Subgroups — C₂×C₄×Q₈×Dic₅₃ #27542289

Order 13568.

3966 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	C₄×Q₈×Dic₅₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×Dic₅₃	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal
5	C₂×C₄²×Dic₅₃	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₅₃	C₄	1	32	yes	direct product factor
8	C₂×Q₈×Dic₅₃	C₂²	1	16	yes	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	6	yes	—
10	C₂²×C₄×Dic₅₃	C₂²	1	3	yes	—
11	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×Dic₅₃	C₂²	1	48	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	24	yes	—
16	C₂×#13133124	C₂²	1	6	yes	—
17	C₄²×Dic₅₃	C₂²	1	24	yes	—
18	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₅₃	C₄	1	16	yes	complemented
20	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
23	C₂³×Dic₅₃	C₂³	1	6	yes	—
24	C₂³×Dic₅₃	C₂³	1	2	yes	—
25	Q₈×Dic₅₃	C₂×C₄	1	128	yes	direct product factor
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	48	yes	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
28	C₂×C₄×Dic₅₃	—	2	12	no	—
29	C₂×C₄×Dic₅₃	—	2	24	no	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
31	C₂×C₄×Dic₅₃	Q₈	1	8	yes	direct product factor
32	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
34	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
36	#13133124	C₂³	1	24	yes	—
37	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	—
38	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
39	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₃	C₂×C₄	1	64	yes	complemented
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
45	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
48	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂×Q₈×C₅₃	C₄²	1	64	yes	complemented
50	C₂×Q₈×C₅₃	C₂²×C₄	1	32	yes	—
51	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
52	C₂²×C₄×C₅₃	—	2	12	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₅₃	#41	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
62	C₂²×Dic₅₃	—	2	12	no	—
63	C₂²×Dic₅₃	—	2	12	no	—
64	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
65	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
66	C₂²×Dic₅₃	C₂×Q₈	1	8	yes	—
67	C₂²×Dic₅₃	#41	1	8	yes	—
68	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
69	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	48	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	48	yes	—
72	C₄⋊Dic₅₃	C₂²×C₄	1	96	yes	—
73	C₄×Dic₅₃	—	2	24	no	—
74	C₄×Dic₅₃	—	2	48	no	—
75	C₄×Dic₅₃	C₂²×C₄	1	96	yes	—
76	C₄×Dic₅₃	C₂×Q₈	1	16	yes	direct product factor
77	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
78	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
79	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
80	C₂³×C₅₃	C₂×#41	1	4	yes	—
81	C₂³×C₅₃	C₂×#41	1	2	yes	—
82	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
83	Q₈×C₅₃	C₂×C₄²	1	64	yes	complemented
84	C₂×C₄×C₅₃	—	2	24	no	—
85	C₂×C₄×C₅₃	—	2	12	no	—
86	C₂×C₄×C₅₃	C₂×C₄²	1	48	yes	—
87	C₂×C₄×C₅₃	C₄×Q₈	1	16	yes	complemented
88	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
89	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
90	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
91	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
92	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
93	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
94	C₂×Dic₅₃	—	2	48	no	—
95	C₂×Dic₅₃	—	2	12	no	—
