Subgroups — C₄×#14582660 #25001771

Order 11008.

1214 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13063284	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582660	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063284	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₄₃	—	2	2	no	—
7	C₂²×C₄×Dic₄₃	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₄₃	C₄	1	4	yes	complemented
9	C₄×C₂²⋊Dic₄₃	—	2	2	no	—
10	#14582745	—	2	1	no	—
11	C₄²×Dic₄₃	—	2	2	no	—
12	C₄²×Dic₄₃	—	2	1	no	—
13	C₄²×Dic₄₃	C₄	1	4	yes	complemented
14	#14582660	C₄	1	16	yes	direct product factor
15	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₄₃	—	2	2	no	—
18	C₄×M₁₆×C₄₃	C₄	1	4	yes	complemented
19	C₂×#97×C₄₃	—	2	1	no	—
20	C₂×C₄×C₈×C₄₃	—	2	1	no	—
21	C₂³×Dic₄₃	—	2	2	no	—
22	#13063284	—	2	4	no	—
23	#13063284	C₂×C₄	1	8	yes	—
24	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄×Dic₄₃	—	2	16	no	—
27	C₂×C₄×Dic₄₃	—	2	8	no	—
28	C₂×C₄×Dic₄₃	—	2	2	no	—
29	C₂×C₄×Dic₄₃	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	4	no	—
31	C₂×C₄×Dic₄₃	—	2	4	no	—
32	C₂×C₄×Dic₄₃	C₂×C₄	1	8	yes	—
33	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
34	C₄₃⋊#95	—	2	4	no	—
35	C₂³×C₄×C₄₃	D₄	1	2	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₃	—	2	2	no	—
38	C₂×M₁₆×C₄₃	—	2	4	no	—
39	C₂×M₁₆×C₄₃	C₂×C₄	1	8	yes	—
40	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
49	#97×C₄₃	—	2	4	no	—
50	C₄×C₈×C₄₃	—	2	4	no	—
51	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	8	no	—
53	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	4	yes	—
65	C₂²×Dic₄₃	—	2	16	no	—
66	C₂²×Dic₄₃	—	2	4	no	—
67	C₂²×Dic₄₃	—	2	4	no	—
68	C₂²×Dic₄₃	—	2	8	no	—
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	8	no	—
71	M₁₆×C₄₃	C₄²	1	8	yes	complemented
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	8	no	—
74	C₄²×C₄₃	C₄²	1	4	yes	—
75	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
80	C₂²⋊Dic₄₃	—	2	8	no	—
81	C₄⋊Dic₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	C₄²	1	8	yes	complemented
83	C₄×Dic₄₃	—	2	32	no	—
84	C₄×Dic₄₃	—	2	4	no	—
85	C₄×Dic₄₃	—	2	8	no	—
86	C₄×Dic₄₃	—	2	8	no	—
87	C₄×Dic₄₃	—	4	8	no	—
88	C₄×Dic₄₃	C₄²	1	8	yes	complemented
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₃	C₄×D₄	1	4	yes	—
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	8	no	—
96	C₂×C₄×C₄₃	—	2	8	no	—
97	C₂×C₄×C₄₃	#95	1	4	yes	—
98	C₂×C₄×C₄₃	#95	1	4	yes	—
99	C₂×C₄×C₄₃	#104	1	8	yes	—
100	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#117	1	2	yes	—
108	C₂×C₄×C₄₃	#117	1	2	yes	—
109	C₂×C₄×C₄₃	#117	1	2	yes	—
110	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
113	C₂×Dic₄₃	—	2	16	no	—
114	C₂×Dic₄₃	—	2	32	no	—
115	C₂×Dic₄₃	—	2	8	no	—
116	C₂×Dic₄₃	—	2	8	no	—
117	C₂×Dic₄₃	—	4	8	no	—
118	C₈×C₄₃	—	2	8	no	—
119	C₄×#339	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	—	2	8	no	—
123	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
124	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
125	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
126	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
127	C₂²×C₄₃	C₂×#104	1	4	yes	—
128	C₄×C₄₃	—	2	4	no	—
129	C₄×C₄₃	#339	1	8	yes	complemented
130	C₄×C₄₃	C₄³	1	1	yes	characteristic
131	C₄×C₄₃	#376	1	1	yes	characteristic
132	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
133	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
134	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
135	Dic₄₃	—	2	16	no	—
136	Dic₄₃	—	4	8	no	—
137	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
138	C₄×#117	—	43	1	no	—
139	C₂×C₄³	—	43	1	no	—
140	C₂×#339	—	43	4	no	—
141	C₂×C₄₃	—	2	2	no	—
142	C₂×C₄₃	—	2	2	no	—
143	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
144	C₂×C₄₃	C₂×#339	1	2	yes	—
145	C₂×C₄₃	C₄×#104	1	4	yes	—
146	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
147	C₂×#117	—	43	2	no	—
148	C₂²×C₄²	—	43	2	no	—
149	C₂²×C₄²	—	86	2	no	—
150	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
151	C₄×M₁₆	—	2	2	no	—
152	C₄×M₁₆	Dic₄₃	1	4	yes	complemented
153	C₂×#97	—	2	1	no	—
154	C₂×C₄×C₈	—	2	1	no	—
155	C₄×C₄⋊C₄	—	43	4	no	—
156	C₄×C₂²⋊C₄	—	86	2	no	—
157	#376	—	86	1	no	—
158	C₄³	—	43	4	no	—
159	C₄³	—	86	2	no	—
160	C₄³	—	86	1	no	—
