Subgroups — #28311375

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 170 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	#20289753	C₂	1	4	yes	maximal
3	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	C₂²	1	4	yes	—
5	C₄⋊#11719910	—	2	4	no	—
6	C₂×C₄×#88322	C₄	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄³×C₅₃	C₄	1	1	yes	characteristic
11	C₂×#11719910	—	2	4	no	—
12	C₂×#11719910	—	2	8	no	—
13	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂²×#88322	C₂³	1	1	yes	characteristic
16	C₄×#88322	—	2	2	no	—
17	C₄×#88322	—	2	4	no	—
18	C₄×#88322	—	2	4	no	—
19	C₄×#88322	D₄	1	2	yes	—
20	C₄×#88322	Q₈	1	2	yes	—
21	C₂×C₄²×C₅₃	C₈	1	2	yes	—
22	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₈	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	—	2	4	no	—
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	—	2	2	no	—
35	C₄²×C₅₃	C₁₆	1	4	yes	complemented
36	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₄²×C₅₃	M₁₆	1	2	yes	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	—	2	2	no	—
41	C₂×#88322	C₄⋊C₄	1	2	yes	—
42	C₂×#88322	M₁₆	1	4	yes	—
43	C₂×#88322	C₂×D₄	1	1	yes	characteristic
44	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
45	C₂×#88322	#41	1	4	yes	—
46	#11719910	—	4	4	no	—
47	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
56	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
58	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
63	#88322	—	2	2	no	—
64	#88322	—	2	4	no	—
65	#88322	#124	1	2	yes	—
66	#88322	#125	1	2	yes	—
67	#12879	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₃	#431	1	2	yes	—
70	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	—	2	4	no	—
76	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	—
77	C₄×C₅₃	#382	1	1	yes	characteristic
78	C₄×C₅₃	#423	1	1	yes	characteristic
79	C₄×C₅₃	#432	1	2	yes	—
80	C₂×C₄⋊C₁₆	—	53	2	no	—
81	C₄²×C₈	—	53	1	no	—
82	#1374	—	53	4	no	—
83	C₂×C₅₃	#1831	1	1	yes	characteristic
84	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₅₃	#2150	1	1	yes	characteristic
86	C₂×C₅₃	#2162	1	2	yes	—
87	C₂×C₅₃	#2163	1	2	yes	—
88	C₂²×C₁₆	—	53	4	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	—	53	2	no	—
91	C₂×C₄×C₈	—	53	1	no	—
92	C₂×C₄×C₈	—	53	2	no	—
93	C₄³	#6157	1	1	yes	characteristic
94	C₄⋊C₁₆	—	106	4	no	—
95	C₅₃	#12879	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	53	1	no	—
97	C₂²×C₈	—	53	1	no	—
98	C₂²×C₈	—	53	2	no	—
99	C₂×C₄²	#88324	1	2	yes	—
100	C₂×C₄²	#88324	1	1	yes	characteristic
101	C₂×C₄²	#88324	1	1	yes	characteristic
102	C₂×C₄²	C₂×#6157	1	2	yes	—
103	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
104	C₂×C₁₆	—	106	4	no	—
105	C₂×C₁₆	—	106	8	no	—
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	106	4	no	—
109	C₄×C₈	—	106	4	no	—
110	C₄×C₈	—	106	2	no	—
111	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
112	C₂²×C₄	C₂×#88324	1	2	yes	—
113	C₂²×C₄	C₂×#88324	1	2	yes	—
114	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
115	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
116	C₂×C₈	—	53	1	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	106	2	no	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	#11719910	1	4	yes	complemented
129	C₄²	#11719936	1	2	yes	—
130	C₄²	C₄⋊#6157	1	2	yes	—
131	C₁₆	—	212	4	no	—
132	C₂³	C₂²×#88324	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄⋊#88324	1	2	yes	—
139	C₂×C₄	C₄⋊#88324	1	2	yes	—
140	C₂×C₄	C₂×#11719910	1	4	yes	—
141	C₂×C₄	C₂×#11719910	1	2	yes	—
142	C₂×C₄	#13133291	1	4	yes	—
143	C₂×C₄	#13133291	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
145	C₂×C₄	#13133296	1	2	yes	—
146	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
147	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
148	C₈	—	53	2	no	—
149	C₈	—	53	2	no	—
150	C₈	—	106	2	no	—
151	C₈	—	106	4	no	—
152	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
153	C₂²	#16811639	1	2	yes	—
154	C₂²	C₂²×#11719910	1	1	yes	characteristic
155	C₂²	C₂×#13133291	1	2	yes	—
156	C₂²	C₂×#13133291	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	4	no	—
159	C₄	—	2	2	no	—
160	C₄	C₄⋊#11719910	1	4	yes	—
161	C₄	#16811567	1	1	yes	characteristic
162	C₄	#16811628	1	1	yes	characteristic
163	C₄	#16811645	1	2	yes	—
164	C₂	#20290630	1	1	yes	characteristic
165	C₂	C₂×C₄⋊#11719910	1	1	yes	characteristic
166	C₂	#20291047	1	1	yes	characteristic
167	C₂	#20291072	1	2	yes	—
168	C₂	#20291076	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₄³×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	C₄³	#6157
hypocenter	C₅₃	#12879
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
35. C₄²×C₅₃	C₁₆	semidirect
95. C₅₃	#12879	semidirect
128. C₄²	#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
#20289753	2
C₄²×#88322	2
#12879	53