	C₂×C₄	—	47	16	no	—
250	C₂×C₄	—	376	8	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
255	C₂×C₄	#13093980	1	4	yes	—
256	C₂×C₄	#13093981	1	4	yes	—
257	C₂×C₄	#13094010	1	4	yes	—
258	C₂×C₄	#13094014	1	8	yes	—
259	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×#10625352	1	2	yes	—
262	C₂×C₄	C₂×#10625350	1	2	yes	—
263	C₈	—	2	4	no	—
264	C₈	—	94	4	no	—
265	C₂²	#15688987	1	4	yes	—
266	C₂²	#106×D₄₇	1	4	yes	—
267	C₂²	#107×D₄₇	1	4	yes	—
268	C₂²	#15689468	1	8	yes	—
269	C₂²	#15689472	1	8	yes	—
270	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
271	C₂²	C₂×#13094010	1	2	yes	—
272	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
273	C₂²	C₂×SD₁₆×D₄₇	1	2	yes	—
274	C₂²	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
275	C₄	—	8	4	no	—
276	C₄	—	47	8	no	—
277	C₄	—	47	8	no	—
278	C₄	—	376	4	no	—
279	C₄	#15688987	1	4	yes	—
280	C₄	C₂×#13093980	1	1	yes	characteristic
281	C₄	C₂×#13093981	1	1	yes	characteristic
282	C₄	C₂×#13094014	1	2	yes	—
283	C₂	#20243514	1	8	yes	direct product factor
284	C₂	C₂×#15688987	1	1	yes	characteristic
285	C₂	C₂×#106×D₄₇	1	1	yes	characteristic
286	C₂	C₂×#107×D₄₇	1	1	yes	characteristic
287	C₂	C₂×#15689468	1	2	yes	—
288	C₂	C₂×#15689472	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	C₂×#1834
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
3. #20243514	C₂	direct (N × H)
12. C₂×C₈×Dic₄₇	C₄	semidirect
39. C₈×Dic₄₇	C₂×C₄	semidirect
145. C₂×C₄₇	#1834	semidirect
165. C₄₇	C₂×#1834	semidirect
283. C₂	#20243514	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
C₂×#15688992	2
#20243514	2
C₂×#15688886	2
C₂×#15688885	2
C₂×C₄₇×#340	2
C₂×#1834	47

Subgroups — C2×#20243514 #26216460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20243514 #26216460