161	#339	—	43	16	no	—
162	C₄₃	C₄×#339	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	86	2	no	—
164	C₂³×C₄	D₁₇₂	1	2	yes	—
165	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	C₄×D₄₃	1	8	yes	—
169	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
170	C₂²×C₈	—	2	2	no	—
171	#117	—	43	8	no	—
172	#117	—	86	4	no	—
173	C₂×C₄⋊C₄	—	43	4	no	—
174	C₂×C₂²⋊C₄	—	86	2	no	—
175	C₂×C₄²	—	43	8	no	—
176	C₂×C₄²	—	43	4	no	—
177	C₂×C₄²	—	86	16	no	—
178	C₂×C₄²	—	86	8	no	—
179	C₂×C₄²	—	86	4	no	—
180	C₂×C₄²	—	86	2	no	—
181	C₂×C₄²	—	86	2	no	—
182	C₂×C₄²	—	86	4	no	—
183	C₂×C₄²	Dic₈₆	1	2	yes	—
184	C₂×C₄²	C₄×D₄₃	1	2	yes	—
185	C₂×C₄²	C₄×D₄₃	1	2	yes	—
186	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
187	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	4	no	—
192	#95	—	86	4	no	—
193	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	86	16	no	—
196	C₂²×C₄	—	86	4	no	—
197	C₂²×C₄	—	86	8	no	—
198	C₂²×C₄	—	86	4	no	—
199	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
200	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
201	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	4	yes	—
202	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
203	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
207	C₂²×C₄	#10621898	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	8	no	—
213	M₁₆	C₄×Dic₄₃	1	8	yes	complemented
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	8	no	—
216	C₄⋊C₄	—	43	8	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₂²⋊C₄	—	86	8	no	—
219	C₄²	—	43	8	no	—
220	C₄²	—	86	32	no	—
221	C₄²	—	86	4	no	—
222	C₄²	—	86	8	no	—
223	C₄²	—	86	8	no	—
224	C₄²	—	172	8	no	—
225	C₄²	C₄×Dic₄₃	1	4	yes	—
226	C₄²	Dic₄₃⋊C₄	1	2	yes	—
227	C₄²	Dic₄₃⋊C₄	1	4	yes	—
228	C₄²	Dic₄₃⋊C₄	1	2	yes	—
229	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
230	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
231	C₂³	—	2	4	no	—
232	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
233	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
234	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
235	C₂³	C₄×D₁₇₂	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	86	16	no	—
240	C₂×C₄	—	86	32	no	—
241	C₂×C₄	—	86	8	no	—
242	C₂×C₄	—	86	8	no	—
243	C₂×C₄	—	172	8	no	—
244	C₂×C₄	C₄₃⋊#95	1	4	yes	—
245	C₂×C₄	C₄₃⋊#95	1	4	yes	—
246	C₂×C₄	#13063251	1	8	yes	—
247	C₂×C₄	C₄²×D₄₃	1	2	yes	—
248	C₂×C₄	C₄²×D₄₃	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
252	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
254	C₂×C₄	#13063282	1	2	yes	—
255	C₂×C₄	#13063282	1	2	yes	—
256	C₂×C₄	#13063284	1	2	yes	—
257	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
258	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
259	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
260	C₈	—	2	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
265	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
266	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	2	yes	—
267	C₂²	C₄×C₄⋊Dic₄₃	1	2	yes	—
268	C₂²	C₂×#13063251	1	4	yes	—
269	C₄	—	2	4	no	—
270	C₄	—	86	16	no	—
271	C₄	—	172	8	no	—
272	C₄	#14582660	1	8	yes	direct product factor
273	C₄	C₄²×Dic₄₃	1	1	yes	characteristic
274	C₄	#14582745	1	1	yes	characteristic
275	C₄	C₄×C₂²⋊Dic₄₃	1	2	yes	—
276	C₄	C₄×Dic₄₃⋊C₄	1	2	yes	—
277	C₄	C₄×Dic₄₃⋊C₄	1	2	yes	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	C₄×C₄₃⋊#95	1	1	yes	characteristic
281	C₂	C₂×#14582660	1	2	yes	—
282	C₂	C₄×#13063251	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₄³
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	C₄×#339
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
8. C₄×C₄⋊Dic₄₃	C₄	semidirect
13. C₄²×Dic₄₃	C₄	semidirect
14. #14582660	C₄	direct (N × H)
18. C₄×M₁₆×C₄₃	C₄	semidirect
71. M₁₆×C₄₃	C₄²	semidirect
82. C₄⋊Dic₄₃	C₄²	semidirect
88. C₄×Dic₄₃	C₄²	semidirect
129. C₄×C₄₃	#339	semidirect
152. C₄×M₁₆	Dic₄₃	semidirect
162. C₄₃	C₄×#339	semidirect
213. M₁₆	C₄×Dic₄₃	semidirect
272. C₄	#14582660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13063284	2
C₂×C₄²×Dic₄₃	2
C₂×#14582660	2
C₂×C₄×M₁₆×C₄₃	2
C₄×#339	43

Subgroups — C4×#14582660 #25001771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582660 #25001771