Subgroups — C₂²×C₄×Dic₄₃⋊C₄ #25254776

Order 11008.

5758 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₄₃⋊C₄	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₄₃⋊C₄	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄₃⋊#95	C₂	1	2	yes	maximal
6	C₂³×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₄₃	C₂²	1	2	yes	—
9	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
10	C₂²×Dic₄₃⋊C₄	C₄	1	32	yes	direct product factor
11	C₂²×Dic₄₃⋊C₄	C₂²	1	24	yes	—
12	C₂²×C₄×Dic₄₃	—	2	8	no	—
13	C₂²×C₄×Dic₄₃	C₄	1	16	yes	complemented
14	C₂²×C₄×Dic₄₃	C₂²	1	12	yes	—
15	C₂²×C₄×Dic₄₃	C₂²	1	12	yes	—
16	C₄×Dic₄₃⋊C₄	C₂²	1	64	yes	direct product factor
17	C₂×C₄₃⋊#95	C₂²	1	24	yes	—
18	C₂⁴×C₄×C₄₃	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₃	—	2	8	no	—
21	C₂²×C₄²×C₄₃	C₂²	1	12	yes	—
22	C₂³×Dic₄₃	—	2	8	no	—
23	C₂³×Dic₄₃	—	2	8	no	—
24	C₂³×Dic₄₃	—	2	8	no	—
25	C₂³×Dic₄₃	C₂×C₄	1	8	yes	—
26	C₂³×Dic₄₃	C₂×C₄	1	16	yes	—
27	C₂³×Dic₄₃	C₂×C₄	1	8	yes	—
28	C₂³×Dic₄₃	C₂×C₄	1	16	yes	—
29	C₂³×Dic₄₃	C₂³	1	12	yes	—
30	C₂³×Dic₄₃	C₂³	1	6	yes	—
31	C₂³×Dic₄₃	C₂³	1	6	yes	—
32	C₂×Dic₄₃⋊C₄	C₂×C₄	1	192	yes	direct product factor
33	C₂×Dic₄₃⋊C₄	C₂³	1	32	yes	—
34	C₂×C₄×Dic₄₃	—	2	48	no	—
35	C₂×C₄×Dic₄₃	C₂×C₄	1	96	yes	complemented
36	C₂×C₄×Dic₄₃	C₂³	1	16	yes	—
37	C₂×C₄×Dic₄₃	C₂³	1	16	yes	—
38	C₄₃⋊#95	C₂³	1	32	yes	—
39	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	—	2	8	no	—
41	C₂³×C₄×C₄₃	—	2	8	no	—
42	C₂³×C₄×C₄₃	C₂×C₄	1	16	yes	—
43	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₄₃	D₄	1	8	yes	—
45	C₂³×C₄×C₄₃	Q₈	1	8	yes	—
46	C₂³×C₄×C₄₃	C₂³	1	12	yes	—
47	C₂³×C₄×C₄₃	C₂³	1	6	yes	—
48	C₂×C₄²×C₄₃	—	2	48	no	—
49	C₂×C₄²×C₄₃	C₂³	1	16	yes	—
50	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
51	C₂⁴×C₄₃	C₂²×C₄	1	8	yes	—
52	C₂⁴×C₄₃	C₂×D₄	1	4	yes	—
53	C₂⁴×C₄₃	C₂×Q₈	1	4	yes	—
54	C₂⁴×C₄₃	#41	1	8	yes	—
55	C₂⁴×C₄₃	C₂⁴	1	3	yes	—
56	C₂²×C₄×C₄₃	—	2	64	no	—
57	C₂²×C₄×C₄₃	—	2	24	no	—
58	C₂²×C₄×C₄₃	—	2	24	no	—
59	C₂²×C₄×C₄₃	C₄⋊C₄	1	64	yes	complemented
60	C₂²×C₄×C₄₃	C₂²×C₄	1	48	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	24	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	24	yes	—
63	C₂²×C₄×C₄₃	C₂×Q₈	1	24	yes	—
64	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
66	C₂²×Dic₄₃	—	2	32	no	—
67	C₂²×Dic₄₃	—	2	64	no	—
68	C₂²×Dic₄₃	—	2	24	no	—
69	C₂²×Dic₄₃	—	2	24	no	—
70	C₂²×Dic₄₃	—	2	24	no	—
71	C₂²×Dic₄₃	C₄²	1	64	yes	complemented
72	C₂²×Dic₄₃	C₂²×C₄	1	24	yes	—
