Subgroups — #28318875

Order 13568.

990 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	C₂×Q₈×#88324	C₂	1	2	yes	maximal
2	C₄⋊C₄×#88324	C₂	1	6	yes	maximal
3	C₄×C₄⋊#88324	C₂	1	3	yes	maximal
4	#20290429	C₂	1	3	yes	maximal
5	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	C₄	1	16	yes	complemented
7	C₂×C₄⋊#88324	C₂²	1	6	yes	—
8	C₂×C₄×#88324	C₂²	1	6	yes	—
9	C₂×C₄×#88324	C₂²	1	6	yes	—
10	#16811489	C₂²	1	1	yes	characteristic
11	#16811489	C₂²	1	3	yes	—
12	#16811489	C₂²	1	6	yes	—
13	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₅₃	C₂²	1	3	yes	—
17	C₄×C₄⋊Dic₅₃	C₄	1	3	yes	—
18	C₄²×Dic₅₃	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂²×#88324	C₂³	1	6	yes	—
21	C₂²×#88324	C₂³	1	2	yes	—
22	C₄⋊#88324	C₂×C₄	1	24	yes	—
23	C₄×#88324	—	4	6	no	—
24	C₄×#88324	C₂×C₄	1	24	yes	—
25	Q₈×Dic₅₃	C₈	1	8	yes	complemented
26	Q₈×Dic₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
35	#13133124	C₂×C₄	1	6	yes	—
36	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	C₈	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
40	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
41	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×#88324	—	2	6	no	—
51	C₂×#88324	—	2	2	no	—
52	C₂×#88324	—	2	2	no	—
53	C₂×#88324	—	2	6	no	—
54	C₂×#88324	C₂²×C₄	1	4	yes	—
55	C₂×#88324	C₂²×C₄	1	12	yes	—
56	C₄⋊C₄×C₅₃	C₂×C₈	1	12	yes	—
57	C₄²×C₅₃	C₂×C₈	1	12	yes	—
58	C₄⋊Dic₅₃	C₄²	1	12	yes	—
59	C₄⋊Dic₅₃	C₂×C₈	1	12	yes	—
60	C₄×Dic₅₃	—	2	3	no	—
61	C₄×Dic₅₃	—	2	2	no	—
62	C₄×Dic₅₃	—	2	3	no	—
63	C₄×Dic₅₃	—	4	3	no	—
64	C₄×Dic₅₃	—	4	3	no	—
65	C₄×Dic₅₃	C₂×C₈	1	12	yes	—
66	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
67	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
68	Q₈×C₅₃	C₄×C₈	1	16	yes	complemented
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	3	no	—
71	C₂×C₄×C₅₃	—	2	3	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
76	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	3	no	—
80	C₂×Dic₅₃	—	2	3	no	—
81	C₂×Dic₅₃	—	2	3	no	—
82	C₂×Dic₅₃	—	2	3	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	C₂×C₄²	1	6	yes	—
85	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
86	C₂×Dic₅₃	C₂²×C₈	1	6	yes	—
87	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
88	C₂×Dic₅₃	H(5,𝔽₂)	1	1	yes	characteristic
89	C₂×Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
90	C₂×Dic₅₃	2₋¹⁺⁴	1	2	yes	—
91	#88324	—	2	4	no	—
92	#88324	—	2	12	no	—
93	#20569	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
95	C₂²×C₅₃	#519	1	1	yes	characteristic
96	C₂²×C₅₃	#520	1	1	yes	characteristic
97	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
98	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
99	C₂²×C₅₃	#568	1	1	yes	characteristic
100	C₂²×C₅₃	#568	1	1	yes	characteristic
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	6	no	—
103	C₄×C₅₃	C₂×C₄×C₈	1	12	yes	—
104	Dic₅₃	—	2	6	no	—
105	Dic₅₃	—	2	2	no	—
106	Dic₅₃	—	2	6	no	—
107	Dic₅₃	#519	1	2	yes	—
108	Dic₅₃	#520	1	2	yes	—
109	C₂×C₈×Q₈	—	53	2	no	—
110	C₄²×Q₈	—	53	1	no	—
111	C₈×C₄⋊C₄	—	53	6	no	—
112	C₄×C₄⋊C₈	—	53	3	no	—
113	#1744	—	53	3	no	—
114	C₂×C₅₃	#2276	1	1	yes	characteristic
115	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
116	C₂×C₅₃	#2875	1	1	yes	characteristic
117	C₂×C₅₃	#2982	1	1	yes	characteristic
118	C₂×C₅₃	#2982	1	1	yes	characteristic
119	C₂×C₅₃	#2988	1	1	yes	characteristic
120	C₂×C₅₃	#2988	1	1	yes	characteristic
121	C₂×C₄×Q₈	—	53	1	no	—
122	C₂×C₄×Q₈	—	53	1	no	—
123	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
124	C₈×Q₈	—	53	16	no	—
125	C₂×C₄⋊C₈	—	53	6	no	—
126	C₂×C₄×C₈	—	53	6	no	—
