Subgroups — C₂×C₄×Dic₄₂₄ #27537368

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₁₂	C₂	1	2	yes	maximal
3	C₄×Dic₄₂₄	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810428	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₆	—	2	2	no	—
9	C₂×Dic₄₂₄	C₄	1	8	yes	direct product factor
10	C₂×Dic₄₂₄	C₂²	1	4	yes	—
11	C₄×Dic₂₁₂	—	2	4	no	—
12	C₄×Dic₂₁₂	C₂²	1	8	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133142	—	2	2	no	—
15	#16810449	C₂²	1	4	yes	—
16	#16810428	C₂²	1	8	yes	—
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
21	C₂²×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	4	no	—
24	C₂×Dic₂₁₂	C₂×C₄	1	8	yes	—
25	C₂×Dic₂₁₂	C₂³	1	4	yes	—
26	C₄×Dic₁₀₆	—	2	8	no	—
27	C₄×Dic₁₀₆	—	4	4	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×#11719919	—	4	2	no	—
30	C₂×C₄×Dic₅₃	—	4	2	no	—
31	Dic₄₂₄	C₂×C₄	1	16	yes	direct product factor
32	#13133151	—	2	2	no	—
33	#13133151	C₂³	1	4	yes	—
34	#13133142	—	2	8	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₂×C₁₆×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₄×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	2	8	no	—
50	C₂×Dic₁₀₆	—	2	4	no	—
51	C₂×Dic₁₀₆	—	4	2	no	—
52	C₂×Dic₁₀₆	—	4	4	no	—
53	C₂×C₈×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	2	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₄²×C₅₃	D₈	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄⋊Dic₅₃	—	2	4	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	#11719919	—	4	8	no	—
64	C₄×Dic₅₃	—	4	8	no	—
65	Dic₂₁₂	—	2	8	no	—
66	Dic₂₁₂	—	2	4	no	—
67	Dic₂₁₂	C₂²×C₄	1	8	yes	—
68	C₁₆×C₅₃	—	2	2	no	—
69	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	Q₃₂	1	4	yes	complemented
73	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₅₃	#135	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₅₃	—	4	4	no	—
80	C₂×Dic₅₃	—	4	8	no	—
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	4	no	—
83	Dic₁₀₆	—	2	8	no	—
84	Dic₁₀₆	—	4	4	no	—
85	Dic₁₀₆	—	4	8	no	—
86	C₈×C₅₃	—	2	2	no	—
87	C₈×C₅₃	C₄×D₄	1	2	yes	—
88	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
90	C₂×C₄×Q₃₂	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	C₄×D₈	1	2	yes	—
92	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
93	C₂²×C₅₃	#508	1	2	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	C₄×D₈	1	2	yes	—
97	C₄×C₅₃	C₂×Q₃₂	1	4	yes	complemented
98	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
99	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
100	Dic₅₃	—	4	8	no	—
101	Dic₅₃	—	8	4	no	—
102	C₂²×Q₃₂	—	53	1	no	—
103	C₂×C₄×Q₁₆	—	53	2	no	—
104	C₄×Q₃₂	—	53	8	no	—
105	C₂×#366	—	53	1	no	—
106	C₂×#358	—	53	2	no	—
107	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
108	C₂×C₅₃	C₄×Q₃₂	1	4	yes	complemented
109	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
111	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
112	C₂²×Q₁₆	—	53	2	no	—
113	C₂×C₄×Q₈	—	106	2	no	—
114	C₂×Q₃₂	—	53	8	no	—
115	C₂×Q₃₂	—	53	4	no	—
116	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
117	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
118	C₄×Q₁₆	—	53	8	no	—
119	C₄×Q₁₆	—	106	4	no	—
120	C₂×#107	—	53	2	no	—
121	C₂×#103	—	106	2	no	—
122	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
123	#366	—	53	4	no	—
124	#358	—	53	8	no	—
125	C₄×C₁₆	C₂×D₅₃	1	4	yes	—
