Subgroups — #28328541

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	6	yes	maximal, complemented
2	Q₈×#11719910	C₂	1	2	yes	maximal, complemented
3	#20291076	C₂	1	3	yes	maximal
4	#20291072	C₂	1	3	yes	maximal
5	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719910	C₂²	1	6	yes	—
7	C₄⋊#11719910	C₂²	1	6	yes	—
8	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
9	C₂²⋊#11719910	C₂²	1	3	yes	—
10	#16811511	C₂²	1	6	yes	—
11	C₄×#11719910	C₂²	1	6	yes	—
12	Q₈×#88322	C₄	1	3	yes	—
13	Q₈×#88322	C₄	1	1	yes	characteristic
14	Q₈×#88322	C₂²	1	1	yes	characteristic
15	Q₈×#88322	C₂²	1	3	yes	—
16	C₂×C₄⋊#88322	C₄	1	3	yes	—
17	C₂×C₄×#88322	C₂²	1	3	yes	—
18	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#11719910	C₂³	1	2	yes	—
20	C₂×#11719910	C₂³	1	6	yes	—
21	C₂²×#88322	C₂×C₄	1	3	yes	—
22	C₂²×#88322	C₂³	1	1	yes	characteristic
23	C₄⋊#88322	C₂×C₄	1	3	yes	—
24	C₄⋊#88322	C₂×C₄	1	6	yes	—
25	C₄⋊#88322	C₂×C₄	1	3	yes	—
26	C₄×#88322	C₂×C₄	1	3	yes	—
27	C₄×#88322	C₂×C₄	1	3	yes	—
28	C₄×#88322	C₂³	1	3	yes	—
29	C₄×#88322	C₂³	1	3	yes	—
30	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
31	C₄×Q₈×C₅₃	C₈	1	3	yes	—
32	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
33	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	C₈	1	3	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×Q₈×C₅₃	—	2	4	no	—
38	C₂×Q₈×C₅₃	C₂×C₈	1	3	yes	—
39	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₄⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
44	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
45	C₄⋊C₄×C₅₃	C₂×C₈	1	3	yes	—
46	C₄²×C₅₃	C₂×C₈	1	3	yes	—
47	C₄²×C₅₃	C₂×C₈	1	3	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	4	3	no	—
52	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂²×C₄	1	3	yes	—
54	C₂×#88322	C₂²×C₄	1	3	yes	—
55	C₂×#88322	C₂⁴	1	1	yes	characteristic
56	#11719910	—	2	2	no	—
57	#11719910	—	2	6	no	—
58	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₅₃	—	2	6	no	—
60	Q₈×C₅₃	M₃₂	1	4	yes	—
61	C₂×C₄×C₅₃	—	2	6	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	6	no	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
71	#88322	—	2	1	no	—
72	#88322	—	2	3	no	—
73	#88322	—	2	3	no	—
74	#88322	2₋¹⁺⁴	1	2	yes	—
75	#33498	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	1	no	—
77	C₂²×C₅₃	—	2	1	no	—
78	C₂²×C₅₃	#504	1	2	yes	—
79	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₅₃	—	2	3	no	—
81	C₄×C₅₃	—	2	3	no	—
82	C₄×C₅₃	—	2	3	no	—
83	C₄×C₅₃	—	2	1	no	—
84	C₄×C₅₃	C₂×M₃₂	1	6	yes	—
85	C₄×C₅₃	#519	1	1	yes	characteristic
86	C₄×C₅₃	#568	1	1	yes	characteristic
87	C₂×C₈×Q₈	—	53	1	no	—
88	#2183	—	53	6	no	—
89	Q₈×C₁₆	—	53	2	no	—
90	#2163	—	53	3	no	—
91	#2162	—	53	3	no	—
92	C₂×C₅₃	—	2	2	no	—
93	C₂×C₅₃	#2982	1	1	yes	characteristic
