Subgroups — #28328445

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
2	#20291094	C₂	1	6	yes	maximal, complemented
3	#20291075	C₂	1	3	yes	maximal, complemented
4	C₄×D₅₃⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	C₈×#11719949	C₂	1	1	yes	maximal, characteristic
6	C₂×D₅₃⋊C₁₆	C₂²	1	6	yes	—
7	C₄⋊#11719910	C₂²	1	6	yes	—
8	#16811518	C₂²	1	6	yes	—
9	#16811509	C₂²	1	3	yes	—
10	#16811509	C₂²	1	1	yes	characteristic
11	C₄×#11719910	C₂²	1	6	yes	—
12	C₄×#11719949	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₂²	1	1	yes	characteristic
14	C₈×D₂₁₂	C₄	1	3	yes	—
15	#16810688	C₂²	1	3	yes	—
16	#16810675	C₄	1	3	yes	—
17	C₄×C₈×D₅₃	C₂²	1	3	yes	—
18	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	D₅₃⋊C₁₆	—	2	12	no	—
20	C₂×#11719910	C₂³	1	6	yes	—
21	C₂×#11719910	C₂³	1	2	yes	—
22	C₂×#11719949	C₈	1	1	yes	characteristic
23	C₂×C₈×D₅₃	C₂×C₄	1	3	yes	—
24	C₂×C₈×D₅₃	C₂³	1	3	yes	—
25	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
26	C₄×D₂₁₂	C₂×C₄	1	3	yes	—
27	#13133179	C₈	1	3	yes	—
28	#13133175	C₈	1	3	yes	—
29	C₄²×D₅₃	C₂×C₄	1	3	yes	—
30	C₄⋊#88322	C₂×C₄	1	3	yes	—
31	#13133129	C₂×C₄	1	6	yes	—
32	C₄×#88322	C₂³	1	3	yes	—
33	C₈×Dic₅₃	C₂×C₄	1	3	yes	—
34	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
38	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
39	#11719949	C₁₆	1	8	yes	complemented
40	C₂×D₂₁₂	C₂×C₈	1	3	yes	—
41	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
42	C₂×C₄×D₅₃	C₂×C₈	1	3	yes	—
43	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
47	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
48	#11719921	C₂×C₈	1	6	yes	—
49	C₄⋊Dic₅₃	C₂×C₈	1	3	yes	—
50	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
51	C₄×Dic₅₃	C₂×C₈	1	3	yes	—
52	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂²×C₄	1	3	yes	—
54	C₂×#88322	C₂⁴	1	1	yes	characteristic
55	C₈×D₅₃	—	2	6	no	—
56	C₈×D₅₃	—	2	6	no	—
57	#11719910	—	2	6	no	—
58	#11719910	#41	1	4	yes	complemented
59	C₂²×D₅₃	C₂²×C₈	1	3	yes	—
60	Q₈×C₅₃	C₂×C₁₆	1	4	yes	complemented
61	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×Dic₅₃	C₂²×C₈	1	3	yes	—
65	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
67	D₂₁₂	C₂×C₁₆	1	12	yes	complemented
68	C₄×D₅₃	—	2	3	no	—
69	C₄×D₅₃	—	2	3	no	—
70	C₄×D₅₃	—	2	6	no	—
71	C₄×D₅₃	C₂×C₁₆	1	12	yes	complemented
72	C₈×C₅₃	—	2	3	no	—
73	C₈×C₅₃	#131	1	2	yes	—
74	#88322	—	2	3	no	—
75	#88322	C₂×#41	1	2	yes	—
76	C₁₆×#41	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
78	C₂×D₅₃	—	2	3	no	—
79	C₂×D₅₃	—	2	3	no	—
80	C₂×D₅₃	C₂²×C₁₆	1	6	yes	—
81	C₄×C₅₃	—	2	3	no	—
82	C₄×C₅₃	C₂²×C₁₆	1	6	yes	—
83	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
84	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
85	Dic₅₃	—	2	3	no	—
86	Dic₅₃	C₂²×C₁₆	1	2	yes	—
87	Dic₅₃	#504	1	2	yes	—
88	C₈×#41	—	53	1	no	—
89	Q₈×C₁₆	—	53	2	no	—
90	D₄×C₁₆	—	53	6	no	—
91	#2162	—	53	3	no	—
92	C₂×C₄×C₁₆	—	53	3	no	—
93	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
94	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
95	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