126	C₅₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	106	2	no	—
128	C₂×Q₁₆	—	53	8	no	—
129	C₂×Q₁₆	—	53	4	no	—
130	C₂×Q₁₆	—	106	2	no	—
131	C₂×Q₁₆	—	106	4	no	—
132	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
134	C₄×Q₈	—	106	8	no	—
135	C₄×Q₈	—	212	4	no	—
136	C₂×C₄⋊C₄	—	106	2	no	—
137	C₂×C₄⋊C₄	—	212	2	no	—
138	C₂×C₄²	—	212	2	no	—
139	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
140	Q₃₂	—	53	16	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
143	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
144	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
145	#107	—	53	4	no	—
146	#107	—	106	2	no	—
147	#103	—	106	8	no	—
148	C₄×C₈	D₂₁₂	1	2	yes	—
149	C₄×C₈	C₂²×D₅₃	1	2	yes	—
150	C₂×Q₈	—	106	8	no	—
151	C₂×Q₈	—	106	4	no	—
152	C₂×Q₈	—	212	2	no	—
153	C₂×Q₈	—	212	4	no	—
154	C₂²×C₄	—	212	2	no	—
155	C₂²×C₄	—	212	2	no	—
156	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
157	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
159	Q₁₆	—	53	8	no	—
160	Q₁₆	—	106	8	no	—
161	Q₁₆	—	106	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
164	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
165	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
166	C₂×C₈	#11719945	1	2	yes	—
167	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄⋊C₄	—	212	8	no	—
171	C₄²	—	212	8	no	—
172	C₄²	D₄₂₄	1	2	yes	—
173	C₄²	C₂×D₂₁₂	1	2	yes	—
174	C₁₆	—	2	2	no	—
175	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
176	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
177	Q₈	—	106	8	no	—
178	Q₈	—	212	4	no	—
179	Q₈	—	212	8	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	424	4	no	—
182	C₂×C₄	—	212	4	no	—
183	C₂×C₄	—	212	4	no	—
184	C₂×C₄	—	212	8	no	—
185	C₂×C₄	Dic₄₂₄	1	4	yes	direct product factor
186	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
187	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
188	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
189	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
190	C₂×C₄	#13133239	1	2	yes	—
191	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	C₄×D₂₁₂	1	2	yes	—
194	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
195	C₈	C₂×#11719945	1	1	yes	characteristic
196	C₂²	C₄×D₄₂₄	1	2	yes	—
197	C₂²	C₂×Dic₄₂₄	1	2	yes	—
198	C₂²	#16811055	1	2	yes	—
199	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	424	4	no	—
202	C₄	—	212	8	no	—
203	C₄	C₄×D₄₂₄	1	2	yes	—
204	C₄	C₂×Dic₄₂₄	1	4	yes	direct product factor
205	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
206	C₄	C₂×#13133239	1	1	yes	characteristic
207	C₂	C₄×Dic₄₂₄	1	4	yes	direct product factor
208	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
209	C₂	C₂²×Dic₄₂₄	1	1	yes	characteristic
210	C₂	C₂×#16811055	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₄₂₄	C₂	direct (N × H)
9. C₂×Dic₄₂₄	C₄	direct (N × H)
31. Dic₄₂₄	C₂×C₄	direct (N × H)
72. C₂×C₄×C₅₃	Q₃₂	semidirect
97. C₄×C₅₃	C₂×Q₃₂	semidirect
108. C₂×C₅₃	C₄×Q₃₂	semidirect
126. C₅₃	C₂×C₄×Q₃₂	semidirect
185. C₂×C₄	Dic₄₂₄	direct (N × H)
204. C₄	C₂×Dic₄₂₄	direct (N × H)
207. C₂	C₄×Dic₄₂₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₄	2
C₂×C₄×Dic₂₁₂	2
C₄×Dic₄₂₄	2
C₂×#16810449	2
C₂×#16810428	2
C₂×C₄×C₁₆×C₅₃	2
C₂×C₄×Q₃₂	53

Subgroups — C2×C4×Dic424 #27537368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Dic₄₂₄ #27537368