Subgroups — #28298300

Order 13568.

262 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal
2	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719910	—	2	2	no	—
4	C₄⋊#11719910	—	2	2	no	—
5	C₄⋊#11719910	C₄	1	4	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₈×#88322	—	2	2	no	—
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719910	—	2	4	no	—
11	C₂×#11719910	—	4	2	no	—
12	C₂×#11719910	—	4	2	no	—
13	C₂²×#88322	—	2	1	no	—
14	C₂²×#88322	D₄	1	1	yes	characteristic
15	C₂²×#88322	Q₈	1	1	yes	characteristic
16	C₄×#88322	—	2	1	no	—
17	C₄×#88322	C₈	1	2	yes	—
18	C₄×#88322	C₂×C₄	1	2	yes	—
19	C₄×#88322	D₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₈	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	—	2	1	no	—
23	C₄×C₈×C₅₃	—	2	1	no	—
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₁₆	1	4	yes	complemented
30	C₄²×C₅₃	C₄²	1	1	yes	characteristic
31	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
34	C₂×#88322	—	2	4	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	—	2	1	no	—
37	C₂×#88322	—	2	1	no	—
38	C₂×#88322	C₂²⋊C₄	1	2	yes	—
39	C₂×#88322	C₄⋊C₄	1	2	yes	—
40	C₂×#88322	D₈	1	1	yes	characteristic
41	C₂×#88322	SD₁₆	1	2	yes	—
42	C₂×#88322	Q₁₆	1	1	yes	characteristic
43	#11719910	—	4	4	no	—
44	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
58	C₈×C₅₃	—	2	4	no	—
59	#88322	—	2	4	no	—
60	#88322	—	2	2	no	—
61	#88322	#102	1	2	yes	—
62	#88322	#103	1	2	yes	—
63	#7151	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	#334	1	1	yes	characteristic
65	C₂²×C₅₃	#335	1	1	yes	characteristic
66	C₂²×C₅₃	#336	1	1	yes	characteristic
67	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
68	C₂²×C₅₃	#364	1	2	yes	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	#334	1	1	yes	characteristic
72	C₄×C₅₃	#335	1	1	yes	characteristic
73	C₄×C₅₃	#340	1	1	yes	characteristic
74	C₄×C₅₃	#341	1	1	yes	characteristic
75	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
76	C₄×C₅₃	#346	1	1	yes	characteristic
77	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
78	C₂×C₄⋊C₁₆	—	53	2	no	—
79	C₂×C₈²	—	53	1	no	—
80	C₂×C₅₃	#1277	1	1	yes	characteristic
81	C₂×C₅₃	#1367	1	2	yes	—
82	C₂×C₅₃	#1370	1	2	yes	—
83	C₂×C₅₃	#1374	1	1	yes	characteristic
84	C₂×C₅₃	#1376	1	1	yes	characteristic
85	C₂²×C₁₆	—	106	2	no	—
86	C₂×C₄×C₈	—	53	1	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	C₂×C₄×C₈	#6157	1	1	yes	characteristic
89	C₄⋊C₁₆	—	53	4	no	—
90	C₄⋊C₁₆	—	106	2	no	—
91	C₈²	—	106	2	no	—
92	C₅₃	#7151	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	53	1	no	—
94	C₂²×C₈	—	53	1	no	—
95	C₂²×C₈	—	106	1	no	—
96	C₂²×C₈	#88324	1	2	yes	—
97	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
98	C₂×C₁₆	—	106	4	no	—
99	C₂×C₁₆	—	212	2	no	—
100	C₂×C₁₆	—	212	2	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	—	53	2	no	—
104	C₄×C₈	—	53	2	no	—
105	C₄×C₈	—	53	2	no	—
106	C₄×C₈	—	106	1	no	—
107	C₂²×C₄	#11719936	1	1	yes	characteristic
108	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
109	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	106	4	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	1	no	—
120	C₂×C₈	—	106	1	no	—
121	C₂×C₈	#11719910	1	4	yes	complemented
122	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
123	C₄²	C₄×#6157	1	1	yes	characteristic
124	C₄²	#11719940	1	1	yes	characteristic
125	C₄²	C₂²⋊#6157	1	1	yes	characteristic
126	C₁₆	—	212	4	no	—
127	C₂³	C₄⋊#88324	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
130	C₂×C₄	C₄×#88324	1	1	yes	characteristic
131	C₂×C₄	#13133272	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
133	C₂×C₄	#13133274	1	1	yes	characteristic
134	C₂×C₄	#13133284	1	1	yes	characteristic
135	C₂×C₄	#13133285	1	1	yes	characteristic
136	C₂×C₄	#13133286	1	1	yes	characteristic
137	C₂×C₄	#13133287	1	1	yes	characteristic
138	C₂×C₄	#13133288	1	1	yes	characteristic
139	C₂×C₄	C₂×#11719910	1	2	yes	—
140	C₂×C₄	#13133291	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	53	2	no	—
143	C₈	—	53	2	no	—
144	C₈	—	106	4	no	—
145	C₈	—	106	2	no	—
146	C₂²	#16811484	1	1	yes	characteristic
147	C₂²	#16811486	1	1	yes	characteristic
148	C₂²	#16811489	1	1	yes	characteristic
149	C₂²	C₄⋊#11719910	1	2	yes	—
150	C₂²	#16811545	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	#16811485	1	1	yes	characteristic
154	C₄	#16811487	1	1	yes	characteristic
155	C₄	#16811497	1	1	yes	characteristic
156	C₄	#16811499	1	1	yes	characteristic
157	C₄	C₄×#11719910	1	1	yes	characteristic
158	C₄	#16811511	1	1	yes	characteristic
159	C₄	C₂²⋊#11719910	1	2	yes	—
160	C₂	#20289545	1	1	yes	characteristic
161	C₂	#20289737	1	2	yes	—
162	C₂	#20289744	1	2	yes	—
163	C₂	#20289753	1	1	yes	characteristic
164	C₂	#20289760	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6157
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₂×C₄×#88322	C₂²
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7151
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
29. C₂×C₈×C₅₃	C₁₆	semidirect
92. C₅₃	#7151	semidirect
121. C₂×C₈	#11719910	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719910	2
C₂×C₈×#88322	2
#7151	53