73	C₂²×Dic₄₃	C₂²×C₄	1	48	yes	—
74	C₂²×Dic₄₃	C₂²×C₄	1	24	yes	—
75	C₂²×Dic₄₃	C₂²×C₄	1	48	yes	—
76	C₂²×Dic₄₃	C₂⁴	1	4	yes	—
77	C₂²×Dic₄₃	C₂⁴	1	8	yes	—
78	C₂²×Dic₄₃	C₂⁴	1	4	yes	—
79	C₄²×C₄₃	—	2	32	no	—
80	Dic₄₃⋊C₄	C₂²×C₄	1	128	yes	direct product factor
81	C₄×Dic₄₃	—	2	32	no	—
82	C₄×Dic₄₃	C₂²×C₄	1	64	yes	complemented
83	C₂³×C₄₃	C₂×C₄²	1	16	yes	—
84	C₂³×C₄₃	C₂×C₄⋊C₄	1	16	yes	—
85	C₂³×C₄₃	#117	1	16	yes	—
86	C₂³×C₄₃	C₄×D₄	1	32	yes	—
87	C₂³×C₄₃	C₄×Q₈	1	32	yes	—
88	C₂³×C₄₃	C₂³×C₄	1	6	yes	—
89	C₂³×C₄₃	C₂³×C₄	1	12	yes	—
90	C₂³×C₄₃	C₂²×D₄	1	6	yes	—
91	C₂³×C₄₃	C₂²×Q₈	1	6	yes	—
92	C₂³×C₄₃	C₂×#41	1	12	yes	—
93	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	2	96	no	—
95	C₂×C₄×C₄₃	—	2	8	no	—
96	C₂×C₄×C₄₃	—	2	8	no	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	96	yes	complemented
98	C₂×C₄×C₄₃	C₂³×C₄	1	16	yes	—
99	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
100	C₂×C₄×C₄₃	C₂²×D₄	1	8	yes	—
101	C₂×C₄×C₄₃	C₂²×Q₈	1	8	yes	—
102	C₂×Dic₄₃	—	2	48	no	—
103	C₂×Dic₄₃	—	2	96	no	—
104	C₂×Dic₄₃	—	2	8	no	—
105	C₂×Dic₄₃	—	2	8	no	—
106	C₂×Dic₄₃	—	2	8	no	—
107	C₂×Dic₄₃	C₂×C₄²	1	96	yes	complemented
108	C₂×Dic₄₃	C₂³×C₄	1	8	yes	—
109	C₂×Dic₄₃	C₂³×C₄	1	16	yes	—
110	C₂×Dic₄₃	C₂³×C₄	1	8	yes	—
111	C₂×Dic₄₃	C₂³×C₄	1	16	yes	—
112	C₂²×C₄×C₄⋊C₄	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₄×C₄⋊C₄	1	64	yes	complemented
114	C₂²×C₄₃	C₂²×C₄²	1	12	yes	—
115	C₂²×C₄₃	C₂²×C₄⋊C₄	1	12	yes	—
116	C₂²×C₄₃	C₂×#117	1	12	yes	—
117	C₂²×C₄₃	C₂×C₄×D₄	1	24	yes	—
118	C₂²×C₄₃	C₂×C₄×Q₈	1	24	yes	—
119	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
120	C₂²×C₄₃	C₂⁴×C₄	1	2	yes	—
121	C₂²×C₄₃	C₂³×D₄	1	1	yes	characteristic
122	C₂²×C₄₃	C₂³×Q₈	1	1	yes	characteristic
123	C₂²×C₄₃	C₂²×#41	1	2	yes	—
124	C₄×C₄₃	—	2	16	no	—
125	C₄×C₄₃	C₂²×C₄⋊C₄	1	16	yes	complemented
126	Dic₄₃	—	2	8	no	—
127	Dic₄₃	—	2	16	no	—
128	Dic₄₃	C₂²×C₄²	1	16	yes	complemented
129	C₂³×C₄⋊C₄	—	43	2	no	—
130	C₂³×C₄²	—	43	1	no	—
131	C₂³×C₄²	—	43	1	no	—
132	C₂³×C₄²	D₄₃	1	1	yes	characteristic
133	C₂×C₄×C₄⋊C₄	—	43	24	no	—
134	C₂²×#95	—	43	2	no	—
135	C₂×C₄₃	C₂×C₄×C₄⋊C₄	1	24	yes	complemented
136	C₂×C₄₃	C₂³×C₄²	1	1	yes	characteristic
137	C₂×C₄₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
139	C₂×C₄₃	C₂²×C₄×D₄	1	2	yes	—
140	C₂×C₄₃	C₂²×C₄×Q₈	1	2	yes	—
141	C₂⁴×C₄	—	43	1	no	—
142	C₂⁴×C₄	—	43	2	no	—
143	C₂⁴×C₄	—	43	1	no	—
144	C₂⁴×C₄	C₂×D₄₃	1	2	yes	—
145	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	—	43	24	no	—
