Subgroups — C₂²×C₄×Dic₂₁₂ #28248109

Order 13568.

2830 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₀₆	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₂₁₂	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13133142	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂²×Dic₂₁₂	C₄	1	16	yes	direct product factor
9	C₂²×Dic₂₁₂	C₂²	1	12	yes	—
10	C₂×C₄×Dic₁₀₆	—	2	12	no	—
11	C₂×C₄×Dic₁₀₆	C₂²	1	24	yes	—
12	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂²×#11719919	—	2	2	no	—
14	C₂²×C₄×Dic₅₃	—	2	2	no	—
15	C₄×Dic₂₁₂	C₂²	1	64	yes	direct product factor
16	C₂×#13133151	C₂²	1	12	yes	—
17	C₂×#13133142	C₂²	1	24	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
22	C₂²×Dic₁₀₆	—	2	6	no	—
23	C₂²×Dic₁₀₆	—	2	8	no	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	16	yes	—
25	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
26	C₂³×Dic₅₃	—	2	2	no	—
27	C₂³×Dic₅₃	—	2	2	no	—
28	C₂×Dic₂₁₂	C₂×C₄	1	96	yes	direct product factor
29	C₂×Dic₂₁₂	C₂³	1	16	yes	—
30	C₄×Dic₁₀₆	—	2	48	no	—
31	C₄×Dic₁₀₆	C₂³	1	32	yes	—
32	C₂×C₄⋊Dic₅₃	—	2	6	no	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
34	C₂×#11719919	—	2	24	no	—
35	C₂×C₄×Dic₅₃	—	2	24	no	—
36	#13133151	C₂³	1	16	yes	—
37	#13133142	C₂³	1	32	yes	—
38	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	—	2	4	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
45	C₂×C₄²×C₅₃	D₄	1	6	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
47	C₄×C₈×C₅₃	C₂³	1	16	yes	—
48	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	Q₁₆	1	8	yes	complemented
51	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
55	C₂²×C₄×C₅₃	#41	1	4	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
57	C₂²×Dic₅₃	—	2	16	no	—
58	C₂²×Dic₅₃	—	2	12	no	—
59	C₂²×Dic₅₃	—	2	12	no	—
60	C₂²×Dic₅₃	—	4	8	no	—
61	C₂×Dic₁₀₆	—	2	48	no	—
62	C₂×Dic₁₀₆	—	2	12	no	—
63	C₂×Dic₁₀₆	—	2	24	no	—
64	C₂×Dic₁₀₆	C₂²×C₄	1	48	yes	—
65	C₂×Dic₁₀₆	C₂⁴	1	8	yes	—
66	C₂×C₈×C₅₃	—	2	12	no	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	24	yes	—
68	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
69	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
70	C₄²×C₅₃	C₂×D₄	1	12	yes	—
71	C₄²×C₅₃	C₂⁴	1	4	yes	—
72	C₄⋊Dic₅₃	—	2	12	no	—
73	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
74	#11719919	—	2	32	no	—
75	C₄×Dic₅₃	—	2	32	no	—
76	Dic₂₁₂	C₂²×C₄	1	64	yes	direct product factor
77	C₂³×C₅₃	C₄×D₄	1	4	yes	—
78	C₂³×C₅₃	C₂×Q₁₆	1	4	yes	—
79	C₂³×C₅₃	#135	1	4	yes	—
80	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
81	C₂×C₄×C₅₃	—	2	12	no	—
82	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
83	C₂×C₄×C₅₃	C₂×Q₁₆	1	24	yes	complemented
84	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
86	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
87	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
