Subgroups — C₄×#15689080 #26213619

Order 12032.

1526 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689080	C₂	1	4	yes	maximal
2	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄×#13093981	C₂	1	1	yes	maximal, characteristic
5	#20243076	C₂	1	4	yes	maximal, complemented
6	C₄×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	2	yes	—
11	C₂²×C₈×D₄₇	C₂²	1	2	yes	—
12	C₂×#13093981	C₂²	1	2	yes	—
13	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
14	#15689080	C₄	1	32	yes	direct product factor
15	C₄×#10625324	C₄	1	8	yes	complemented
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×#13093934	C₂²	1	4	yes	—
18	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₇⋊C₈	C₂²	1	4	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
22	C₂×C₈×Dic₄₇	C₂²	1	4	yes	—
23	C₄×C₈×D₄₇	C₄	1	8	yes	complemented
24	C₄×Dic₄₇⋊C₄	C₄	1	4	yes	—
25	C₄×(C₂×D₄₇)⋊C₄	C₄	1	4	yes	—
26	#15688969	C₂²	1	2	yes	—
27	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×#97×C₄₇	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
34	C₂×#10625324	C₂×C₄	1	8	yes	—
35	C₂×#10625324	C₂×C₄	1	16	yes	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
39	C₂×C₈×D₄₇	C₂×C₄	1	16	yes	—
40	#13093981	C₂×C₄	1	16	yes	—
41	C₂×Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
42	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
47	C₄²×D₄₇	C₂×C₄	1	4	yes	—
48	C₄²×D₄₇	C₂×C₄	1	4	yes	—
49	#13093934	C₂×C₄	1	8	yes	—
50	#13093934	C₂×C₄	1	16	yes	complemented
51	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
52	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
53	C₄×C₄₇⋊C₈	C₂×C₄	1	16	yes	complemented
54	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
55	C₈×Dic₄₇	C₂×C₄	1	16	yes	complemented
56	C₄₇⋊#95	C₂×C₄	1	8	yes	—
57	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#97×C₄₇	C₂×C₄	1	16	yes	complemented
63	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
65	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
70	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
71	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×D₄₇	—	2	8	no	—
73	C₂×C₄×D₄₇	C₄²	1	8	yes	—
74	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₄×D₄₇	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₇	C₂²×C₄	1	8	yes	—
85	(C₂×D₄₇)⋊C₄	C₄²	1	16	yes	—
86	Dic₄₇⋊C₄	C₄²	1	16	yes	—
87	C₄×Dic₄₇	—	2	8	no	—
88	C₄×Dic₄₇	C₄²	1	8	yes	—
89	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₄₇	C₂²×C₄	1	8	yes	—
91	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
94	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
96	#10625324	C₄²	1	32	yes	complemented
97	C₈×D₄₇	C₄²	1	32	yes	complemented
98	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
99	C₂²×D₄₇	—	2	4	no	—
100	C₂²×D₄₇	C₂×C₄²	1	4	yes	—
101	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	C₂×C₄²	1	8	yes	—
104	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
105	C₂×C₄×C₄₇	#131	1	8	yes	—