147	C₂²×C₄⋊C₄	—	43	32	no	—
148	C₂²×C₄²	—	2	8	no	—
149	C₂²×C₄²	—	43	12	no	—
150	C₂²×C₄²	—	43	16	no	—
151	C₂²×C₄²	—	43	12	no	—
152	C₂²×C₄²	—	86	8	no	—
153	C₂²×C₄²	C₂×D₄₃	1	12	yes	—
154	C₄×C₄⋊C₄	—	43	64	no	—
155	C₂×#95	—	43	24	no	—
156	C₄₃	C₂²×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
157	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	—	43	16	no	—
161	C₂³×C₄	—	43	8	no	—
162	C₂³×C₄	—	43	6	no	—
163	C₂³×C₄	—	43	16	no	—
164	C₂³×C₄	—	43	8	no	—
165	C₂³×C₄	—	43	12	no	—
166	C₂³×C₄	—	43	6	no	—
167	C₂³×C₄	—	86	8	no	—
168	C₂³×C₄	—	86	8	no	—
169	C₂³×C₄	—	86	8	no	—
170	C₂³×C₄	Dic₈₆	1	8	yes	—
171	C₂³×C₄	C₄×D₄₃	1	16	yes	—
172	C₂³×C₄	C₄×D₄₃	1	8	yes	—
173	C₂³×C₄	C₂²⋊D₄₃	1	8	yes	—
174	C₂³×C₄	C₂²×D₄₃	1	12	yes	—
175	C₂³×C₄	C₂²×D₄₃	1	6	yes	—
176	C₂×C₄⋊C₄	—	43	192	no	—
177	C₂×C₄⋊C₄	—	43	32	no	—
178	C₂×C₄²	—	2	48	no	—
179	C₂×C₄²	—	43	96	no	—
180	C₂×C₄²	—	43	16	no	—
181	C₂×C₄²	—	43	16	no	—
182	C₂×C₄²	—	86	48	no	—
183	C₂×C₄²	C₂²×D₄₃	1	16	yes	—
184	#95	—	43	32	no	—
185	C₂⁴	C₂×Dic₈₆	1	4	yes	—
186	C₂⁴	C₂×C₄×D₄₃	1	4	yes	—
187	C₂⁴	C₂×C₄×D₄₃	1	8	yes	—
188	C₂⁴	#10621898	1	8	yes	—
189	C₂⁴	C₂×C₂²⋊D₄₃	1	4	yes	—
190	C₂⁴	C₂³×D₄₃	1	3	yes	—
191	C₂²×C₄	—	2	64	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	43	64	no	—
195	C₂²×C₄	—	43	24	no	—
196	C₂²×C₄	—	43	48	no	—
197	C₂²×C₄	—	43	4	no	—
198	C₂²×C₄	—	43	24	no	—
199	C₂²×C₄	—	43	48	no	—
200	C₂²×C₄	—	43	4	no	—
201	C₂²×C₄	—	43	8	no	—
202	C₂²×C₄	—	86	32	no	—
203	C₂²×C₄	—	86	64	no	—
204	C₂²×C₄	—	86	24	no	—
205	C₂²×C₄	—	86	24	no	—
206	C₂²×C₄	—	86	24	no	—
207	C₂²×C₄	Dic₄₃⋊C₄	1	64	yes	direct product factor
208	C₂²×C₄	C₂×Dic₈₆	1	24	yes	—
209	C₂²×C₄	C₂×C₄×D₄₃	1	48	yes	—
210	C₂²×C₄	C₂×C₄×D₄₃	1	24	yes	—
211	C₂²×C₄	C₂×C₂²⋊D₄₃	1	24	yes	—
212	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
213	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
214	C₄⋊C₄	—	43	128	no	—
215	C₄²	—	2	32	no	—
216	C₄²	—	43	64	no	—
217	C₄²	—	86	32	no	—
218	C₂³	C₄²×D₄₃	1	16	yes	—
219	C₂³	C₂×Dic₄₃⋊C₄	1	16	yes	—
220	C₂³	#13063282	1	16	yes	—
221	C₂³	C₄×C₂²⋊D₄₃	1	32	yes	—
222	C₂³	C₄×Dic₈₆	1	32	yes	—
223	C₂³	C₂²×C₄×D₄₃	1	6	yes	—
224	C₂³	C₂²×C₄×D₄₃	1	12	yes	—
225	C₂³	C₂²×C₂²⋊D₄₃	1	6	yes	—
226	C₂³	C₂²×Dic₈₆	1	6	yes	—
227	C₂³	C₂×#10621898	1	12	yes	—
228	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
229	C₂×C₄	—	2	96	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	43	96	no	—
233	C₂×C₄	—	43	8	no	—
234	C₂×C₄	—	43	16	no	—
