Subgroups — #28327902

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	2	yes	maximal
2	#20290924	C₂	1	8	yes	maximal, complemented
3	C₄×#13133291	C₂	1	4	yes	maximal, complemented
4	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₄	1	16	yes	complemented
6	C₂×#13133291	C₂²	1	4	yes	—
7	C₂×#13133291	C₂²	1	4	yes	—
8	C₂²×#11719910	C₂²	1	4	yes	—
9	#16811511	C₂²	1	8	yes	complemented
10	C₄×#11719910	C₂²	1	8	yes	complemented
11	C₂²×#11719917	C₄	1	1	yes	characteristic
12	C₂³×#88322	C₂²	1	1	yes	characteristic
13	#16810579	C₄	1	4	yes	—
14	#16810579	C₂²	1	4	yes	—
15	C₂×#13133128	C₄	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
18	#13133291	C₂×C₄	1	16	yes	—
19	#13133291	C₂×C₄	1	16	yes	complemented
20	C₂×#11719910	C₂×C₄	1	16	yes	complemented
21	C₂×#11719910	C₂³	1	8	yes	—
22	C₂×#11719917	C₈	1	8	yes	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
24	C₂²×#88322	C₂×C₄	1	8	yes	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	2	yes	—
30	#13133128	C₂×C₄	1	8	yes	—
31	C₄×#88322	C₂×C₄	1	4	yes	—
32	C₄×#88322	C₂³	1	4	yes	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#117×C₅₃	C₈	1	4	yes	—
35	#117×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
37	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
48	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
49	C₄²×C₅₃	C₂×C₈	1	4	yes	—
50	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
51	#11719917	C₂×C₈	1	16	yes	—
52	C₂×#88322	C₄²	1	4	yes	—
53	C₂×#88322	C₄²	1	8	yes	—
54	C₂×#88322	C₂×C₈	1	8	yes	—
55	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
56	C₂×#88322	C₂²×C₄	1	2	yes	—
57	C₂×#88322	C₂²×C₄	1	4	yes	—
58	C₂×#88322	C₂²×C₄	1	4	yes	—
59	C₂×#88322	C₂²×C₄	1	4	yes	—
60	C₂×#88322	C₂²×C₄	1	4	yes	—
61	C₂×#88322	C₂⁴	1	1	yes	characteristic
62	#11719910	C₂²×C₄	1	16	yes	—
63	C₂³×C₅₃	—	4	2	no	—
64	C₂³×C₅₃	C₄×C₈	1	4	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₅₃	—	4	4	no	—
68	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
69	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
70	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
78	#88322	C₂×C₄²	1	2	yes	—
79	#88322	C₂×C₄²	1	4	yes	—
80	#88322	C₂²×C₈	1	8	yes	—
81	#88322	C₂³×C₄	1	2	yes	—
82	#33324	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	2	4	no	—
85	C₂²×C₅₃	—	4	4	no	—
86	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
87	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
88	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₅₃	—	2	8	no	—
90	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
91	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
92	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
93	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
94	C₂²×M₃₂	—	53	2	no	—
95	C₂×#405	—	53	1	no	—
96	#2108	—	53	8	no	—
97	C₄×M₃₂	—	53	4	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	—	2	4	no	—
100	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
101	C₂×C₅₃	#3407	1	2	yes	—
102	C₂²×M₁₆	—	53	1	no	—
103	C₂³×C₈	—	53	1	no	—
104	C₂×#117	#6157	1	1	yes	characteristic
105	C₂×M₃₂	—	53	4	no	—
106	C₂×M₃₂	—	53	4	no	—
107	C₂×M₃₂	—	53	16	no	—
108	C₂²×C₁₆	—	53	4	no	—
109	#405	—	53	4	no	—
110	#405	—	53	4	no	—
111	C₂×#97	—	53	2	no	—
112	C₂×C₄×C₈	—	53	2	no	—
113	#346	—	53	8	no	—
114	C₄×C₁₆	—	53	8	no	—
115	C₅₃	#33324	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
117	C₂×M₁₆	—	53	4	no	—
118	C₂×M₁₆	—	53	8	no	—
119	C₂²×C₈	—	53	8	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	—	53	2	no	—
123	C₂²×C₈	—	53	2	no	—
124	C₂²×C₈	—	53	2	no	—
125	#117	#88324	1	4	yes	—
126	#117	C₂×#6157	1	4	yes	—
127	C₂×C₄⋊C₄	#88324	1	2	yes	—
128	C₂×C₂²⋊C₄	#88324	1	2	yes	—
129	C₂×C₄²	C₂×#6157	1	2	yes	—
130	M₃₂	—	53	16	no	—
131	M₃₂	—	53	16	no	—
132	C₂×C₁₆	—	53	8	no	—
133	C₂×C₁₆	—	53	16	no	—
134	#97	—	53	8	no	—
135	C₄×C₈	—	53	4	no	—
136	C₄×C₈	—	53	4	no	—
137	C₂⁴	C₂×#88324	1	1	yes	characteristic
138	C₂²×C₄	D₅₃⋊C₈	1	4	yes	—
139	C₂²×C₄	C₄×#6157	1	4	yes	—
140	C₂²×C₄	C₂×#88324	1	4	yes	—
141	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
142	C₂²×C₄	C₂×#88324	1	2	yes	—
143	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
144	C₂²×C₄	C₂²×#6157	1	2	yes	—
145	M₁₆	—	53	16	no	—
146	C₂×C₈	—	53	4	no	—
147	C₂×C₈	—	53	4	no	—
148	C₂×C₈	—	53	4	no	—
149	C₂×C₈	—	53	4	no	—
150	C₂×C₈	—	53	8	no	—
151	C₂×C₈	—	53	8	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	1	no	—
154	C₂×C₈	—	53	4	no	—
155	C₂×C₈	—	53	2	no	—
156	C₄⋊C₄	C₂×#88324	1	8	yes	—
157	C₂²⋊C₄	C₂×#88324	1	8	yes	—
158	C₄²	C₂×#88324	1	4	yes	—
159	C₄²	C₂²×#6157	1	4	yes	—
160	C₁₆	—	53	16	no	—
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#88324	1	4	yes	—
163	C₂³	C₂²×#88324	1	2	yes	—
164	C₂³	C₂²×#88324	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#88324	1	4	yes	—
167	C₂×C₄	C₄×#88324	1	8	yes	—
168	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
169	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
170	C₂×C₄	C₂²×#88324	1	8	yes	—
171	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
172	C₂×C₄	C₂²×#88324	1	2	yes	—
173	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
174	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
175	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
176	C₈	—	53	4	no	—
177	C₈	—	53	2	no	—
178	C₈	—	53	2	no	—
179	C₈	—	53	8	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#88324	1	2	yes	—
184	C₂²	C₂×C₄×#88324	1	4	yes	—
185	C₂²	C₂³×#88324	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#88324	1	2	yes	—
188	C₄	C₂×C₄×#88324	1	4	yes	—
189	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
190	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
194	C₂	#20291682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂×#117×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	C₂×#117	#6157
hypocenter	C₅₃	#33324
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

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
5. C₂×#13133291	C₄	semidirect
9. #16811511	C₂²	semidirect
10. C₄×#11719910	C₂²	semidirect
19. #13133291	C₂×C₄	semidirect
20. C₂×#11719910	C₂×C₄	semidirect
115. C₅₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133291	2
#20290924	2
C₄×#13133291	2
C₂×#16810579	2
#33324	53