Subgroups — #28328486

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11719910	C₂	1	8	yes	maximal, complemented
2	#20291072	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#11719910	C₂	1	4	yes	maximal, complemented
4	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	8	yes	complemented
6	C₄⋊#11719910	C₂²	1	4	yes	complemented
7	C₂²⋊#11719910	C₂²	1	4	yes	—
8	C₂²⋊#11719910	C₂²	1	8	yes	complemented
9	C₄×#11719910	C₂²	1	4	yes	complemented
10	C₂³×#88322	C₂²	1	2	yes	—
11	D₄×#88322	C₄	1	4	yes	—
12	D₄×#88322	C₂²	1	4	yes	—
13	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊#88322	C₄	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#11719910	—	4	8	no	—
18	C₂×#11719910	C₂³	1	8	yes	—
19	C₂²×#88322	—	4	4	no	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	2	yes	—
22	C₂²×#88322	C₂×C₄	1	4	yes	—
23	C₂²×#88322	C₂³	1	1	yes	characteristic
24	C₂²×#88322	C₂³	1	4	yes	—
25	C₄⋊#88322	C₂×C₄	1	4	yes	—
26	C₂²⋊#88322	C₂×C₄	1	8	yes	—
27	C₄×#88322	C₂×C₄	1	2	yes	—
28	C₄×#88322	C₂³	1	2	yes	—
29	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×D₄×C₅₃	C₈	1	4	yes	—
32	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
37	C₂×D₄×C₅₃	C₁₆	1	8	yes	complemented
38	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
48	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
49	C₄²×C₅₃	C₂×C₈	1	2	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	8	no	—
53	C₂×#88322	—	4	4	no	—
54	C₂×#88322	—	4	2	no	—
55	C₂×#88322	—	4	1	no	—
56	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
57	C₂×#88322	C₂²×C₄	1	2	yes	—
58	C₂×#88322	C₂²×C₄	1	4	yes	—
59	C₂×#88322	C₂⁴	1	1	yes	characteristic
60	#11719910	—	2	8	no	—
61	C₂³×C₅₃	—	4	2	no	—
62	C₂³×C₅₃	—	4	2	no	—
63	C₂³×C₅₃	C₂×C₁₆	1	8	yes	complemented
64	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
66	D₄×C₅₃	C₂×C₁₆	1	16	yes	complemented
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
80	#88322	—	2	1	no	—
81	#88322	—	2	4	no	—
82	#88322	—	2	2	no	—
83	#88322	H(5,𝔽₂)	1	2	yes	—
84	#33486	—	53	1	no	maximal, Sylow
85	C₂²×C₅₃	—	2	1	no	—
86	C₂²×C₅₃	—	2	4	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	4	no	—
89	C₂²×C₅₃	—	4	8	no	—
90	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
91	C₂²×C₅₃	C₂²×C₁₆	1	8	yes	—
92	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	C₂²×C₁₆	1	4	yes	—
97	C₄×C₅₃	#518	1	1	yes	characteristic
98	C₄×C₅₃	#568	1	1	yes	characteristic
99	C₂×C₈×D₄	—	53	1	no	—
100	D₄×C₁₆	—	53	8	no	—
101	#2162	—	53	2	no	—
102	C₂×C₂²⋊C₁₆	—	53	4	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	2	8	no	—
105	C₂×C₅₃	#2972	1	1	yes	characteristic
106	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
107	C₂×C₅₃	#3407	1	1	yes	characteristic
108	C₂³×C₈	—	53	2	no	—
109	C₂×C₄×D₄	#6157	1	1	yes	characteristic
110	C₂²×C₁₆	—	53	8	no	—
111	C₈×D₄	—	53	4	no	—
112	C₈×D₄	—	53	4	no	—
113	C₂×C₄⋊C₈	—	53	1	no	—
114	C₂×C₂²⋊C₈	—	53	2	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₄⋊C₁₆	—	53	4	no	—
117	C₂²⋊C₁₆	—	53	4	no	—
118	C₂²⋊C₁₆	—	53	8	no	—
119	C₄×C₁₆	—	53	4	no	—
120	C₅₃	#33486	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₄	#88324	1	1	yes	characteristic