88	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
89	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₅₃	—	2	48	no	—
91	C₂×Dic₅₃	—	2	8	no	—
92	C₂×Dic₅₃	—	2	8	no	—
93	C₂×Dic₅₃	—	4	24	no	—
94	Dic₁₀₆	—	2	48	no	—
95	Dic₁₀₆	—	2	8	no	—
96	Dic₁₀₆	C₂³×C₄	1	16	yes	—
97	C₈×C₅₃	—	2	4	no	—
98	C₈×C₅₃	C₂³×C₄	1	8	yes	—
99	C₂²×C₄×Q₁₆	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₄×Q₁₆	1	16	yes	complemented
101	C₂²×C₅₃	C₂×C₄×D₄	1	6	yes	—
102	C₂²×C₅₃	C₂²×Q₁₆	1	6	yes	—
103	C₂²×C₅₃	C₂×#135	1	6	yes	—
104	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
105	C₄×C₅₃	—	2	4	no	—
106	C₄×C₅₃	C₂×C₄×D₄	1	6	yes	—
107	C₄×C₅₃	C₂²×Q₁₆	1	8	yes	complemented
108	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
110	Dic₅₃	—	2	16	no	—
111	Dic₅₃	—	4	8	no	—
112	C₂³×Q₁₆	—	53	1	no	—
113	C₂²×C₄×Q₈	—	53	2	no	—
114	C₂×C₄×Q₁₆	—	53	24	no	—
115	C₂²×#107	—	53	1	no	—
116	C₂²×#103	—	53	2	no	—
117	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
118	C₂×C₅₃	C₂×C₄×Q₁₆	1	12	yes	complemented
119	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
120	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
122	C₂³×Q₈	—	53	2	no	—
123	C₂²×Q₁₆	—	53	16	no	—
124	C₂²×Q₁₆	—	53	12	no	—
125	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
126	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	—	53	24	no	—
128	C₂×C₄×Q₈	—	106	12	no	—
129	C₂²×C₄⋊C₄	—	53	2	no	—
130	C₂²×C₄⋊C₄	—	106	2	no	—
131	C₂²×C₄²	—	106	2	no	—
132	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
133	C₄×Q₁₆	—	53	64	no	—
134	C₂×#107	—	53	12	no	—
135	C₂×#103	—	53	24	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	12	yes	—
137	C₅₃	C₂²×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	53	16	no	—
139	C₂²×Q₈	—	53	12	no	—
140	C₂²×Q₈	—	106	6	no	—
141	C₂²×Q₈	—	106	8	no	—
142	C₂³×C₄	—	106	2	no	—
143	C₂³×C₄	—	106	2	no	—
144	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
145	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
146	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
147	C₂×Q₁₆	—	53	96	no	—
148	C₂×Q₁₆	—	53	16	no	—
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	C₄×D₅₃	1	8	yes	—
151	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
152	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
153	C₄×Q₈	—	53	32	no	—
154	C₄×Q₈	—	106	48	no	—
155	C₂×C₄⋊C₄	—	53	12	no	—
156	C₂×C₄⋊C₄	—	106	6	no	—
157	C₂×C₄⋊C₄	—	106	24	no	—
158	C₂×C₄²	—	106	24	no	—
159	C₂×C₄²	D₂₁₂	1	6	yes	—
160	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
161	#107	—	53	16	no	—
162	#103	—	53	32	no	—
163	C₄×C₈	C₂²×D₅₃	1	16	yes	—
164	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
165	C₂×Q₈	—	53	48	no	—
166	C₂×Q₈	—	53	8	no	—
167	C₂×Q₈	—	106	48	no	—
168	C₂×Q₈	—	106	12	no	—
169	C₂×Q₈	—	106	24	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	106	16	no	—
172	C₂²×C₄	—	106	12	no	—
173	C₂²×C₄	—	106	12	no	—