106	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂³×C₄	1	4	yes	—
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
112	C₂×Dic₄₇	C₂×C₄²	1	8	yes	—
113	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
114	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
115	C₄×D₄₇	—	2	16	no	—
116	C₄×D₄₇	C₂×C₄²	1	8	yes	—
117	C₄×D₄₇	C₂×C₄²	1	8	yes	—
118	C₈×C₄₇	C₂×C₄²	1	8	yes	—
119	C₈×C₄₇	C₂×C₄²	1	8	yes	—
120	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
121	C₄₇⋊C₈	C₂×C₄²	1	8	yes	—
122	C₄×#405	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
124	C₂²×C₄₇	C₂²×C₄²	1	2	yes	—
125	C₂²×C₄₇	C₂×#131	1	4	yes	—
126	C₂×D₄₇	—	2	4	no	—
127	C₂×D₄₇	—	2	8	no	—
128	C₂×D₄₇	C₄³	1	4	yes	—
129	C₄×C₄₇	—	2	8	no	—
130	C₄×C₄₇	#405	1	8	yes	complemented
131	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
132	C₄×C₄₇	C₂²×C₄²	1	2	yes	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	—	2	8	no	—
135	Dic₄₇	C₄³	1	4	yes	—
136	C₂×#405	—	47	4	no	—
137	C₂×C₄×M₁₆	—	47	1	no	—
138	C₂²×C₄×C₈	—	47	1	no	—
139	C₄×#117	—	47	1	no	—
140	#1726	—	47	4	no	—
141	C₄×#97	—	47	1	no	—
142	C₄×#97	D₄₇	1	1	yes	characteristic, complemented
143	C₄²×C₈	—	47	1	no	—
144	C₄²×C₈	—	47	1	no	—
145	C₂×C₄₇	C₂×C₄³	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#405	1	2	yes	—
147	C₂×C₄₇	C₄×#131	1	4	yes	—
148	D₄₇	—	2	4	no	—
149	C₂²×M₁₆	—	47	2	no	—
150	C₂³×C₈	—	47	2	no	—
151	C₂×#117	—	47	2	no	—
152	C₂²×C₄²	—	47	1	no	—
153	#405	—	47	32	no	—
154	C₄×M₁₆	—	47	8	no	—
155	C₂×#97	—	47	2	no	—
156	C₂×#97	—	47	4	no	—
157	C₂×#97	C₂×D₄₇	1	4	yes	—
158	C₂×C₄×C₈	—	47	2	no	—
159	C₂×C₄×C₈	—	47	4	no	—
160	C₂×C₄×C₈	—	47	4	no	—
161	C₂×C₄×C₈	—	47	1	no	—
162	C₂×C₄×C₈	—	47	1	no	—
163	C₂×C₄×C₈	—	47	8	no	—
164	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	47	4	no	—
167	C₄×C₂²⋊C₄	—	47	4	no	—
168	#376	—	47	2	no	—
169	C₄³	—	47	1	no	—
170	C₄³	C₂×D₄₇	1	1	yes	characteristic
171	C₄₇	C₄×#405	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	—	47	1	no	—
173	C₂³×C₄	—	47	2	no	—
174	C₂×M₁₆	—	47	8	no	—
175	C₂×M₁₆	—	47	16	no	—
176	C₂²×C₈	—	47	2	no	—
177	C₂²×C₈	—	47	2	no	—
178	C₂²×C₈	—	47	8	no	—
179	C₂²×C₈	—	47	16	no	—
180	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
182	#117	—	47	16	no	—
183	C₂×C₄⋊C₄	—	47	4	no	—
184	C₂×C₂²⋊C₄	—	47	4	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	1	no	—
188	C₂×C₄²	—	47	4	no	—
189	C₂×C₄²	—	47	4	no	—
190	C₂×C₄²	—	47	4	no	—
191	C₂×C₄²	C₄×D₄₇	1	4	yes	—
192	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
193	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
194	#97	—	47	8	no	—
195	#97	—	47	16	no	—
196	#97	C₄×D₄₇	1	16	yes	complemented
197	C₄×C₈	—	47	8	no	—
198	C₄×C₈	—	47	16	no	—
199	C₄×C₈	—	47	4	no	—
200	C₄×C₈	—	47	4	no	—
201	C₄×C₈	—	47	16	no	—
202	C₄×C₈	C₄×D₄₇	1	4	yes	—
203	C₄×C₈	C₄×D₄₇	1	4	yes	—
204	#95	—	47	8	no	—
205	C₂⁴	—	47	1	no	—
206	C₂²×C₄	—	47	1	no	—
207	C₂²×C₄	—	47	2	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	47	8	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	—	47	4	no	—
