Subgroups — C₂×Q₈×D₄₂₄ #28264507

Order 13568.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
2	Q₈×D₄₂₄	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811028	C₂	1	3	yes	maximal, complemented
4	C₂×#16810904	C₂	1	6	yes	maximal, complemented
5	C₂×C₄×D₄₂₄	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×D₅₃	—	2	2	no	—
8	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
9	Q₈×D₂₁₂	—	2	8	no	—
10	Q₈×D₂₁₂	C₂²	1	16	yes	complemented
11	C₂×#13133217	C₂²	1	6	yes	—
12	C₂×#13133197	—	2	6	no	—
13	C₂×C₄×D₂₁₂	C₂²	1	6	yes	—
14	#16811028	C₂²	1	24	yes	complemented
15	#16810904	C₂²	1	48	yes	complemented
16	C₄×D₄₂₄	C₂²	1	24	yes	—
17	C₂×#13133151	C₂²	1	3	yes	—
18	C₂×#13133151	C₂²	1	6	yes	—
19	C₂×#13133139	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₂²	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂²×Dic₁₀₆	—	2	6	no	—
25	C₂×Q₈×D₅₃	—	2	16	no	—
26	C₂×Q₈×D₅₃	—	4	16	no	—
27	C₂²×D₂₁₂	C₂³	1	2	yes	—
28	C₂²×C₄×D₅₃	—	2	6	no	—
29	C₂×D₄₂₄	Q₈	1	8	yes	direct product factor
30	C₂×D₄₂₄	C₂³	1	4	yes	—
31	#13133217	—	2	12	no	—
32	#13133217	C₂³	1	24	yes	—
33	#13133197	—	2	48	no	—
34	C₄×D₂₁₂	—	2	12	no	—
35	C₄×D₂₁₂	C₂³	1	24	yes	—
36	C₂×C₄⋊Dic₅₃	—	2	6	no	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
38	C₂×#11719921	—	2	6	no	—
39	#13133151	C₂³	1	12	yes	—
40	#13133151	C₂³	1	24	yes	—
41	#13133139	C₂³	1	24	yes	—
42	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	D₄	1	4	yes	—
46	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
49	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
50	C₄×C₈×C₅₃	C₂³	1	12	yes	—
51	C₂³×D₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	D₈	1	8	yes	complemented
53	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	2	6	no	—
58	Q₈×D₅₃	—	4	64	no	—
59	C₂×D₂₁₂	—	2	2	no	—
60	C₂×D₂₁₂	—	2	4	no	—
61	C₂×D₂₁₂	C₂×Q₈	1	8	yes	complemented
62	C₂×D₂₁₂	C₂⁴	1	4	yes	—
63	C₂×C₄×D₅₃	—	2	24	no	—
64	C₂×C₄×D₅₃	—	4	24	no	—
65	C₂×Dic₁₀₆	—	2	24	no	—
66	C₂×Dic₁₀₆	—	4	24	no	—
67	C₂×C₈×C₅₃	—	2	6	no	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	complemented
69	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
70	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
71	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
72	C₄²×C₅₃	C₂×D₄	1	6	yes	—
73	C₄²×C₅₃	C₂⁴	1	6	yes	—
74	#11719921	—	2	24	no	—
75	C₄⋊Dic₅₃	—	2	24	no	—
76	C₄⋊Dic₅₃	—	2	6	no	—
77	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
78	D₄₂₄	C₂×Q₈	1	16	yes	direct product factor
79	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂²×D₅₃	—	2	8	no	—
81	C₂²×D₅₃	—	2	4	no	—
82	C₂²×D₅₃	—	4	4	no	—
83	C₂²×D₅₃	—	4	4	no	—
84	Q₈×C₅₃	C₂×D₈	1	16	yes	complemented
85	C₂×C₄×C₅₃	—	2	6	no	—
86	C₂×C₄×C₅₃	C₂×D₈	1	12	yes	—
87	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
88	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
90	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