96	D₅₃	—	2	6	no	—
97	C₄×#41	—	53	1	no	—
98	C₂²×C₁₆	—	53	6	no	—
99	C₈×Q₈	—	53	1	no	—
100	C₈×Q₈	#6157	1	1	yes	characteristic
101	C₈×D₄	—	53	3	no	—
102	C₈×D₄	—	53	3	no	—
103	#431	—	53	3	no	—
104	C₂×C₄×C₈	—	53	3	no	—
105	C₄⋊C₁₆	—	53	6	no	—
106	C₂²⋊C₁₆	—	53	6	no	—
107	C₄×C₁₆	—	53	1	no	—
108	C₄×C₁₆	—	53	3	no	—
109	C₄×C₁₆	—	53	6	no	—
110	C₅₃	C₁₆×#41	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	—	53	1	no	—
112	C₂²×C₈	—	53	3	no	—
113	C₂²×C₈	—	53	3	no	—
114	C₄×Q₈	—	53	1	no	—
115	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
116	C₄×D₄	—	53	3	no	—
117	C₄×D₄	—	53	3	no	—
118	#117	—	53	3	no	—
119	C₂×C₄²	—	53	3	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	6	no	—
122	C₂×C₁₆	—	106	12	no	—
123	C₄⋊C₈	—	53	3	no	—
124	C₄⋊C₈	C₂×#6157	1	3	yes	—
125	C₂²⋊C₈	—	53	6	no	—
126	C₄×C₈	—	53	3	no	—
127	C₄×C₈	—	53	1	no	—
128	C₄×C₈	—	53	3	no	—
129	C₄×C₈	C₂×#6157	1	3	yes	—
130	#41	—	53	8	no	—
131	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
132	C₂×D₄	—	53	3	no	—
133	C₂²×C₄	—	53	3	no	—
134	C₂²×C₄	—	53	3	no	—
135	C₂²×C₄	—	53	3	no	—
136	C₂×C₈	—	53	1	no	—
137	C₂×C₈	—	53	3	no	—
138	C₂×C₈	—	106	6	no	—
139	C₂×C₈	—	106	6	no	—
140	C₂×C₈	C₂²×#6157	1	3	yes	—
141	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
142	C₄⋊C₄	—	53	3	no	—
143	C₄⋊C₄	D₅₃⋊C₈	1	3	yes	—
144	C₂²⋊C₄	—	53	6	no	—
145	C₄²	—	53	3	no	—
146	C₄²	—	53	3	no	—
147	C₄²	—	53	1	no	—
148	C₄²	C₂²×#6157	1	3	yes	—
149	C₁₆	—	53	4	no	—
150	C₁₆	—	106	6	no	—
151	C₂³	—	53	3	no	—
152	Q₈	D₅₃⋊C₁₆	1	4	yes	complemented
153	D₄	—	53	12	no	—
154	C₂×C₄	—	53	1	no	—
155	C₂×C₄	—	53	1	no	—
156	C₂×C₄	—	53	12	no	—
157	C₂×C₄	—	53	3	no	—
158	C₂×C₄	—	106	6	no	—
159	C₂×C₄	—	106	3	no	—
160	C₂×C₄	—	106	3	no	—
161	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
162	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
163	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
164	C₈	—	2	3	no	—
165	C₈	—	53	2	no	—
166	C₈	—	106	3	no	—
167	C₈	#13133305	1	2	yes	—
168	C₂²	—	53	6	no	—
169	C₂²	—	106	3	no	—
170	C₂²	—	106	3	no	—
171	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
172	C₄	—	2	3	no	—
173	C₄	—	53	2	no	—
174	C₄	—	53	2	no	—
175	C₄	—	106	3	no	—
176	C₄	C₂×D₅₃⋊C₁₆	1	6	yes	—
177	C₄	#41×#6157	1	1	yes	characteristic
178	C₄	C₂×#13133305	1	1	yes	characteristic
179	C₂	—	106	6	no	—
180	C₂	#20291521	1	1	yes	characteristic
181	C₂	C₂²×D₅₃⋊C₁₆	1	1	yes	characteristic
182	C₂	D₅₃⋊#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₈×Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#11719949	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₈×Q₈	#6157
hypocenter	C₅₃	C₁₆×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#11719910	C₂	semidirect
2. #20291094	C₂	semidirect
3. #20291075	C₂	semidirect
4. C₄×D₅₃⋊C₁₆	C₂	semidirect
39. #11719949	C₁₆	semidirect
58. #11719910	#41	semidirect
60. Q₈×C₅₃	C₂×C₁₆	semidirect
67. D₂₁₂	C₂×C₁₆	semidirect
71. C₄×D₅₃	C₂×C₁₆	semidirect
110. C₅₃	C₁₆×#41	semidirect
152. Q₈	D₅₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719910	2
#20291094	2
#20291075	2
C₄×D₅₃⋊C₁₆	2
C₈×#11719949	2
C₁₆×#41	53