Subgroups — C₄×#15689230 #26214780

Order 12032.

1086 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×#15689230	C₂	1	2	yes	maximal
2	C₄×Q₈×Dic₄₇	C₂	1	2	yes	maximal
3	#20243287	C₂	1	2	yes	maximal
4	C₄×#13093956	C₂	1	1	yes	maximal, characteristic
5	#20243220	C₂	1	4	yes	maximal
6	C₄×#13093948	C₂	1	2	yes	maximal
7	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	—
10	#15689230	C₄	1	16	yes	direct product factor
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×#13093948	C₂²	1	4	yes	—
13	C₂×#13093934	C₂²	1	2	yes	—
14	C₂×#13093934	C₂²	1	2	yes	—
15	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₄₇	—	2	4	no	—
17	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₇	—	2	2	no	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄²×Dic₄₇	—	2	2	no	—
21	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
22	#15688886	C₂²	1	2	yes	—
23	#15688885	C₂²	1	4	yes	—
24	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₄₇	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₄₇	C₄	1	8	yes	complemented
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₄₇	—	2	8	no	—
32	Q₈×Dic₄₇	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊Dic₄₇	—	2	4	no	—
34	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
35	C₂×C₄×Dic₄₇	—	2	4	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	4	no	—
38	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
40	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
41	#13093956	C₂×C₄	1	8	yes	—
42	#13093948	C₂×C₄	1	16	yes	—
43	#13093934	—	2	4	no	—
44	#13093934	C₂×C₄	1	8	yes	—
45	C₄×C₄₇⋊C₈	—	2	2	no	—
46	C₄₇⋊#95	—	2	2	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
49	C₂×Q₁₆×C₄₇	C₂×C₄	1	8	yes	—
50	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₄₇	—	2	4	no	—
54	C₄×Q₈×C₄₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	2	no	—
58	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
59	#107×C₄₇	C₂×C₄	1	4	yes	—
60	#103×C₄₇	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	4	no	—
64	C₂×Q₈×C₄₇	C₂²×C₄	1	8	yes	—
65	C₂×Q₈×C₄₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
73	Q₁₆×C₄₇	C₄²	1	16	yes	complemented
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄⋊C₄×C₄₇	—	2	8	no	—
79	C₄⋊C₄×C₄₇	—	2	2	no	—
80	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
81	C₄²×C₄₇	—	2	8	no	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	#41	1	2	yes	—
84	C₄⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	16	no	—
86	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
87	C₄×Dic₄₇	—	2	4	no	—
88	C₄×Dic₄₇	—	2	16	no	—
89	C₄×Dic₄₇	—	4	2	no	—
90	C₄×Dic₄₇	—	4	8	no	—
91	C₄×Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
93	C₄×Dic₄₇	#41	1	4	yes	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
97	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₄₇	—	2	8	no	—
99	Q₈×C₄₇	—	2	4	no	—
100	Q₈×C₄₇	C₂×C₄²	1	8	yes	—
101	C₂×C₄×C₄₇	—	2	8	no	—
102	C₂×C₄×C₄₇	—	2	4	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₇	#137	1	4	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
112	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
113	C₂×Dic₄₇	—	2	4	no	—
114	C₂×Dic₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	2	8	no	—
116	C₂×Dic₄₇	—	4	4	no	—
117	C₂×Dic₄₇	—	4	4	no	—
118	C₂×Dic₄₇	C₄×D₄	1	8	yes	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	C₂×C₄²	1	4	yes	—
121	C₄₇⋊C₈	—	2	4	no	—
122	C₄×#441	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	C₂×C₄×D₄	1	2	yes	—
124	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×#137	1	2	yes	—
126	C₂²×C₄₇	#575	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	8	no	—
129	C₄×C₄₇	—	4	4	no	—
130	C₄×C₄₇	#441	1	4	yes	complemented
131	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₄₇	C₄×#41	1	2	yes	—
133	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
134	Dic₄₇	—	2	8	no	—
135	Dic₄₇	—	4	8	no	—
136	C₂×#441	—	47	2	no	—
137	C₂×C₄×Q₁₆	D₄₇	1	1	yes	characteristic
138	C₄²×Q₈	—	47	2	no	—
139	#1776	—	47	2	no	—
140	C₄×#106	—	47	1	no	—
141	#1763	—	47	4	no	—
142	C₄×#103	—	47	2	no	—
143	C₄×#97	—	47	1	no	—
144	C₂×C₄₇	C₄²×D₄	1	1	yes	characteristic
145	C₂×C₄₇	C₂×#441	1	1	yes	characteristic
146	C₂×C₄₇	#2943	1	1	yes	characteristic
147	C₂×C₄₇	C₄×#137	1	4	yes	—
