Subgroups — #28327912

Order 13568.

818 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	C₂×#16811681	C₂	1	2	yes	maximal
2	#20290924	C₂	1	6	yes	maximal, complemented
3	C₄×#13133291	C₂	1	3	yes	maximal, complemented
4	C₂×#16811511	C₂	1	3	yes	maximal, complemented
5	C₄×#13133224	C₂	1	1	yes	maximal, characteristic
6	#16811681	C₄	1	16	yes	complemented
7	C₂×#13133291	C₂²	1	6	yes	—
8	C₂²×#11719910	C₂²	1	6	yes	—
9	#16811511	C₂²	1	1	yes	characteristic
10	#16811511	C₂²	1	3	yes	—
11	#16811511	C₂²	1	6	yes	—
12	C₄×#11719910	C₂²	1	6	yes	—
13	C₂×#13133224	C₄	1	1	yes	characteristic
14	C₂×#13133224	C₂²	1	1	yes	characteristic
15	#16810579	C₄	1	3	yes	—
16	#16810579	C₂²	1	3	yes	—
17	C₄×#11719917	C₄	1	3	yes	—
18	C₂×C₄×#88322	C₂²	1	3	yes	—
19	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
20	#13133291	C₂×C₄	1	24	yes	complemented
21	C₂×#11719910	C₂×C₄	1	24	yes	complemented
22	C₂×#11719910	C₂³	1	2	yes	—
23	C₂×#11719910	C₂³	1	6	yes	—
24	#13133224	C₈	1	8	yes	—
25	#13133224	C₂×C₄	1	8	yes	—
26	C₂×#11719917	C₂×C₄	1	3	yes	—
27	C₂×#11719917	C₂×C₄	1	3	yes	—
28	C₂²×#88322	C₂×C₄	1	3	yes	—
29	C₂²×#88322	C₂³	1	3	yes	—
30	#13133128	C₂×C₄	1	6	yes	—
31	C₄×#88322	C₂×C₄	1	3	yes	—
32	C₄×#88322	C₂×C₄	1	3	yes	—
33	C₄×#88322	C₂³	1	1	yes	characteristic
34	C₄×#88322	C₂³	1	3	yes	—
35	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	C₈	1	2	yes	—
37	C₄×D₄×C₅₃	C₈	1	6	yes	—
38	#117×C₅₃	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
40	#41×C₅₃	C₄²	1	4	yes	—
41	#41×C₅₃	C₂×C₈	1	4	yes	—
42	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄×C₅₃	C₂×C₈	1	3	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₄⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
47	C₂²⋊C₄×C₅₃	C₂×C₈	1	6	yes	—
48	C₄²×C₅₃	C₂×C₈	1	6	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
51	#11719917	C₄²	1	12	yes	—
52	#11719917	C₂×C₈	1	12	yes	—
53	C₂×#88322	C₂×C₈	1	12	yes	—
54	C₂×#88322	C₂²×C₄	1	3	yes	—
55	C₂×#88322	C₂²×C₄	1	12	yes	—
56	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
57	C₂×#88322	C₂²×C₄	1	3	yes	—
58	C₂×#88322	C₂⁴	1	1	yes	characteristic
59	#11719910	C₂²×C₄	1	4	yes	—
60	#11719910	C₂²×C₄	1	12	yes	—
61	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
62	Q₈×C₅₃	C₄×C₈	1	4	yes	—
63	D₄×C₅₃	C₄×C₈	1	12	yes	—
64	C₂×C₄×C₅₃	—	4	6	no	—
65	C₂×C₄×C₅₃	C₂×C₄²	1	6	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
70	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
71	#88322	C₂×C₄²	1	6	yes	—
72	#88322	C₂²×C₈	1	2	yes	—
73	#88322	C₂²×C₈	1	6	yes	—
74	#88322	C₂³×C₄	1	2	yes	—
75	#33326	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	6	no	—
77	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
78	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
79	C₄×C₅₃	—	2	2	no	—
80	C₄×C₅₃	—	2	6	no	—
81	C₄×C₅₃	C₂×C₄×C₈	1	6	yes	—
82	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
83	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
84	C₂×#504	—	53	2	no	—
85	C₄×#131	—	53	1	no	—