122	C₂³×C₄	C₂×#6157	1	2	yes	—
123	C₂²×C₈	—	53	1	no	—
124	C₂²×C₈	—	53	4	no	—
125	C₂²×C₈	—	53	2	no	—
126	C₂²×C₈	—	53	4	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	212	4	no	—
129	C₄×D₄	#88324	1	4	yes	—
130	C₄×D₄	C₂×#6157	1	4	yes	—
131	C₂×C₄⋊C₄	#88324	1	1	yes	characteristic
132	C₂×C₂²⋊C₄	#88324	1	2	yes	—
133	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
134	C₂×C₁₆	—	53	8	no	—
135	C₂×C₁₆	—	212	8	no	—
136	C₄⋊C₈	—	53	4	no	—
137	C₂²⋊C₈	—	53	8	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₂⁴	C₂×#88324	1	2	yes	—
141	C₂×D₄	#11719910	1	8	yes	complemented
142	C₂×D₄	C₂×#88324	1	4	yes	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
146	C₂²×C₄	C₂×#88324	1	2	yes	—
147	C₂²×C₄	C₂×#88324	1	4	yes	—
148	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
149	C₂²×C₄	C₂²×#6157	1	4	yes	—
150	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	2	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	106	8	no	—
156	C₂×C₈	—	106	2	no	—
157	C₂×C₈	—	212	2	no	—
158	C₂×C₈	—	212	4	no	—
159	C₂×C₈	—	212	1	no	—
160	C₄⋊C₄	C₂×#88324	1	4	yes	—
161	C₂²⋊C₄	C₂×#88324	1	8	yes	—
162	C₄²	C₂×#88324	1	2	yes	—
163	C₄²	C₂²×#6157	1	2	yes	—
164	C₁₆	—	106	8	no	—
165	C₂³	—	4	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂×#11719910	1	8	yes	complemented
168	C₂³	C₂²×#88324	1	4	yes	—
169	C₂³	C₂²×#88324	1	1	yes	characteristic
170	D₄	C₂×#11719910	1	16	yes	complemented
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	C₂×#11719910	1	4	yes	complemented
179	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
180	C₂×C₄	C₂²×#88324	1	2	yes	—
181	C₂×C₄	C₂²×#88324	1	4	yes	—
182	C₂×C₄	C₂²×#88324	1	2	yes	—
183	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
184	C₈	—	53	2	no	—
185	C₈	—	106	4	no	—
186	C₈	—	106	1	no	—
187	C₈	—	106	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	8	no	—
193	C₂²	C₂²×#11719910	1	2	yes	—
194	C₂²	C₂²×#11719910	1	8	yes	—
195	C₂²	C₂³×#88324	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	4	no	—
199	C₄	C₂²×#11719910	1	4	yes	—
200	C₄	#16811690	1	1	yes	characteristic
201	C₄	#16811701	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	#20291562	1	1	yes	characteristic
205	C₂	C₂³×#11719910	1	1	yes	characteristic
206	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂³×C₁₆
Fitting	C₂×C₄×D₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×C₄×D₄	#6157
hypocenter	C₅₃	#33486
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. D₄×#11719910	C₂	semidirect
2. #20291072	C₂	semidirect
3. C₂×C₂²⋊#11719910	C₂	semidirect
5. C₂²×#11719910	C₂²	semidirect
6. C₄⋊#11719910	C₂²	semidirect
8. C₂²⋊#11719910	C₂²	semidirect
9. C₄×#11719910	C₂²	semidirect
37. C₂×D₄×C₅₃	C₁₆	semidirect
63. C₂³×C₅₃	C₂×C₁₆	semidirect
66. D₄×C₅₃	C₂×C₁₆	semidirect
74. C₂×C₄×C₅₃	C₂×C₁₆	semidirect
120. C₅₃	#33486	semidirect
141. C₂×D₄	#11719910	semidirect
167. C₂³	C₂×#11719910	semidirect
170. D₄	C₂×#11719910	semidirect
178. C₂×C₄	C₂×#11719910	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11719910	2
#20291072	2
C₂×C₂²⋊#11719910	2
C₂×D₄×#88322	2
#33486	53