Subgroups — C₄×#88324⋊C₈ #28300032

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#88324⋊C₈	C₂	1	4	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88324	C₄	1	4	yes	—
6	C₂×C₄×#88324	C₄	1	4	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#88324⋊C₈	C₄	1	16	yes	direct product factor
10	C₂×C₄×#88322	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
12	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	C₂×C₄	1	4	yes	—
16	C₂²×#88324	C₂×C₄	1	4	yes	—
17	C₄×#88324	—	2	8	no	—
18	C₄×#88324	C₈	1	16	yes	complemented
19	C₄×#88324	C₂×C₄	1	8	yes	—
20	C₄×#88324	C₂×C₄	1	8	yes	—
21	C₂²×#88322	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₄×#88322	—	2	2	no	—
27	C₈×Dic₅₃	—	2	4	no	—
28	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₄²	1	2	yes	—
35	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×#88324	—	2	8	no	—
38	C₂×#88324	C₄²	1	8	yes	—
39	C₂×#88324	C₄²	1	8	yes	—
40	C₂×#88324	C₂×C₈	1	16	yes	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	C₄²	1	4	yes	—
44	C₄²×C₅₃	M₁₆	1	2	yes	—
45	C₄²×C₅₃	M₁₆	1	2	yes	—
46	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
47	C₄×Dic₅₃	M₁₆	1	4	yes	—
48	C₄×Dic₅₃	M₁₆	1	8	yes	—
49	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	C₄²	1	4	yes	—
53	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
55	C₂×C₄×C₅₃	#97	1	4	yes	—
56	C₂×C₄×C₅₃	#97	1	4	yes	—
57	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
62	C₂×Dic₅₃	#97	1	4	yes	—
63	C₂×Dic₅₃	#97	1	4	yes	—
64	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
65	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
66	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
67	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
68	#88324	—	2	8	no	—
69	#88324	C₄×C₈	1	16	yes	complemented
70	C₈×C₅₃	—	2	4	no	—
71	#88322	—	2	4	no	—
72	C₄×#322	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	C₄³	1	1	yes	characteristic
74	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
75	C₂²×C₅₃	C₂×#97	1	2	yes	—
76	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
77	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
78	C₄×C₅₃	#322	1	8	yes	complemented
79	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
80	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
81	Dic₅₃	#322	1	8	yes	complemented
82	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
83	Dic₅₃	C₄×M₁₆	1	4	yes	—
84	C₄²×C₈	—	53	1	no	—
85	C₄²×C₈	—	53	1	no	—
86	C₄²×C₈	—	53	1	no	—
87	C₂×#322	—	53	4	no	—
88	C₂×C₅₃	C₂×#322	1	2	yes	—
89	C₂×C₅₃	C₂×#322	1	2	yes	—
90	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
91	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
92	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
93	C₂×C₄×C₈	—	53	4	no	—
94	C₂×C₄×C₈	—	53	2	no	—
95	C₂×C₄×C₈	—	53	2	no	—
96	C₂×C₄×C₈	—	53	2	no	—
97	C₂×C₄×C₈	—	53	4	no	—
98	C₂×C₄×C₈	—	53	2	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₂×C₄×C₈	#6157	1	1	yes	characteristic
101	C₄³	—	53	1	no	—
102	#322	—	53	16	no	—
103	C₅₃	C₄×#322	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₈	—	53	4	no	—
105	C₂²×C₈	—	53	4	no	—
106	C₂²×C₈	—	53	2	no	—
107	C₂²×C₈	C₂×#6157	1	2	yes	—
108	C₂×C₄²	—	53	2	no	—
109	C₂×C₄²	—	53	2	no	—
110	C₂×C₄²	—	53	1	no	—
111	C₂×C₄²	—	53	1	no	—
112	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	53	8	no	—
115	C₄×C₈	—	53	8	no	—
116	C₄×C₈	—	53	16	no	—
117	C₄×C₈	—	53	8	no	—
118	C₄×C₈	—	106	4	no	—
119	C₄×C₈	—	106	8	no	—
120	C₄×C₈	—	106	2	no	—
121	C₂²×C₄	—	53	2	no	—
122	C₂²×C₄	—	53	1	no	—
123	C₂²×C₄	—	53	1	no	—
124	C₂²×C₄	C₄×#6157	1	2	yes	—
125	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	53	8	no	—
129	C₂×C₈	—	53	16	no	—
130	C₂×C₈	—	53	8	no	—
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	106	8	no	—
133	C₂×C₈	—	106	2	no	—
134	C₂×C₈	—	106	2	no	—
135	C₂×C₈	C₄×#6157	1	4	yes	—
136	C₄²	—	53	8	no	—
137	C₄²	—	53	8	no	—
138	C₄²	—	53	4	no	—
139	C₄²	—	53	4	no	—
140	C₄²	#11719936	1	2	yes	—
141	C₄²	#11719940	1	2	yes	—
142	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
143	C₂×C₄	—	53	2	no	—
144	C₂×C₄	—	53	2	no	—
145	C₂×C₄	—	53	4	no	—
146	C₂×C₄	—	53	4	no	—
147	C₂×C₄	—	53	4	no	—
148	C₂×C₄	—	53	4	no	—
149	C₂×C₄	—	53	4	no	—
150	C₂×C₄	C₈×#6157	1	4	yes	—
151	C₂×C₄	#13133271	1	4	yes	—
152	C₂×C₄	#13133272	1	4	yes	—
153	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
154	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
155	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
156	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
157	C₈	—	2	4	no	—
158	C₈	—	53	16	no	—
159	C₈	—	106	8	no	—
160	C₈	—	106	4	no	—
161	C₂²	C₄²×#6157	1	1	yes	characteristic
162	C₂²	C₂×C₈×#6157	1	2	yes	—
163	C₂²	C₂×#13133271	1	2	yes	—
164	C₂²	C₂×#13133272	1	1	yes	characteristic
165	C₂²	#16811582	1	1	yes	characteristic
166	C₄	—	53	4	no	—
167	C₄	—	53	4	no	—
168	C₄	—	53	8	no	—
169	C₄	#88324⋊C₈	1	8	yes	direct product factor
170	C₄	C₄×#11719940	1	2	yes	—
171	C₄	C₄×#11719936	1	2	yes	—
172	C₂	C₂×#88324⋊C₈	1	2	yes	—
173	C₂	#20289911	1	2	yes	—
174	C₂	C₄×C₈×#6157	1	1	yes	characteristic
175	C₂	C₄×#13133271	1	1	yes	characteristic
176	C₂	C₄×#13133272	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₄²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₄×#322
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. #88324⋊C₈	C₄	direct (N × H)
18. C₄×#88324	C₈	semidirect
69. #88324	C₄×C₈	semidirect
78. C₄×C₅₃	#322	semidirect
81. Dic₅₃	#322	semidirect
103. C₅₃	C₄×#322	semidirect
169. C₄	#88324⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×#88324	2
C₂×#88324⋊C₈	2
C₄×C₈×Dic₅₃	2
C₄×#322	53

Subgroups — C4×#88324⋊C8 #28300032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#88324⋊C₈ #28300032