92	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
93	C₂×Dic₅₃	—	2	12	no	—
94	C₂×Dic₅₃	—	4	12	no	—
95	D₂₁₂	—	2	8	no	—
96	D₂₁₂	—	2	4	no	—
97	D₂₁₂	C₂²×Q₈	1	8	yes	complemented
98	C₄×D₅₃	—	4	48	no	—
99	Dic₁₀₆	—	4	48	no	—
100	C₈×C₅₃	—	2	6	no	—
101	C₈×C₅₃	C₂²×Q₈	1	4	yes	complemented
102	C₂×Q₈×D₈	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	D₄×Q₈	1	2	yes	—
104	C₂²×C₅₃	C₂²×D₈	1	2	yes	—
105	C₂²×C₅₃	#578	1	2	yes	—
106	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
107	C₂×D₅₃	—	2	8	no	—
108	C₂×D₅₃	—	4	16	no	—
109	C₂×D₅₃	—	4	4	no	—
110	C₂×D₅₃	—	4	4	no	—
111	C₄×C₅₃	—	2	6	no	—
112	C₄×C₅₃	D₄×Q₈	1	2	yes	—
113	C₄×C₅₃	C₂²×D₈	1	12	yes	—
114	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
116	Dic₅₃	—	4	12	no	—
117	C₂×D₄×Q₈	—	53	2	no	—
118	Q₈×D₈	—	53	16	no	—
119	C₂×#500	—	53	3	no	—
120	C₂×#474	—	53	6	no	—
121	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
122	C₂×C₄×D₈	—	53	3	no	—
123	C₂×C₅₃	Q₈×D₈	1	4	yes	complemented
124	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
125	C₂×C₅₃	C₂³×D₈	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
127	D₅₃	—	4	8	no	—
128	C₂³×Q₈	—	106	2	no	—
129	C₂²×D₈	—	53	1	no	—
130	D₄×Q₈	—	53	16	no	—
131	D₄×Q₈	—	106	8	no	—
132	C₂×C₄⋊Q₈	—	53	6	no	—
133	C₂×C₂²⋊Q₈	—	106	6	no	—
134	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₄×D₄	—	53	6	no	—
136	#500	—	53	24	no	—
137	#474	—	53	48	no	—
138	C₈×Q₈	C₂×D₅₃	1	8	yes	complemented
139	C₄×D₈	—	53	24	no	—
140	C₂×#107	—	53	3	no	—
141	C₂×#107	—	53	6	no	—
142	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
143	C₂×#102	—	53	6	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
145	C₅₃	C₂×Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	106	16	no	—
147	C₂²×Q₈	—	106	6	no	—
148	C₂²×Q₈	—	212	16	no	—
149	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
150	C₂²×D₄	—	53	2	no	—
151	C₂³×C₄	—	106	6	no	—
152	C₂×D₈	—	53	8	no	—
153	C₂×D₈	—	53	4	no	—
154	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	53	24	no	—
157	C₄⋊Q₈	—	106	12	no	—
158	C₂²⋊Q₈	—	106	48	no	—
159	C₄×Q₈	D₂₁₂	1	4	yes	—
160	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
161	C₄×D₄	—	53	24	no	—
162	C₄×D₄	—	106	12	no	—
163	C₂×C₄⋊C₄	—	53	6	no	—
164	C₂×C₄⋊C₄	—	106	6	no	—
165	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
166	C₂×C₂²⋊C₄	—	106	6	no	—
167	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
168	#107	—	53	12	no	—
169	#107	—	53	24	no	—
170	C₄⋊C₈	C₂²×D₅₃	1	12	yes	—
171	#102	—	53	24	no	—
172	C₄×C₈	C₂²×D₅₃	1	12	yes	—
173	C₂⁴	—	106	2	no	—
174	C₂×Q₈	—	106	24	no	—
175	C₂×Q₈	—	212	64	no	—
176	C₂×Q₈	—	212	24	no	—
177	C₂×Q₈	D₄₂₄	1	8	yes	direct product factor
178	C₂×Q₈	C₂×D₂₁₂	1	4	yes	—
179	C₂×D₄	—	53	8	no	—
180	C₂×D₄	—	53	4	no	—
181	C₂×D₄	—	106	2	no	—
182	C₂×D₄	—	106	4	no	—
183	C₂²×C₄	—	106	6	no	—
184	C₂²×C₄	—	106	24	no	—
185	C₂²×C₄	—	212	24	no	—
186	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
