Subgroups — C₈×C₄⋊#88322 #27275213

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×C₈×#88322	C₂	1	2	yes	maximal
3	#20272542	C₂	1	2	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₄	1	4	yes	—
6	C₂×C₄×#88322	C₂²	1	4	yes	—
7	#16810378	C₄	1	4	yes	—
8	C₈×#88322	—	2	4	no	—
9	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	C₂×C₄	1	8	yes	—
15	C₄⋊#88322	C₈	1	16	yes	direct product factor
16	C₄×#88322	—	2	4	no	—
17	C₄×#88322	—	2	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₅₃	—	2	4	no	—
26	C₄×C₈×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
28	C₄×C₈×C₅₃	D₄	1	2	yes	—
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
40	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
41	C₄²×C₅₃	—	2	2	no	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	C₂×C₈	1	8	yes	—
44	C₄²×C₅₃	C₂×C₈	1	4	yes	—
45	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₅₃	#41	1	2	yes	—
48	C₂×#88322	—	2	8	no	—
49	C₂×#88322	—	2	8	no	—
50	C₂×#88322	C₂×C₈	1	16	yes	—
51	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
56	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
57	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	#131	1	4	yes	—
67	C₈×C₅₃	—	2	4	no	—
68	C₈×C₅₃	C₄⋊C₈	1	8	yes	complemented
69	#88322	—	2	16	no	—
70	C₈×C₄⋊C₈	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
73	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	#405	1	2	yes	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₈²	1	4	yes	complemented
79	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
80	C₄×C₅₃	#431	1	2	yes	—
81	C₄×C₅₃	C₈×D₄	1	4	yes	—
82	C₄×C₅₃	C₈×Q₈	1	4	yes	—
83	C₄×C₄⋊C₈	—	53	2	no	—
84	C₄²×C₈	D₅₃	1	1	yes	characteristic
85	C₂×C₈²	—	53	2	no	—
86	#1273	—	53	2	no	—
87	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
88	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
89	C₂×C₅₃	#1450	1	1	yes	characteristic
90	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₅₃	C₈×C₄⋊C₄	1	2	yes	—
92	C₂×C₄⋊C₈	—	53	4	no	—
93	C₂×C₄×C₈	—	53	4	no	—
94	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
95	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
98	C₄³	C₂×D₅₃	1	1	yes	characteristic
99	#336	—	53	4	no	—
100	C₈²	—	106	4	no	—
101	C₅₃	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	—	53	8	no	—
103	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
104	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
105	C₂×C₄²	C₄×D₅₃	1	2	yes	—
106	C₂×C₄²	C₄×D₅₃	1	2	yes	—
107	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
109	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
110	C₄⋊C₈	—	53	16	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	106	4	no	—
114	C₄×C₈	—	106	4	no	—
115	C₄×C₈	#88322	1	8	yes	complemented
116	C₄×C₈	Dic₁₀₆	1	2	yes	—
117	C₄×C₈	D₂₁₂	1	2	yes	—
118	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
119	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	53	16	no	—
126	C₂×C₈	—	106	8	no	—
127	C₂×C₈	—	106	8	no	—
128	C₂×C₈	C₂×#88322	1	4	yes	—
129	C₂×C₈	C₂×#88322	1	4	yes	—
130	C₂×C₈	#11719917	1	4	yes	—
131	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₈×D₅₃	1	8	yes	—
135	C₄²	C₂×#88322	1	4	yes	—
136	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
138	C₄²	#11719945	1	2	yes	—
139	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
144	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
145	C₂×C₄	C₄×#88322	1	2	yes	—
146	C₂×C₄	C₄×#88322	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₄	#13133176	1	1	yes	characteristic
149	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
150	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
151	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
152	C₂×C₄	C₂²×#88322	1	2	yes	—
153	C₂×C₄	C₂×#11719917	1	2	yes	—
154	C₂×C₄	#13133223	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	106	16	no	—
157	C₈	C₄⋊#88322	1	8	yes	direct product factor
158	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
160	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
161	C₂²	C₄×#11719917	1	1	yes	characteristic
162	C₂²	#16810578	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	C₈×#88322	1	4	yes	complemented
166	C₄	C₂×C₄⋊#88322	1	2	yes	—
167	C₄	#16810676	1	2	yes	—
168	C₄	C₈×D₂₁₂	1	4	yes	—
169	C₄	C₈×Dic₁₀₆	1	4	yes	—
170	C₂	C₂×C₈×#88322	1	1	yes	characteristic
171	C₂	C₈×#11719917	1	2	yes	—
172	C₂	#20272998	1	1	yes	characteristic
173	C₂	C₄×C₄⋊#88322	1	1	yes	characteristic
174	C₂	C₈×C₄⋊Dic₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
15. C₄⋊#88322	C₈	direct (N × H)
27. C₄×C₈×C₅₃	C₈	semidirect
68. C₈×C₅₃	C₄⋊C₈	semidirect
78. C₄×C₅₃	C₈²	semidirect
101. C₅₃	C₈×C₄⋊C₈	semidirect
115. C₄×C₈	#88322	semidirect
157. C₈	C₄⋊#88322	direct (N × H)
165. C₄	C₈×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₂×C₈×#88322	2
#20272542	2
C₄²×C₈×C₅₃	2
C₈×C₄⋊C₈	53

Subgroups — C8×C4⋊#88322 #27275213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊#88322 #27275213