94	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
95	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
96	C₂×C₄×Q₈	#6157	1	1	yes	characteristic
97	C₈×Q₈	—	53	1	no	—
98	C₈×Q₈	—	53	3	no	—
99	C₈×Q₈	—	53	3	no	—
100	C₈×Q₈	—	53	1	no	—
101	C₂×C₄⋊C₈	—	53	3	no	—
102	C₂×C₄×C₈	—	53	3	no	—
103	C₄⋊C₁₆	—	53	6	no	—
104	C₄⋊C₁₆	—	53	6	no	—
105	C₂²⋊C₁₆	—	53	1	no	—
106	C₂²⋊C₁₆	—	53	3	no	—
107	#346	—	53	6	no	—
108	C₄×C₁₆	—	53	6	no	—
109	C₅₃	#33498	1	1	yes	characteristic, complemented, Sylow
110	C₂²×Q₈	#88324	1	1	yes	characteristic
111	C₂²×C₈	—	53	1	no	—
112	C₂²×C₈	—	53	3	no	—
113	C₄×Q₈	#88324	1	3	yes	—
114	C₄×Q₈	#88324	1	1	yes	characteristic
115	C₄×Q₈	C₂×#6157	1	3	yes	—
116	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
117	C₂×C₄⋊C₄	#88324	1	3	yes	—
118	C₂×C₄²	C₂×#6157	1	3	yes	—
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	53	6	no	—
121	C₄⋊C₈	—	53	3	no	—
122	C₄⋊C₈	—	53	6	no	—
123	C₄⋊C₈	—	53	3	no	—
124	C₄×C₈	—	53	3	no	—
125	C₄×C₈	—	53	3	no	—
126	C₄×C₈	—	53	3	no	—
127	C₄×C₈	—	53	3	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	C₂×#88324	1	3	yes	—
130	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
131	C₂²×C₄	C₂×#88324	1	3	yes	—
132	C₂²×C₄	C₂×#88324	1	3	yes	—
133	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	1	no	—
136	C₂×C₈	—	53	3	no	—
137	C₂×C₈	—	53	3	no	—
138	C₂×C₈	—	106	2	no	—
139	C₂×C₈	—	212	3	no	—
140	C₄⋊C₄	C₂×#88324	1	6	yes	—
141	C₄⋊C₄	C₂×#88324	1	3	yes	—
142	C₄⋊C₄	C₂×#88324	1	3	yes	—
143	C₄²	C₂×#88324	1	3	yes	—
144	C₄²	C₂×#88324	1	3	yes	—
145	C₄²	C₂²×#6157	1	3	yes	—
146	C₄²	C₂²×#6157	1	3	yes	—
147	C₁₆	—	106	2	no	—
148	C₁₆	—	106	6	no	—
149	C₂³	C₂²×#88324	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	#13133291	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
157	C₂×C₄	C₂²×#88324	1	3	yes	—
158	C₂×C₄	C₂²×#88324	1	3	yes	—
159	C₂×C₄	C₂²×#88324	1	3	yes	—
160	C₂×C₄	C₂²×#88324	1	3	yes	—
161	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
162	C₈	—	53	2	no	—
163	C₈	—	106	1	no	—
164	C₈	—	106	3	no	—
165	C₈	—	106	3	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#16811681	1	2	yes	—
169	C₂²	C₂³×#88324	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	3	no	—
174	C₄	C₂×#13133291	1	6	yes	—
175	C₄	#16811691	1	1	yes	characteristic
176	C₄	#16811703	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	#20291608	1	1	yes	characteristic
179	C₂	C₂²×#13133291	1	1	yes	characteristic
180	C₂	C₂×#16811681	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	#33498
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
2. Q₈×#11719910	C₂	semidirect
109. C₅₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
Q₈×#11719910	2
#20291076	2
#20291072	2
C₂×Q₈×#88322	2
#33498	53