86	#2108	—	53	6	no	—
87	C₄×M₃₂	—	53	3	no	—
88	C₂×#346	—	53	3	no	—
89	C₂×C₅₃	—	2	6	no	—
90	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
91	C₂×C₅₃	#3407	1	2	yes	—
92	C₂×#131	—	53	1	no	—
93	C₂×#131	—	53	1	no	—
94	C₄×#41	#6157	1	1	yes	characteristic
95	#504	—	53	16	no	—
96	C₂×M₃₂	—	53	6	no	—
97	C₂²×C₁₆	—	53	6	no	—
98	#405	—	53	3	no	—
99	#405	—	53	3	no	—
100	C₄×M₁₆	—	53	3	no	—
101	C₂×C₄×C₈	—	53	3	no	—
102	#346	—	53	1	no	—
103	#346	—	53	3	no	—
104	#346	—	53	6	no	—
105	C₄×C₁₆	—	53	6	no	—
106	C₅₃	#33326	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	C₂×#6157	1	1	yes	characteristic
108	#131	—	53	8	no	—
109	#131	—	53	8	no	—
110	C₂×M₁₆	—	53	3	no	—
111	C₂×M₁₆	—	53	3	no	—
112	C₂²×C₈	—	53	3	no	—
113	C₂²×C₈	—	53	3	no	—
114	C₄×Q₈	#88324	1	2	yes	—
115	C₄×D₄	#88324	1	6	yes	—
116	#117	C₂×#6157	1	3	yes	—
117	C₂×C₄²	C₂×#6157	1	3	yes	—
118	M₃₂	—	53	24	no	—
119	C₂×C₁₆	—	53	6	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	24	no	—
122	#97	—	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	1	no	—
127	#41	D₅₃⋊C₈	1	4	yes	—
128	#41	C₄×#6157	1	4	yes	—
129	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
130	C₂×D₄	C₂×#88324	1	3	yes	—
131	C₂²×C₄	C₂×#88324	1	6	yes	—
132	C₂²×C₄	C₂²×#6157	1	3	yes	—
133	M₁₆	—	53	12	no	—
134	M₁₆	—	53	12	no	—
135	C₂×C₈	—	53	12	no	—
136	C₂×C₈	—	53	12	no	—
137	C₂×C₈	—	53	1	no	—
138	C₂×C₈	—	53	1	no	—
139	C₂×C₈	—	53	3	no	—
140	C₂×C₈	—	53	3	no	—
141	C₄⋊C₄	C₂×#88324	1	6	yes	—
142	C₂²⋊C₄	C₂×#88324	1	6	yes	—
143	C₄²	C₂×#88324	1	6	yes	—
144	C₄²	C₂²×#6157	1	1	yes	characteristic
145	C₄²	C₂²×#6157	1	3	yes	—
146	C₁₆	—	53	4	no	—
147	C₁₆	—	53	12	no	—
148	C₂³	C₂²×#88324	1	3	yes	—
149	Q₈	C₄×#88324	1	4	yes	—
150	D₄	C₄×#88324	1	12	yes	—
151	C₂×C₄	—	4	6	no	—
152	C₂×C₄	C₂×C₄×#6157	1	6	yes	—
153	C₂×C₄	C₂²×#88324	1	2	yes	—
154	C₂×C₄	C₂²×#88324	1	6	yes	—
155	C₂×C₄	C₂²×#88324	1	3	yes	—
156	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
157	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
158	C₈	—	53	6	no	—
159	C₈	—	53	2	no	—
160	C₈	—	53	2	no	—
161	C₈	—	53	6	no	—
162	C₂²	—	2	6	no	—
163	C₂²	C₂×C₄×#88324	1	6	yes	—
164	C₂²	C₂³×#88324	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	6	no	—
167	C₄	C₂×C₄×#88324	1	6	yes	—
168	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
169	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
170	C₂	—	2	6	no	—
171	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
172	C₂	#20291682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₄×#41×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	C₄×#41	#6157
hypocenter	C₅₃	#33326
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #20290924	C₂	semidirect
3. C₄×#13133291	C₂	semidirect
4. C₂×#16811511	C₂	semidirect
6. #16811681	C₄	semidirect
20. #13133291	C₂×C₄	semidirect
21. C₂×#11719910	C₂×C₄	semidirect
106. C₅₃	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811681	2
#20290924	2
C₄×#13133291	2
C₂×#16811511	2
C₄×#13133224	2
#33326	53