96	C₂×Dic₅₃	—	2	12	no	—
97	C₂×Dic₅₃	C₄×Q₈	1	32	yes	direct product factor
98	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
99	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
100	C₂×Dic₅₃	C₂²×Q₈	1	8	yes	—
101	C₂×Dic₅₃	C₂×#41	1	8	yes	—
102	C₂×C₄²×Q₈	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
104	C₂²×C₅₃	C₂×C₄×Q₈	1	8	yes	—
105	C₂²×C₅₃	C₂×C₄×Q₈	1	4	yes	—
106	C₂²×C₅₃	C₄×#41	1	8	yes	—
107	C₂²×C₅₃	C₄×#41	1	4	yes	—
108	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
109	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
110	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
111	C₂²×C₅₃	C₂²×#41	1	2	yes	—
112	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
113	C₄×C₅₃	—	2	12	no	—
114	C₄×C₅₃	C₂²×C₄²	1	24	yes	—
115	C₄×C₅₃	C₂×C₄×Q₈	1	8	yes	complemented
116	Dic₅₃	—	2	24	no	—
117	Dic₅₃	C₂×C₄×Q₈	1	16	yes	direct product factor
118	C₂²×C₄×Q₈	—	53	2	no	—
119	C₂²×C₄×Q₈	D₅₃	1	1	yes	characteristic
120	C₄²×Q₈	—	53	16	no	—
121	C₂×C₄×C₄⋊C₄	—	53	3	no	—
122	C₂×C₄×C₄⋊C₄	—	53	6	no	—
123	C₂×C₄³	—	53	3	no	—
124	C₂×C₅₃	C₄²×Q₈	1	8	yes	complemented
125	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
126	C₂×C₅₃	C₂²×C₄×Q₈	1	2	yes	—
127	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
128	C₂×C₅₃	C₂×C₄×#41	1	2	yes	—
129	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
130	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂×C₄×Q₈	—	53	32	no	—
132	C₂×C₄×Q₈	—	53	16	no	—
133	C₂×C₄×Q₈	Dic₅₃	1	16	yes	direct product factor
134	C₂×C₄×Q₈	C₂×D₅₃	1	8	yes	—
135	C₂²×C₄⋊C₄	—	53	6	no	—
136	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
137	C₂²×C₄²	—	53	6	no	—
138	C₂²×C₄²	—	53	1	no	—
139	C₂²×C₄²	—	53	6	no	—
140	C₂²×C₄²	—	53	3	no	—
141	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
142	C₄×C₄⋊C₄	—	53	48	no	—
143	C₄×C₄⋊C₄	—	53	24	no	—
144	C₂×#95	—	53	6	no	—
145	C₄³	—	53	24	no	—
146	C₅₃	C₂×C₄²×Q₈	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	C₄×D₅₃	1	16	yes	—
148	C₂²×Q₈	C₂×Dic₅₃	1	8	yes	—
149	C₂²×Q₈	C₂²×D₅₃	1	4	yes	—
150	C₂³×C₄	—	53	2	no	—
151	C₂³×C₄	—	53	6	no	—
152	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
153	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
154	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
155	C₄×Q₈	—	53	128	no	—
156	C₄×Q₈	C₂×Dic₅₃	1	64	yes	direct product factor
157	C₂×C₄⋊C₄	—	53	48	no	—
158	C₂×C₄⋊C₄	—	53	24	no	—
159	C₂×C₄⋊C₄	C₂×Dic₅₃	1	24	yes	—
160	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
161	C₂×C₄²	—	53	48	no	—
162	C₂×C₄²	—	53	8	no	—
163	C₂×C₄²	—	53	24	no	—
164	C₂×C₄²	—	53	4	no	—
165	C₂×C₄²	—	53	24	no	—
166	C₂×C₄²	—	53	12	no	—
167	C₂×C₄²	—	106	24	no	—
168	C₂×C₄²	—	106	12	no	—
169	C₂×C₄²	C₂×Dic₅₃	1	24	yes	—
170	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
171	#95	—	53	24	no	—
172	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
173	C₂×Q₈	C₄×Dic₅₃	1	64	yes	direct product factor
174	C₂×Q₈	C₂×C₄×D₅₃	1	32	yes	—
175	C₂×Q₈	C₂²×Dic₅₃	1	16	yes	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	—	53	8	no	—
178	C₂²×C₄	—	53	8	no	—
179	C₂²×C₄	—	53	8	no	—
180	C₂²×C₄	—	53	4	no	—
181	C₂²×C₄	—	53	24	no	—
182	C₂²×C₄	—	53	12	no	—
183	C₂²×C₄	—	106	12	no	—
184	C₂²×C₄	—	106	12	no	—
185	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
186	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
187	C₂²×C₄	#11719949	1	4	yes	—
188	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
