Subgroups — #28328537

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	8	yes	maximal, complemented
2	#20291076	C₂	1	4	yes	maximal
3	#20291072	C₂	1	2	yes	maximal
4	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719910	C₂²	1	4	yes	—
6	C₄⋊#11719910	C₂²	1	8	yes	—
7	C₂²⋊#11719910	C₂²	1	4	yes	—
8	#16811511	C₂²	1	8	yes	—
9	C₄×#11719910	C₂²	1	4	yes	—
10	Q₈×#88322	C₄	1	4	yes	—
11	Q₈×#88322	C₂²	1	4	yes	—
12	C₂×C₄⋊#88322	C₄	1	2	yes	—
13	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719910	C₂³	1	8	yes	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	C₄⋊#88322	C₂×C₄	1	8	yes	—
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	C₄×#88322	C₂×C₄	1	2	yes	—
24	C₄×#88322	C₂×C₄	1	4	yes	—
25	C₄×#88322	C₂³	1	2	yes	—
26	C₄×#88322	C₂³	1	4	yes	—
27	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
28	C₄×Q₈×C₅₃	C₈	1	4	yes	—
29	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×Q₈×C₅₃	—	2	4	no	—
35	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
42	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
43	C₄²×C₅₃	C₂×C₈	1	4	yes	—
44	C₄²×C₅₃	C₂×C₈	1	2	yes	—
45	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
46	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	—	4	2	no	—
50	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
51	C₂×#88322	C₂²×C₄	1	2	yes	—
52	C₂×#88322	C₂²×C₄	1	4	yes	—
53	C₂×#88322	C₂⁴	1	1	yes	characteristic
54	#11719910	—	2	8	no	—
55	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	8	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	#88322	—	2	1	no	—
70	#88322	—	2	2	no	—
71	#88322	—	2	4	no	—
72	#88322	2₋¹⁺⁴	1	2	yes	—
73	#33497	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
77	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
83	C₄×C₅₃	#519	1	1	yes	characteristic
84	C₄×C₅₃	#568	1	1	yes	characteristic
85	C₂×C₈×Q₈	—	53	1	no	—
86	#2183	—	53	8	no	—
87	#2163	—	53	4	no	—
88	#2162	—	53	2	no	—
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#2982	1	1	yes	characteristic
91	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
92	C₂×C₅₃	#3407	1	1	yes	characteristic
93	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
94	C₈×Q₈	—	53	4	no	—
95	C₈×Q₈	—	53	4	no	—
96	C₂×C₄⋊C₈	—	53	2	no	—
97	C₂×C₄⋊C₈	—	53	1	no	—
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	—	53	2	no	—
100	C₄⋊C₁₆	—	53	4	no	—
101	C₄⋊C₁₆	—	53	8	no	—
102	C₂²⋊C₁₆	—	53	4	no	—
103	#346	—	53	8	no	—
104	C₄×C₁₆	—	53	4	no	—
105	C₅₃	#33497	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₈	#88324	1	1	yes	characteristic
107	C₂²×C₈	—	53	1	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	—	53	2	no	—
110	C₄×Q₈	#88324	1	4	yes	—
111	C₄×Q₈	C₂×#6157	1	4	yes	—
112	C₂×C₄⋊C₄	#88324	1	2	yes	—
113	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
114	C₂×C₄²	C₂×#6157	1	2	yes	—
115	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
116	C₂×C₁₆	—	53	8	no	—
117	C₄⋊C₈	—	53	8	no	—
118	C₄⋊C₈	—	53	4	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	4	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×#88324	1	4	yes	—
125	C₂²×C₄	C₂×#88324	1	2	yes	—
126	C₂²×C₄	C₂×#88324	1	2	yes	—
127	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
128	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
129	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	53	1	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	4	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	—	212	1	no	—
136	C₂×C₈	—	212	2	no	—
137	C₄⋊C₄	C₂×#88324	1	8	yes	—
138	C₄⋊C₄	C₂×#88324	1	4	yes	—
139	C₄²	C₂×#88324	1	4	yes	—
140	C₄²	C₂×#88324	1	2	yes	—
141	C₄²	C₂²×#6157	1	4	yes	—
142	C₄²	C₂²×#6157	1	2	yes	—
143	C₁₆	—	106	8	no	—
144	C₂³	C₂²×#88324	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	#13133291	1	4	yes	—
152	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
153	C₂×C₄	C₂²×#88324	1	4	yes	—
154	C₂×C₄	C₂²×#88324	1	2	yes	—
155	C₂×C₄	C₂²×#88324	1	4	yes	—
156	C₂×C₄	C₂²×#88324	1	2	yes	—
157	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
158	C₈	—	53	2	no	—
159	C₈	—	106	1	no	—
160	C₈	—	106	2	no	—
161	C₈	—	106	4	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#13133291	1	2	yes	—
165	C₂²	C₂³×#88324	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	C₂×#13133291	1	4	yes	—
171	C₄	#16811691	1	1	yes	characteristic
172	C₄	#16811703	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#20291608	1	1	yes	characteristic
175	C₂	C₂²×#13133291	1	1	yes	characteristic
176	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₅₃	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×Q₈	#6157
hypocenter	C₅₃	#33497
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
105. C₅₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291076	2
#20291072	2
C₂×Q₈×#88322	2
#33497	53