Subgroups — #28328521

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291099	C₂	1	4	yes	maximal, complemented
2	D₄×#11719910	C₂	1	4	yes	maximal, complemented
3	#20291072	C₂	1	2	yes	maximal, complemented
4	#20290945	C₂	1	4	yes	maximal, complemented
5	#20283427	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	4	yes	—
7	C₂²×#11719910	C₂²	1	4	yes	—
8	C₄⋊#11719910	C₂²	1	2	yes	—
9	C₄⋊#11719910	C₂²	1	2	yes	—
10	C₂²⋊#11719910	C₂²	1	4	yes	—
11	C₂²⋊#11719910	C₂²	1	8	yes	—
12	C₄×#11719910	C₂²	1	4	yes	—
13	C₂×#13133224	C₂²	1	2	yes	—
14	#16810701	C₄	1	4	yes	—
15	D₄×#88322	C₂²	1	4	yes	—
16	#16810647	C₄	1	1	yes	characteristic
17	#16810590	C₄	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₅₃×#532	C₄	1	1	yes	characteristic
20	#13133291	—	4	4	no	—
21	C₂×#11719910	—	4	4	no	—
22	C₂×#11719910	C₂³	1	4	yes	—
23	C₂×#11719910	C₂³	1	4	yes	—
24	#13133224	—	4	2	no	—
25	#13133224	—	4	2	no	—
26	C₂×#11719917	C₂×C₄	1	2	yes	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	4	yes	—
31	C₄⋊#88322	C₂×C₄	1	4	yes	—
32	C₂²⋊#88322	C₂×C₄	1	8	yes	—
33	C₄×#88322	C₂×C₄	1	2	yes	—
34	C₄×#88322	C₂³	1	2	yes	—
35	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
37	#127×C₅₃	C₈	1	1	yes	characteristic
38	#124×C₅₃	C₈	1	2	yes	—
39	#121×C₅₃	C₈	1	4	yes	—
40	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#41×C₅₃	—	4	2	no	—
43	#41×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
46	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
47	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
52	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
53	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
56	#11719917	—	4	2	no	—
57	#11719917	—	4	4	no	—
58	C₂×#88322	—	2	1	no	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	4	no	—
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	4	1	no	—
63	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
64	C₂×#88322	C₂²×C₄	1	2	yes	—
65	C₂×#88322	C₂²×C₄	1	4	yes	—
66	C₂×#88322	C₂⁴	1	1	yes	characteristic
67	#11719910	—	2	4	no	—
68	#11719910	—	2	4	no	—
69	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
70	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
71	Q₈×C₅₃	—	4	2	no	—
72	D₄×C₅₃	—	2	8	no	—
73	D₄×C₅₃	—	4	2	no	—
74	D₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	4	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	#88322	—	2	4	no	—
86	#88322	—	2	1	no	—
87	#88322	—	2	2	no	—
88	#88322	H(5,𝔽₂)	1	1	yes	characteristic
89	#88322	H(5,𝔽₂)	1	1	yes	characteristic
90	#33493	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	8	no	—
93	C₂²×C₅₃	—	2	1	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	4	no	—
99	C₄×C₅₃	#504	1	4	yes	—
100	C₄×C₅₃	#518	1	1	yes	characteristic
101	C₄×C₅₃	#568	1	1	yes	characteristic
102	#2931	—	53	1	no	—
103	#2169	—	53	4	no	—
104	D₄×C₁₆	—	53	4	no	—
105	#2162	—	53	2	no	—
106	#2113	—	53	4	no	—
107	C₂×C₅₃	—	4	1	no	—
108	C₂×C₅₃	—	4	4	no	—
109	C₂×C₅₃	#2972	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#504	1	2	yes	—
111	C₂×#131	—	53	2	no	—
112	#532	#6157	1	1	yes	characteristic