235	C₂×C₄	—	43	8	no	—
236	C₂×C₄	—	43	16	no	—
237	C₂×C₄	—	86	48	no	—
238	C₂×C₄	—	86	96	no	—
239	C₂×C₄	—	86	8	no	—
240	C₂×C₄	—	86	8	no	—
241	C₂×C₄	—	86	8	no	—
242	C₂×C₄	C₂×Dic₄₃⋊C₄	1	96	yes	direct product factor
243	C₂×C₄	C₂²×C₄×D₄₃	1	16	yes	—
244	C₂×C₄	C₂²×C₄×D₄₃	1	8	yes	—
245	C₂×C₄	C₂²×C₂²⋊D₄₃	1	8	yes	—
246	C₂×C₄	C₂²×Dic₈₆	1	8	yes	—
247	C₂²	C₄×Dic₄₃⋊C₄	1	64	yes	direct product factor
248	C₂²	C₂×C₄²×D₄₃	1	12	yes	—
249	C₂²	C₂²×Dic₄₃⋊C₄	1	12	yes	—
250	C₂²	C₂×#13063282	1	12	yes	—
251	C₂²	C₂×C₄×C₂²⋊D₄₃	1	24	yes	—
252	C₂²	C₂×C₄×Dic₈₆	1	24	yes	—
253	C₂²	C₂³×C₄×D₄₃	1	1	yes	characteristic
254	C₂²	C₂³×C₄×D₄₃	1	2	yes	—
255	C₂²	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
256	C₂²	C₂³×Dic₈₆	1	1	yes	characteristic
257	C₂²	C₂²×#10621898	1	2	yes	—
258	C₄	—	2	16	no	—
259	C₄	—	43	16	no	—
260	C₄	—	86	8	no	—
261	C₄	—	86	16	no	—
262	C₄	C₂²×Dic₄₃⋊C₄	1	16	yes	direct product factor
263	C₂	C₂×C₄×Dic₄₃⋊C₄	1	24	yes	direct product factor
264	C₂	C₂²×C₄²×D₄₃	1	1	yes	characteristic
265	C₂	C₂³×Dic₄₃⋊C₄	1	1	yes	characteristic
266	C₂	C₂²×#13063282	1	1	yes	characteristic
267	C₂	C₂²×C₄×C₂²⋊D₄₃	1	2	yes	—
268	C₂	C₂²×C₄×Dic₈₆	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄₃
commutator	C₂×C₄₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂³×C₄²×C₄₃	C₂
socle, prime radical	C₂⁵×C₄₃	C₂³
cosocle	C₂³×C₄₃	C₂⁵
hypercenter	C₂³×C₄²	D₄₃
hypocenter	C₄₃	C₂²×C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×Dic₄₃⋊C₄	C₂	direct (N × H)
10. C₂²×Dic₄₃⋊C₄	C₄	direct (N × H)
13. C₂²×C₄×Dic₄₃	C₄	semidirect
16. C₄×Dic₄₃⋊C₄	C₂²	direct (N × H)
32. C₂×Dic₄₃⋊C₄	C₂×C₄	direct (N × H)
35. C₂×C₄×Dic₄₃	C₂×C₄	semidirect
59. C₂²×C₄×C₄₃	C₄⋊C₄	semidirect
71. C₂²×Dic₄₃	C₄²	semidirect
80. Dic₄₃⋊C₄	C₂²×C₄	direct (N × H)
82. C₄×Dic₄₃	C₂²×C₄	semidirect
97. C₂×C₄×C₄₃	C₂×C₄⋊C₄	semidirect
107. C₂×Dic₄₃	C₂×C₄²	semidirect
113. C₂²×C₄₃	C₄×C₄⋊C₄	semidirect
125. C₄×C₄₃	C₂²×C₄⋊C₄	semidirect
128. Dic₄₃	C₂²×C₄²	semidirect
135. C₂×C₄₃	C₂×C₄×C₄⋊C₄	semidirect
156. C₄₃	C₂²×C₄×C₄⋊C₄	semidirect
207. C₂²×C₄	Dic₄₃⋊C₄	direct (N × H)
242. C₂×C₄	C₂×Dic₄₃⋊C₄	direct (N × H)
247. C₂²	C₄×Dic₄₃⋊C₄	direct (N × H)
262. C₄	C₂²×Dic₄₃⋊C₄	direct (N × H)
263. C₂	C₂×C₄×Dic₄₃⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₄₃⋊C₄	2
C₂³×C₄×Dic₄₃	2
C₂×C₄×Dic₄₃⋊C₄	2
C₂²×C₄₃⋊#95	2
C₂³×C₄²×C₄₃	2
C₂²×C₄×C₄⋊C₄	43

Subgroups — C22×C4×Dic43⋊C4 #25254776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Dic₄₃⋊C₄ #25254776