174	C₂²×C₄	—	212	8	no	—
175	C₂²×C₄	Dic₂₁₂	1	8	yes	direct product factor
176	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
177	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
178	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
179	C₂²×C₄	C₂×D₂₁₂	1	6	yes	—
180	C₂²×C₄	#11719945	1	4	yes	—
181	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
182	Q₁₆	—	53	64	no	—
183	C₂×C₈	—	2	12	no	—
184	C₂×C₈	C₂×C₄×D₅₃	1	24	yes	—
185	C₂×C₈	C₂³×D₅₃	1	4	yes	—
186	C₂×C₈	C₂³×D₅₃	1	4	yes	—
187	C₄⋊C₄	—	53	8	no	—
188	C₄⋊C₄	—	106	12	no	—
189	C₄⋊C₄	—	106	32	no	—
190	C₄²	—	106	32	no	—
191	C₄²	C₂×D₂₁₂	1	12	yes	—
192	C₄²	C₂³×D₅₃	1	4	yes	—
193	C₂³	C₄×D₂₁₂	1	4	yes	—
194	C₂³	C₂×Dic₂₁₂	1	4	yes	—
195	C₂³	#13133239	1	4	yes	—
196	C₂³	C₂²×D₂₁₂	1	3	yes	—
197	Q₈	—	53	16	no	—
198	Q₈	—	106	48	no	—
199	Q₈	—	106	8	no	—
200	C₂×C₄	—	2	12	no	—
201	C₂×C₄	—	106	48	no	—
202	C₂×C₄	—	106	8	no	—
203	C₂×C₄	—	106	8	no	—
204	C₂×C₄	—	212	24	no	—
205	C₂×C₄	C₄×D₂₁₂	1	12	yes	—
206	C₂×C₄	C₂×Dic₂₁₂	1	24	yes	direct product factor
207	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
208	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
209	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
210	C₂×C₄	C₂²×D₂₁₂	1	4	yes	—
211	C₂×C₄	C₂×#11719945	1	6	yes	—
212	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
213	C₈	—	2	4	no	—
214	C₈	C₂²×C₄×D₅₃	1	8	yes	—
215	C₂²	C₄×Dic₂₁₂	1	16	yes	direct product factor
216	C₂²	C₂×C₄×D₂₁₂	1	6	yes	—
217	C₂²	C₂²×Dic₂₁₂	1	6	yes	—
218	C₂²	C₂×#13133239	1	6	yes	—
219	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	106	16	no	—
222	C₄	—	212	8	no	—
223	C₄	C₂×C₄×D₂₁₂	1	6	yes	—
224	C₄	C₂²×Dic₂₁₂	1	8	yes	direct product factor
225	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
226	C₄	C₂²×#11719945	1	1	yes	characteristic
227	C₂	C₂×C₄×Dic₂₁₂	1	12	yes	direct product factor
228	C₂	C₂²×C₄×D₂₁₂	1	1	yes	characteristic
229	C₂	C₂³×Dic₂₁₂	1	1	yes	characteristic
230	C₂	C₂²×#13133239	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂²×C₄×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₂₁₂	C₂	direct (N × H)
8. C₂²×Dic₂₁₂	C₄	direct (N × H)
15. C₄×Dic₂₁₂	C₂²	direct (N × H)
28. C₂×Dic₂₁₂	C₂×C₄	direct (N × H)
50. C₂²×C₄×C₅₃	Q₁₆	semidirect
76. Dic₂₁₂	C₂²×C₄	direct (N × H)
83. C₂×C₄×C₅₃	C₂×Q₁₆	semidirect
100. C₂²×C₅₃	C₄×Q₁₆	semidirect
107. C₄×C₅₃	C₂²×Q₁₆	semidirect
118. C₂×C₅₃	C₂×C₄×Q₁₆	semidirect
137. C₅₃	C₂²×C₄×Q₁₆	semidirect
175. C₂²×C₄	Dic₂₁₂	direct (N × H)
206. C₂×C₄	C₂×Dic₂₁₂	direct (N × H)
215. C₂²	C₄×Dic₂₁₂	direct (N × H)
224. C₄	C₂²×Dic₂₁₂	direct (N × H)
227. C₂	C₂×C₄×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
C₂³×Dic₂₁₂	2
C₂²×C₄×Dic₁₀₆	2
C₂×C₄×Dic₂₁₂	2
C₂²×#13133151	2
C₂²×#13133142	2
C₂²×C₄×C₈×C₅₃	2
C₂²×C₄×Q₁₆	53

Subgroups — C22×C4×Dic212 #28248109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×Dic₂₁₂ #28248109