113	C₂×M₃₂	—	53	4	no	—
114	C₂²×C₁₆	—	53	4	no	—
115	#436	—	53	4	no	—
116	C₈×D₄	—	53	4	no	—
117	#423	—	53	1	no	—
118	#408	—	53	2	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₄⋊C₁₆	—	53	2	no	—
121	C₄⋊C₁₆	—	53	2	no	—
122	C₂²⋊C₁₆	—	53	4	no	—
123	C₂²⋊C₁₆	—	53	8	no	—
124	C₄×C₁₆	—	53	4	no	—
125	C₅₃	#33493	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	C₂×#6157	1	2	yes	—
127	#131	—	212	2	no	—
128	#131	—	212	2	no	—
129	C₂×M₁₆	—	53	2	no	—
130	C₂×M₁₆	—	53	4	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	4	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₄⋊Q₈	#88324	1	1	yes	characteristic
135	#127	#88324	1	1	yes	characteristic
136	#124	#88324	1	2	yes	—
137	#121	#88324	1	4	yes	—
138	C₄×D₄	C₂×#6157	1	4	yes	—
139	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
140	M₃₂	—	212	4	no	—
141	C₂×C₁₆	—	53	4	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₂×C₁₆	—	212	4	no	—
144	C₄⋊C₈	—	53	4	no	—
145	C₂²⋊C₈	—	53	8	no	—
146	C₄×C₈	—	53	2	no	—
147	C₄×C₈	—	53	2	no	—
148	#41	—	4	2	no	—
149	#41	—	4	2	no	—
150	C₂×Q₈	C₂×#88324	1	2	yes	—
151	C₂×D₄	C₂×#88324	1	4	yes	—
152	C₂×D₄	C₂×#88324	1	4	yes	—
153	C₂×D₄	C₂×#88324	1	2	yes	—
154	C₂²×C₄	C₂×#88324	1	2	yes	—
155	C₂²×C₄	C₂²×#6157	1	4	yes	—
156	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
157	M₁₆	—	212	2	no	—
158	M₁₆	—	212	4	no	—
159	C₂×C₈	—	53	1	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	53	1	no	—
162	C₂×C₈	—	53	4	no	—
163	C₂×C₈	—	106	4	no	—
164	C₂×C₈	—	106	4	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	212	1	no	—
168	C₄⋊C₄	C₂×#88324	1	4	yes	—
169	C₂²⋊C₄	C₂×#88324	1	8	yes	—
170	C₄²	C₂×#88324	1	1	yes	characteristic
171	C₄²	C₂×#88324	1	1	yes	characteristic
172	C₄²	C₂²×#6157	1	2	yes	—
173	C₁₆	—	106	4	no	—
174	C₁₆	—	106	4	no	—
175	C₂³	C₂²×#88324	1	4	yes	—
176	C₂³	C₂²×#88324	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	D₄	—	2	8	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
187	C₂×C₄	C₂²×#88324	1	2	yes	—
188	C₂×C₄	C₂²×#88324	1	4	yes	—
189	C₂×C₄	C₂²×#88324	1	2	yes	—
190	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
191	C₈	—	53	1	no	—
192	C₈	—	53	1	no	—
193	C₈	—	106	4	no	—
194	C₈	—	106	1	no	—
195	C₈	—	106	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	4	no	—
200	C₂²	C₂³×#88324	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	4	no	—
204	C₄	#16811681	1	4	yes	—
205	C₄	#16811690	1	1	yes	characteristic
206	C₄	#16811701	1	1	yes	characteristic
207	C₂	—	4	1	no	—
208	C₂	—	4	4	no	—
209	C₂	#20291562	1	1	yes	characteristic
210	C₂	C₂×#16811681	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting, prime radical	C₅₃×#532	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#532	#6157
hypocenter	C₅₃	#33493
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20291099	C₂	semidirect
2. D₄×#11719910	C₂	semidirect
3. #20291072	C₂	semidirect
4. #20290945	C₂	semidirect
125. C₅₃	#33493	semidirect

Maximal subgroups

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

subgroup	index
#20291099	2
D₄×#11719910	2
#20291072	2
#20290945	2
#20283427	2
#33493	53