187	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
188	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
189	D₈	—	53	16	no	—
190	C₂×C₈	—	2	6	no	—
191	C₂×C₈	Q₈×D₅₃	1	4	yes	complemented
192	C₂×C₈	C₂³×D₅₃	1	6	yes	—
193	C₂×C₈	C₂³×D₅₃	1	2	yes	—
194	C₄⋊C₄	—	53	12	no	—
195	C₄⋊C₄	—	106	6	no	—
196	C₄⋊C₄	—	106	24	no	—
197	C₄⋊C₄	C₂×D₂₁₂	1	12	yes	—
198	C₂²⋊C₄	—	106	24	no	—
199	C₄²	C₂×D₂₁₂	1	6	yes	—
200	C₄²	C₂³×D₅₃	1	6	yes	—
201	C₂³	—	106	8	no	—
202	C₂³	—	106	4	no	—
203	C₂³	—	212	4	no	—
204	C₂³	—	212	4	no	—
205	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
206	Q₈	—	212	48	no	—
207	Q₈	C₂×D₄₂₄	1	16	yes	direct product factor
208	D₄	—	53	8	no	—
209	D₄	—	106	8	no	—
210	D₄	—	106	4	no	—
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	106	12	no	—
213	C₂×C₄	—	212	12	no	—
214	C₂×C₄	—	212	48	no	—
215	C₂×C₄	C₂×D₄₂₄	1	12	yes	—
216	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
217	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
219	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
220	C₂×C₄	#13133264	1	2	yes	—
221	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂×Q₈×D₅₃	1	4	yes	complemented
224	C₂²	—	106	8	no	—
225	C₂²	—	212	16	no	—
226	C₂²	—	212	4	no	—
227	C₂²	—	212	4	no	—
228	C₂²	Q₈×D₂₁₂	1	2	yes	—
229	C₂²	C₂²×D₄₂₄	1	2	yes	—
230	C₂²	#16811440	1	2	yes	—
231	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	212	12	no	—
234	C₄	Q₈×D₂₁₂	1	2	yes	—
235	C₄	C₂²×D₄₂₄	1	12	yes	—
236	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
237	C₄	C₂×#13133264	1	1	yes	characteristic
238	C₂	—	212	8	no	—
239	C₂	Q₈×D₄₂₄	1	4	yes	direct product factor
240	C₂	C₂×Q₈×D₂₁₂	1	1	yes	characteristic
241	C₂	C₂³×D₄₂₄	1	1	yes	characteristic
242	C₂	C₂×#16811440	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₄₂₄	C₂²
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₂×Q₈×D₈
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₂₁₂	C₂	semidirect
2. Q₈×D₄₂₄	C₂	direct (N × H)
3. C₂×#16811028	C₂	semidirect
4. C₂×#16810904	C₂	semidirect
6. C₂×C₈×Q₈×C₅₃	C₂	semidirect
10. Q₈×D₂₁₂	C₂²	semidirect
14. #16811028	C₂²	semidirect
15. #16810904	C₂²	semidirect
21. C₈×Q₈×C₅₃	C₂²	semidirect
29. C₂×D₄₂₄	Q₈	direct (N × H)
52. C₂×Q₈×C₅₃	D₈	semidirect
61. C₂×D₂₁₂	C₂×Q₈	semidirect
68. C₂×C₈×C₅₃	C₂×Q₈	semidirect
78. D₄₂₄	C₂×Q₈	direct (N × H)
84. Q₈×C₅₃	C₂×D₈	semidirect
97. D₂₁₂	C₂²×Q₈	semidirect
101. C₈×C₅₃	C₂²×Q₈	semidirect
121. C₂×C₈×Q₈	D₅₃	semidirect
123. C₂×C₅₃	Q₈×D₈	semidirect
138. C₈×Q₈	C₂×D₅₃	semidirect
145. C₅₃	C₂×Q₈×D₈	semidirect
177. C₂×Q₈	D₄₂₄	direct (N × H)
191. C₂×C₈	Q₈×D₅₃	semidirect
207. Q₈	C₂×D₄₂₄	direct (N × H)
223. C₈	C₂×Q₈×D₅₃	semidirect
239. C₂	Q₈×D₄₂₄	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×Q₈×D₂₁₂	2
Q₈×D₄₂₄	2
C₂×#16811028	2
C₂×#16810904	2
C₂×C₄×D₄₂₄	2
C₂×C₈×Q₈×C₅₃	2
C₂×Q₈×D₈	53

Subgroups — C2×Q8×D424 #28264507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₄₂₄ #28264507