Subgroups — #28307807

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291072	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic
3	#20289630	C₂	1	2	yes	maximal, complemented
4	C₈×#11719910	C₂	1	2	yes	maximal, complemented
5	#20272996	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719910	C₄	1	2	yes	—
7	C₄⋊#11719910	C₄	1	2	yes	—
8	C₂²⋊#11719910	C₄	1	2	yes	—
9	C₄×#11719910	C₄	1	2	yes	—
10	C₄×#11719910	C₂²	1	2	yes	—
11	C₄×#11719910	C₂²	1	1	yes	characteristic
12	C₄×#11719910	C₂²	1	1	yes	characteristic
13	C₄×#11719917	C₄	1	1	yes	characteristic
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	#16810345	C₄	1	2	yes	—
16	C₈×#88322	C₂²	1	2	yes	—
17	C₅₃×#431	C₄	1	1	yes	characteristic
18	C₂×#11719910	—	2	4	no	—
19	C₂×#11719910	C₂×C₄	1	4	yes	—
20	C₂×#11719910	C₂×C₄	1	2	yes	—
21	C₂×#11719910	C₂×C₄	1	2	yes	—
22	C₂×#11719917	C₈	1	2	yes	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	#13133128	C₈	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₄×#88322	C₂×C₄	1	1	yes	characteristic
29	C₄×#88322	C₂×C₄	1	2	yes	—
30	C₄×#88322	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
33	C₂²⋊C₈×C₅₃	C₈	1	2	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
39	C₄²×C₅₃	C₄²	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	#11719917	C₁₆	1	8	yes	complemented
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	C₄²	1	1	yes	characteristic
47	C₂×#88322	C₄²	1	1	yes	characteristic
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
50	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
51	#11719910	—	2	2	no	—
52	#11719910	—	2	4	no	—
53	#11719910	M₁₆	1	4	yes	complemented
54	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
62	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₈×C₅₃	—	2	4	no	—
66	#88322	—	2	1	no	—
67	#88322	—	2	1	no	—
68	#88322	C₂×C₁₆	1	8	yes	complemented
69	#88322	C₂×M₁₆	1	2	yes	—
70	#88322	#131	1	2	yes	—
71	C₁₆×M₁₆	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
75	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
79	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₅₃	#405	1	1	yes	characteristic
81	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
82	C₄×C₅₃	#504	1	2	yes	—
83	#2162	—	53	1	no	—
84	C₂×C₄×C₁₆	—	53	1	no	—
85	C₈×M₁₆	—	53	1	no	—
86	#1312	—	53	2	no	—
87	C₈×C₁₆	—	53	2	no	—
88	C₂×C₅₃	—	2	2	no	—
89	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
90	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
91	C₂×C₅₃	#2108	1	1	yes	characteristic
92	C₂²×C₁₆	—	53	2	no	—
93	#431	#6157	1	1	yes	characteristic
94	C₄×M₁₆	—	53	1	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	C₄⋊C₁₆	—	53	2	no	—
97	C₂²⋊C₁₆	—	53	2	no	—
98	C₄×C₁₆	—	53	1	no	—
99	C₄×C₁₆	—	53	2	no	—
100	C₄×C₁₆	—	53	1	no	—
101	C₄×C₁₆	—	53	2	no	—
102	#322	—	53	2	no	—
103	C₈²	—	53	2	no	—
104	C₅₃	C₁₆×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	53	2	no	—
106	C₂²×C₈	—	53	1	no	—
107	C₂²×C₈	—	53	1	no	—
108	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
109	C₂×C₁₆	—	53	2	no	—
110	C₂×C₁₆	—	53	2	no	—
111	C₂×C₁₆	—	53	4	no	—
112	C₂×C₁₆	—	106	4	no	—
113	C₄⋊C₈	#88324	1	2	yes	—
114	C₂²⋊C₈	#88324	1	2	yes	—
115	#97	—	53	2	no	—
116	C₄×C₈	—	53	1	no	—
117	C₄×C₈	—	53	1	no	—
118	C₄×C₈	—	53	1	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	1	no	—
121	C₄×C₈	C₂×#6157	1	2	yes	—
122	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
124	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
125	M₁₆	—	53	8	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	4	no	—
129	C₂×C₈	—	53	1	no	—
130	C₂×C₈	—	53	1	no	—
131	C₂×C₈	—	106	2	no	—
132	C₂×C₈	—	106	2	no	—
133	C₂×C₈	C₂×#88324	1	4	yes	—
134	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
135	C₄²	C₄×#6157	1	1	yes	characteristic
136	C₄²	C₂×#88324	1	1	yes	characteristic
137	C₄²	C₂²×#6157	1	1	yes	characteristic
138	C₁₆	—	53	4	no	—
139	C₁₆	—	106	2	no	—
140	C₁₆	—	106	4	no	—
141	C₂³	C₄×#88324	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₄×#88324	1	1	yes	characteristic
146	C₂×C₄	C₄×#88324	1	1	yes	characteristic
147	C₂×C₄	C₄×#88324	1	1	yes	characteristic
148	C₂×C₄	D₅₃⋊C₁₆	1	4	yes	—
149	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
150	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
151	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
152	C₈	—	2	4	no	—
153	C₈	—	53	2	no	—
154	C₈	—	53	2	no	—
155	C₈	—	53	8	no	—
156	C₈	—	106	1	no	—
157	C₈	—	106	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#16811509	1	2	yes	—
161	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	#16811509	1	2	yes	—
165	C₄	M₁₆×#6157	1	1	yes	characteristic
166	C₄	#16811589	1	1	yes	characteristic
167	C₄	C₂×D₅₃⋊C₁₆	1	2	yes	—
168	C₄	#16811681	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	M₁₆×#88324	1	1	yes	characteristic
171	C₂	C₂×#16811509	1	1	yes	characteristic
172	C₂	#20290924	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Fitting	C₅₃×#431	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	#431	#6157
hypocenter	C₅₃	C₁₆×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
3. #20289630	C₂	semidirect
4. C₈×#11719910	C₂	semidirect
43. #11719917	C₁₆	semidirect
53. #11719910	M₁₆	semidirect
68. #88322	C₂×C₁₆	semidirect
104. C₅₃	C₁₆×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20291072	2
C₂×C₄×#11719910	2
#20289630	2
C₈×#11719910	2
#20272996	2
C₁₆×M₁₆	53