Subgroups — C₂×Q₈×Dic₂₁₂ #28264542

Order 13568.

1830 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₀₆	C₂	1	2	yes	maximal
2	Q₈×Dic₂₁₂	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811028	C₂	1	3	yes	maximal
4	C₂×#16810919	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₂₁₂	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
8	Q₈×Dic₁₀₆	—	2	8	no	—
9	Q₈×Dic₁₀₆	C₂²	1	16	yes	—
10	C₂×#13133217	C₂²	1	6	yes	—
11	C₂×#13133216	—	2	6	no	—
12	C₂×Q₈×Dic₅₃	—	2	2	no	—
13	C₂×C₄×Dic₁₀₆	C₂²	1	6	yes	—
14	#16811028	C₂²	1	24	yes	—
15	#16810919	C₂²	1	48	yes	—
16	C₄×Dic₂₁₂	C₂²	1	24	yes	—
17	C₂×#13133151	C₂²	1	3	yes	—
18	C₂×#13133151	C₂²	1	6	yes	—
19	C₂×#13133142	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₂²	1	8	yes	—
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	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
26	C₂×Dic₂₁₂	Q₈	1	8	yes	direct product factor
27	C₂×Dic₂₁₂	C₂³	1	4	yes	—
28	#13133217	—	2	12	no	—
29	#13133217	C₂³	1	24	yes	—
30	#13133216	—	2	48	no	—
31	Q₈×Dic₅₃	—	2	16	no	—
32	C₄×Dic₁₀₆	—	2	12	no	—
33	C₄×Dic₁₀₆	C₂³	1	24	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	6	no	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
36	C₂×#11719919	—	2	6	no	—
37	C₂×C₄×Dic₅₃	—	2	6	no	—
38	#13133151	C₂³	1	12	yes	—
39	#13133151	C₂³	1	24	yes	—
40	#13133142	C₂³	1	24	yes	—
41	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₄×Q₈×C₅₃	D₄	1	4	yes	—
45	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
48	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
49	C₄×C₈×C₅₃	C₂³	1	12	yes	—
50	C₂×Q₈×C₅₃	Q₁₆	1	8	yes	complemented
51	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₅₃	—	2	2	no	—
56	C₂²×Dic₅₃	—	2	6	no	—
57	C₂×Dic₁₀₆	—	2	2	no	—
58	C₂×Dic₁₀₆	—	2	4	no	—
59	C₂×Dic₁₀₆	—	2	24	no	—
60	C₂×Dic₁₀₆	—	8	12	no	—
61	C₂×Dic₁₀₆	C₂×Q₈	1	8	yes	—
62	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
63	C₂×C₈×C₅₃	—	2	6	no	—
64	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
66	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
67	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
68	C₄²×C₅₃	C₂×D₄	1	6	yes	—
69	C₄²×C₅₃	C₂⁴	1	6	yes	—
70	C₄⋊Dic₅₃	—	2	24	no	—
71	C₄⋊Dic₅₃	—	2	6	no	—
72	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
73	#11719919	—	2	24	no	—
74	C₄×Dic₅₃	—	2	24	no	—
75	Dic₂₁₂	C₂×Q₈	1	16	yes	direct product factor
76	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₅₃	C₂×Q₁₆	1	16	yes	complemented
78	C₂×C₄×C₅₃	—	2	6	no	—
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	12	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
83	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
84	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
85	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
86	C₂×Dic₅₃	—	2	8	no	—
87	C₂×Dic₅₃	—	2	4	no	—
88	C₂×Dic₅₃	—	2	12	no	—
89	C₂×Dic₅₃	—	4	12	no	—
90	Dic₁₀₆	—	2	8	no	—
91	Dic₁₀₆	—	2	4	no	—
92	Dic₁₀₆	—	8	24	no	—
93	Dic₁₀₆	C₂²×Q₈	1	8	yes	—
94	C₈×C₅₃	—	2	6	no	—
95	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
96	C₂×Q₈×Q₁₆	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	D₄×Q₈	1	2	yes	—
98	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
99	C₂²×C₅₃	#576	1	2	yes	—
100	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₅₃	—	2	6	no	—
102	C₄×C₅₃	D₄×Q₈	1	2	yes	—
103	C₄×C₅₃	C₂²×Q₁₆	1	12	yes	—
104	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
105	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₅₃	—	2	8	no	—
107	Dic₅₃	—	4	12	no	—
108	C₂×Q₈²	—	53	2	no	—
109	Q₈×Q₁₆	—	53	16	no	—
110	C₂×#500	—	53	3	no	—
111	C₂×#477	—	53	6	no	—
112	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
113	C₂×C₄×Q₁₆	—	53	3	no	—
114	C₂×C₅₃	Q₈×Q₁₆	1	4	yes	complemented
115	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
116	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#576	1	1	yes	characteristic
118	C₂²×Q₁₆	—	53	1	no	—
119	Q₈²	—	53	16	no	—
120	Q₈²	—	106	8	no	—
121	C₂×C₄⋊Q₈	—	53	6	no	—