127	C₂×C₄×C₈	—	53	6	no	—
128	C₄×C₄⋊C₄	—	53	3	no	—
129	C₄×C₄⋊C₄	—	53	3	no	—
130	C₄×C₄⋊C₄	—	53	3	no	—
131	C₄³	—	53	3	no	—
132	#336	—	53	1	no	—
133	#336	—	53	3	no	—
134	#336	—	53	6	no	—
135	C₅₃	#20569	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	C₂×#6157	1	1	yes	characteristic
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	—	53	6	no	—
139	C₄×Q₈	—	53	8	no	—
140	C₄×Q₈	—	53	8	no	—
141	C₄×Q₈	#88324	1	8	yes	complemented
142	C₂×C₄⋊C₄	—	53	3	no	—
143	C₂×C₄⋊C₄	—	53	3	no	—
144	C₂×C₄⋊C₄	C₂×#6157	1	3	yes	—
145	C₂×C₄²	—	53	3	no	—
146	C₂×C₄²	—	53	3	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	53	3	no	—
149	C₂×C₄²	—	53	3	no	—
150	C₂×C₄²	—	53	3	no	—
151	C₂×C₄²	C₂×#6157	1	3	yes	—
152	C₄⋊C₈	—	53	24	no	—
153	C₄×C₈	—	53	24	no	—
154	C₄×C₈	—	212	6	no	—
155	#95	—	53	6	no	—
156	C₂×Q₈	D₅₃⋊C₈	1	4	yes	—
157	C₂×Q₈	C₄×#6157	1	4	yes	—
158	C₂×Q₈	C₂×#88324	1	4	yes	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	3	no	—
161	C₂²×C₄	—	53	3	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	C₂²×#6157	1	3	yes	—
164	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
165	C₂²×C₄	C₂²×#6157	1	3	yes	—
166	C₂×C₈	—	53	4	no	—
167	C₂×C₈	—	53	12	no	—
168	C₂×C₈	—	106	6	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	6	no	—
171	C₂×C₈	—	106	2	no	—
172	C₄⋊C₄	—	53	12	no	—
173	C₄⋊C₄	—	53	12	no	—
174	C₄⋊C₄	C₂×#88324	1	12	yes	—
175	C₄²	—	53	12	no	—
176	C₄²	—	53	12	no	—
177	C₄²	—	106	3	no	—
178	C₄²	—	106	2	no	—
179	C₄²	—	106	3	no	—
180	C₄²	—	212	3	no	—
181	C₄²	—	212	3	no	—
182	C₄²	C₂×#88324	1	12	yes	—
183	C₂³	C₂³×#6157	1	1	yes	characteristic
184	Q₈	C₄×#88324	1	16	yes	complemented
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	3	no	—
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	2	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	6	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	6	no	—
196	C₂×C₄	—	106	1	no	—
197	C₂×C₄	—	106	3	no	—
198	C₂×C₄	—	106	1	no	—
199	C₂×C₄	—	106	3	no	—
200	C₂×C₄	—	106	3	no	—
201	C₂×C₄	—	106	3	no	—
202	C₂×C₄	C₂×C₄×#6157	1	6	yes	—
203	C₂×C₄	C₂²×#88324	1	2	yes	—
204	C₂×C₄	C₂²×#88324	1	6	yes	—
205	C₂×C₄	C₂²×#88324	1	6	yes	—
206	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
207	C₈	—	106	12	no	—
208	C₈	—	106	4	no	—
209	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
210	C₂²	#16811691	1	1	yes	characteristic
211	C₂²	#16811692	1	1	yes	characteristic
212	C₂²	C₂³×#88324	1	1	yes	characteristic
213	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
214	C₂²	#16811702	1	1	yes	characteristic
215	C₂²	#16811703	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	6	no	—
218	C₄	—	53	2	no	—
219	C₄	—	53	2	no	—
220	C₄	—	106	2	no	—
221	C₄	—	106	6	no	—
222	C₄	—	106	6	no	—
223	C₄	C₂×C₄×#88324	1	12	yes	—
224	C₂	#20291155	1	1	yes	characteristic
225	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
226	C₂	#20291287	1	1	yes	characteristic
227	C₂	#20291608	1	1	yes	characteristic
228	C₂	#20291609	1	1	yes	characteristic
229	C₂	#20291630	1	1	yes	characteristic
230	C₂	#20291633	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×C₄×Q₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×Dic₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	#20569
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
6. Q₈×#88324	C₄	semidirect
25. Q₈×Dic₅₃	C₈	semidirect
37. C₄×Q₈×C₅₃	C₈	semidirect
68. Q₈×C₅₃	C₄×C₈	semidirect
135. C₅₃	#20569	semidirect
141. C₄×Q₈	#88324	semidirect
184. Q₈	C₄×#88324	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88324	2
C₄⋊C₄×#88324	2
C₄×C₄⋊#88324	2
#20290429	2
C₄×Q₈×Dic₅₃	2
#20569	53