189	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
190	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
191	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
192	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
193	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
194	C₄⋊C₄	—	53	96	no	—
195	C₄⋊C₄	C₂²×Dic₅₃	1	48	yes	—
196	C₄²	—	53	96	no	—
197	C₄²	—	53	16	no	—
198	C₄²	—	106	48	no	—
199	C₄²	—	106	24	no	—
200	C₄²	C₂²×Dic₅₃	1	48	yes	—
201	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
202	C₂³	C₂³×Dic₅₃	1	2	yes	—
203	C₂³	C₂×Q₈×D₅₃	1	2	yes	—
204	C₂³	#41×D₅₃	1	4	yes	—
205	C₂³	C₂×#11719949	1	2	yes	—
206	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
207	Q₈	C₂×C₄×Dic₅₃	1	64	yes	direct product factor
208	C₂×C₄	—	2	24	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	53	32	no	—
211	C₂×C₄	—	53	8	no	—
212	C₂×C₄	—	53	8	no	—
213	C₂×C₄	—	53	8	no	—
214	C₂×C₄	—	53	24	no	—
215	C₂×C₄	—	106	48	no	—
216	C₂×C₄	—	106	12	no	—
217	C₂×C₄	—	106	12	no	—
218	C₂×C₄	C₂×C₄×Dic₅₃	1	48	yes	—
219	C₂×C₄	Q₈×Dic₅₃	1	16	yes	direct product factor
220	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
221	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
222	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
223	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
224	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
225	C₂×C₄	C₂×#11719949	1	4	yes	—
226	C₂²	C₂²×C₄×Dic₅₃	1	4	yes	—
227	C₂²	C₄×Q₈×D₅₃	1	8	yes	—
228	C₂²	C₂×Q₈×Dic₅₃	1	4	yes	—
229	C₂²	C₄×#11719949	1	8	yes	—
230	C₂²	#41×Dic₅₃	1	4	yes	—
231	C₂²	C₂³×C₄×D₅₃	1	2	yes	—
232	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
233	C₂²	C₂²×Q₈×D₅₃	1	1	yes	characteristic
234	C₂²	C₂×#41×D₅₃	1	2	yes	—
235	C₂²	C₂²×#11719949	1	1	yes	characteristic
236	C₄	—	2	12	no	—
237	C₄	—	53	16	no	—
238	C₄	—	106	24	no	—
239	C₄	C₂²×C₄×Dic₅₃	1	24	yes	—
240	C₄	C₂×Q₈×Dic₅₃	1	8	yes	direct product factor
241	C₂	C₄×Q₈×Dic₅₃	1	8	yes	direct product factor
242	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
243	C₂	C₂×C₄×Q₈×D₅₃	1	2	yes	—
244	C₂	C₂²×Q₈×Dic₅₃	1	1	yes	characteristic
245	C₂	C₂×C₄×#11719949	1	2	yes	—
246	C₂	C₂×#41×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂²×C₄×Q₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂⁴
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×C₄²×Q₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₅₃	C₂	direct (N × H)
7. C₂×Q₈×Dic₅₃	C₄	direct (N × H)
19. C₂×C₄×Q₈×C₅₃	C₄	semidirect
25. Q₈×Dic₅₃	C₂×C₄	direct (N × H)
31. C₂×C₄×Dic₅₃	Q₈	direct (N × H)
43. C₄×Q₈×C₅₃	C₂×C₄	semidirect
49. C₂×Q₈×C₅₃	C₄²	semidirect
76. C₄×Dic₅₃	C₂×Q₈	direct (N × H)
83. Q₈×C₅₃	C₂×C₄²	semidirect
87. C₂×C₄×C₅₃	C₄×Q₈	semidirect
97. C₂×Dic₅₃	C₄×Q₈	direct (N × H)
115. C₄×C₅₃	C₂×C₄×Q₈	semidirect
117. Dic₅₃	C₂×C₄×Q₈	direct (N × H)
124. C₂×C₅₃	C₄²×Q₈	semidirect
133. C₂×C₄×Q₈	Dic₅₃	direct (N × H)
146. C₅₃	C₂×C₄²×Q₈	semidirect
156. C₄×Q₈	C₂×Dic₅₃	direct (N × H)
173. C₂×Q₈	C₄×Dic₅₃	direct (N × H)
207. Q₈	C₂×C₄×Dic₅₃	direct (N × H)
219. C₂×C₄	Q₈×Dic₅₃	direct (N × H)
240. C₄	C₂×Q₈×Dic₅₃	direct (N × H)
241. C₂	C₄×Q₈×Dic₅₃	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×Q₈×Dic₅₃	2
C₄×Q₈×Dic₅₃	2
C₂×C₄⋊C₄×Dic₅₃	2
C₂×C₄×C₄⋊Dic₅₃	2
C₂×C₄²×Dic₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂×C₄²×Q₈	53

Subgroups — C2×C4×Q8×Dic53 #27542289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₈×Dic₅₃ #27542289