148	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	47	4	no	—
150	C₂×C₄×Q₈	C₂×D₄₇	1	2	yes	—
151	#441	—	47	16	no	—
152	C₄×Q₁₆	Dic₄₇	1	8	yes	complemented
153	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#106	—	47	2	no	—
155	C₂×#103	—	47	4	no	—
156	C₂×#103	C₂×D₄₇	1	2	yes	—
157	C₂×#97	—	47	2	no	—
158	C₂×#97	—	47	2	no	—
159	C₂×C₄×C₈	—	47	1	no	—
160	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₄×C₄⋊C₄	—	47	4	no	—
162	C₄×C₄⋊C₄	—	47	2	no	—
163	C₄×C₄⋊C₄	—	94	2	no	—
164	C₄×C₄⋊C₄	—	94	4	no	—
165	C₄³	—	47	1	no	—
166	C₄³	—	94	2	no	—
167	#340	—	47	2	no	—
168	#340	—	47	4	no	—
169	C₄₇	C₄×#441	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
171	C₂×Q₁₆	C₄×D₄₇	1	8	yes	—
172	C₂×Q₁₆	C₂×Dic₄₇	1	4	yes	—
173	C₂²×C₈	—	47	2	no	—
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
176	C₄×Q₈	—	2	4	no	—
177	C₄×Q₈	—	47	16	no	—
178	C₄×Q₈	—	94	8	no	—
179	C₄×Q₈	C₂×Dic₄₇	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	47	4	no	—
182	C₂×C₄⋊C₄	—	94	4	no	—
183	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	—	94	4	no	—
189	C₂×C₄²	—	94	4	no	—
190	C₂×C₄²	—	94	2	no	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	#107	C₂×Dic₄₇	1	4	yes	—
193	#106	—	47	8	no	—
194	#103	—	47	16	no	—
195	#103	C₂×Dic₄₇	1	8	yes	—
196	#97	—	47	8	no	—
197	#97	—	94	4	no	—
198	C₄×C₈	—	94	2	no	—
199	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
200	#95	—	94	4	no	—
201	#95	—	94	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	C₂×C₄×D₄₇	1	8	yes	—
205	C₂×Q₈	C₂²×Dic₄₇	1	4	yes	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	47	4	no	—
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	Q₁₆	C₄×Dic₄₇	1	16	yes	complemented
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	47	4	no	—
216	C₂×C₈	—	94	2	no	—
217	C₂×C₈	—	94	2	no	—
218	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
219	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
220	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	47	8	no	—
224	C₄⋊C₄	—	94	16	no	—
225	C₄⋊C₄	—	94	4	no	—
226	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
227	C₄²	—	2	8	no	—
228	C₄²	—	47	4	no	—
229	C₄²	—	47	4	no	—
230	C₄²	—	94	16	no	—
231	C₄²	—	94	4	no	—
232	C₄²	—	188	2	no	—
233	C₄²	—	188	8	no	—
234	C₄²	—	188	4	no	—
235	C₄²	#10625352	1	2	yes	—
236	C₄²	C₂²×Dic₄₇	1	2	yes	—
237	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
238	Q₈	—	2	8	no	—
239	Q₈	—	2	4	no	—
240	Q₈	C₂×C₄×Dic₄₇	1	8	yes	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	47	8	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	—	94	8	no	—
250	C₂×C₄	—	188	4	no	—
251	C₂×C₄	—	188	4	no	—
252	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
253	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
254	C₂×C₄	#13094056	1	4	yes	—
255	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
256	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
257	C₂×C₄	#41×D₄₇	1	2	yes	—
258	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	94	4	no	—
261	C₈	C₂×C₄×Dic₄₇	1	4	yes	—
262	C₂²	C₄×D₄×D₄₇	1	2	yes	—
263	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
264	C₂²	C₂×#13094056	1	2	yes	—
265	C₂²	#15689920	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	4	no	—
269	C₄	—	94	8	no	—
270	C₄	—	188	8	no	—
271	C₄	#15689230	1	4	yes	direct product factor
272	C₄	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
273	C₄	C₄×#10625352	1	2	yes	—
274	C₄	#41×Dic₄₇	1	1	yes	characteristic
275	C₂	C₄×D₄×Dic₄₇	1	1	yes	characteristic
276	C₂	C₂×#15689230	1	1	yes	characteristic
277	C₂	#20252189	1	1	yes	characteristic
278	C₂	C₄×#13094056	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×Q₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₄₇
hypocenter	C₄₇	C₄×#441
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
10. #15689230	C₄	direct (N × H)
26. C₄×Q₁₆×C₄₇	C₄	semidirect
73. Q₁₆×C₄₇	C₄²	semidirect
130. C₄×C₄₇	#441	semidirect
152. C₄×Q₁₆	Dic₄₇	semidirect
169. C₄₇	C₄×#441	semidirect
213. Q₁₆	C₄×Dic₄₇	semidirect
271. C₄	#15689230	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×#15689230	2
C₄×Q₈×Dic₄₇	2
#20243287	2
C₄×#13093956	2
#20243220	2
C₄×#13093948	2
C₄×#13093934	2
C₂×C₄×Q₁₆×C₄₇	2
C₄×#441	47

Subgroups — C4×#15689230 #26214780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15689230 #26214780