212	C₂²×C₄	—	47	8	no	—
213	C₂²×C₄	—	94	8	no	—
214	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
215	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	M₁₆	—	47	32	no	—
218	C₂×C₈	—	47	4	no	—
219	C₂×C₈	—	47	8	no	—
220	C₂×C₈	—	47	4	no	—
221	C₂×C₈	—	47	8	no	—
222	C₂×C₈	—	47	32	no	—
223	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
224	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
225	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
226	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
227	C₄⋊C₄	—	47	16	no	—
228	C₂²⋊C₄	—	47	16	no	—
229	C₄²	—	2	8	no	—
230	C₄²	—	47	8	no	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	8	no	—
233	C₄²	—	47	2	no	—
234	C₄²	—	94	8	no	—
235	C₄²	C₂×C₄×D₄₇	1	2	yes	—
236	C₄²	C₂×C₄×D₄₇	1	2	yes	—
237	C₄²	C₂×C₄×D₄₇	1	8	yes	—
238	C₂³	—	47	4	no	—
239	C₂³	—	47	2	no	—
240	C₂³	—	94	4	no	—
241	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	8	no	—
246	C₂×C₄	—	47	8	no	—
247	C₂×C₄	—	47	4	no	—
248	C₂×C₄	—	47	8	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	94	4	no	—
251	C₂×C₄	—	94	16	no	—
252	C₂×C₄	C₄²×D₄₇	1	8	yes	—
253	C₂×C₄	C₄²×D₄₇	1	4	yes	—
254	C₂×C₄	#13094032	1	8	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄×D₄₇	1	4	yes	—
258	C₈	—	47	8	no	—
259	C₈	—	47	8	no	—
260	C₈	C₄²×D₄₇	1	8	yes	—
261	C₈	C₄²×D₄₇	1	8	yes	—
262	C₂²	—	47	4	no	—
263	C₂²	—	94	4	no	—
264	C₂²	—	94	8	no	—
265	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
266	C₂²	C₂×C₄²×D₄₇	1	2	yes	—
267	C₂²	C₂×#13094032	1	4	yes	—
268	C₄	—	2	8	no	—
269	C₄	—	47	4	no	—
270	C₄	—	94	4	no	—
271	C₄	—	94	8	no	—
272	C₄	#15689080	1	8	yes	direct product factor
273	C₄	C₂×C₄²×D₄₇	1	2	yes	—
274	C₄	C₂×C₄²×D₄₇	1	2	yes	—
275	C₂	—	94	4	no	—
276	C₂	C₄³×D₄₇	1	1	yes	characteristic
277	C₂	C₂×#15689080	1	2	yes	—
278	C₂	C₄×#13094032	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#97×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#97	D₄₇
hypocenter	C₄₇	C₄×#405
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
5. #20243076	C₂	semidirect
6. C₄×#13093934	C₂	semidirect
7. C₄²×C₄₇⋊C₈	C₂	semidirect
8. C₄×C₈×Dic₄₇	C₂	semidirect
9. C₄×#97×C₄₇	C₂	semidirect
14. #15689080	C₄	direct (N × H)
15. C₄×#10625324	C₄	semidirect
23. C₄×C₈×D₄₇	C₄	semidirect
50. #13093934	C₂×C₄	semidirect
53. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
55. C₈×Dic₄₇	C₂×C₄	semidirect
62. #97×C₄₇	C₂×C₄	semidirect
96. #10625324	C₄²	semidirect
97. C₈×D₄₇	C₄²	semidirect
130. C₄×C₄₇	#405	semidirect
142. C₄×#97	D₄₇	semidirect
171. C₄₇	C₄×#405	semidirect
196. #97	C₄×D₄₇	semidirect
272. C₄	#15689080	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689080	2
C₂×C₄×#10625324	2
C₂×C₄×C₈×D₄₇	2
C₄×#13093981	2
#20243076	2
C₄×#13093934	2
C₄²×C₄₇⋊C₈	2
C₄×C₈×Dic₄₇	2
C₄×#97×C₄₇	2
C₄×#405	47

Subgroups — C4×#15689080 #26213619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689080 #26213619