122	C₂×C₄⋊Q₈	—	106	6	no	—
123	C₂×C₄×Q₈	—	53	6	no	—
124	C₂×C₄×Q₈	—	106	2	no	—
125	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
126	#500	—	53	24	no	—
127	#477	—	53	48	no	—
128	C₈×Q₈	C₂×D₅₃	1	8	yes	—
129	C₄×Q₁₆	—	53	24	no	—
130	C₂×#107	—	53	6	no	—
131	C₂×#107	—	53	3	no	—
132	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
133	C₂×#103	—	53	6	no	—
134	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
135	C₅₃	C₂×Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	53	2	no	—
137	C₂²×Q₈	—	106	6	no	—
138	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
139	C₂×Q₁₆	—	53	8	no	—
140	C₂×Q₁₆	—	53	4	no	—
141	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	53	24	no	—
144	C₄⋊Q₈	—	106	48	no	—
145	C₄⋊Q₈	—	106	12	no	—
146	C₄×Q₈	—	53	24	no	—
147	C₄×Q₈	—	106	16	no	—
148	C₄×Q₈	—	106	12	no	—
149	C₄×Q₈	D₂₁₂	1	4	yes	—
150	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
151	C₂×C₄⋊C₄	—	53	6	no	—
152	C₂×C₄⋊C₄	—	106	6	no	—
153	C₂×C₄⋊C₄	—	106	6	no	—
154	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
155	C₂×C₄²	—	106	6	no	—
156	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
157	#107	—	53	12	no	—
158	#107	—	53	24	no	—
159	C₄⋊C₈	C₂²×D₅₃	1	12	yes	—
160	#103	—	53	24	no	—
161	C₄×C₈	C₂²×D₅₃	1	12	yes	—
162	C₂×Q₈	—	53	8	no	—
163	C₂×Q₈	—	53	4	no	—
164	C₂×Q₈	—	106	2	no	—
165	C₂×Q₈	—	106	24	no	—
166	C₂×Q₈	—	106	4	no	—
167	C₂×Q₈	—	424	12	no	—
168	C₂×Q₈	Dic₂₁₂	1	8	yes	direct product factor
169	C₂×Q₈	C₂×D₂₁₂	1	4	yes	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	—	106	6	no	—
172	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
173	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
174	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
175	Q₁₆	—	53	16	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	Q₈×D₅₃	1	4	yes	—
178	C₂×C₈	C₂³×D₅₃	1	6	yes	—
179	C₂×C₈	C₂³×D₅₃	1	2	yes	—
180	C₄⋊C₄	—	53	12	no	—
181	C₄⋊C₄	—	106	6	no	—
182	C₄⋊C₄	—	106	24	no	—
183	C₄⋊C₄	—	106	24	no	—
184	C₄⋊C₄	C₂×D₂₁₂	1	12	yes	—
185	C₄²	—	106	24	no	—
186	C₄²	C₂×D₂₁₂	1	6	yes	—
187	C₄²	C₂³×D₅₃	1	6	yes	—
188	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
189	Q₈	—	53	8	no	—
190	Q₈	—	106	8	no	—
191	Q₈	—	106	4	no	—
192	Q₈	—	424	24	no	—
193	Q₈	C₂×Dic₂₁₂	1	16	yes	direct product factor
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	106	8	no	—
196	C₂×C₄	—	106	4	no	—
197	C₂×C₄	—	106	12	no	—
198	C₂×C₄	—	212	12	no	—
199	C₂×C₄	C₂×Dic₂₁₂	1	12	yes	—
200	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
201	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
203	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
204	C₂×C₄	#13133264	1	2	yes	—
205	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×Q₈×D₅₃	1	4	yes	—
208	C₂²	Q₈×D₂₁₂	1	2	yes	—
209	C₂²	C₂²×Dic₂₁₂	1	2	yes	—
210	C₂²	#16811428	1	2	yes	—
211	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	106	8	no	—
214	C₄	—	212	12	no	—
215	C₄	Q₈×D₂₁₂	1	2	yes	—
216	C₄	C₂²×Dic₂₁₂	1	12	yes	—
217	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
218	C₄	C₂×#13133264	1	1	yes	characteristic
219	C₂	Q₈×Dic₂₁₂	1	4	yes	direct product factor
220	C₂	C₂×Q₈×D₂₁₂	1	1	yes	characteristic
221	C₂	C₂³×Dic₂₁₂	1	1	yes	characteristic
222	C₂	C₂×#16811428	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, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₂×Q₈×Q₁₆
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
2. Q₈×Dic₂₁₂	C₂	direct (N × H)
26. C₂×Dic₂₁₂	Q₈	direct (N × H)
50. C₂×Q₈×C₅₃	Q₁₆	semidirect
75. Dic₂₁₂	C₂×Q₈	direct (N × H)
77. Q₈×C₅₃	C₂×Q₁₆	semidirect
114. C₂×C₅₃	Q₈×Q₁₆	semidirect
135. C₅₃	C₂×Q₈×Q₁₆	semidirect
168. C₂×Q₈	Dic₂₁₂	direct (N × H)
193. Q₈	C₂×Dic₂₁₂	direct (N × H)
219. C₂	Q₈×Dic₂₁₂	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×Dic₁₀₆	2
Q₈×Dic₂₁₂	2
C₂×#16811028	2
C₂×#16810919	2
C₂×C₄×Dic₂₁₂	2
C₂×C₈×Q₈×C₅₃	2
C₂×Q₈×Q₁₆	53

Subgroups — C2×Q8×Dic212 #28264542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×Dic₂₁₂ #28264542