Subgroups — C₂×#20289738 #28309822

Order 13568.

286 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719910	C₂	1	2	yes	maximal
2	#20289738	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	—	2	2	no	—
5	C₄⋊#11719910	—	2	2	no	—
6	C₄⋊#11719910	C₂²	1	4	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₈×#88322	C₄	1	2	yes	—
10	C₈×#88322	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719910	—	2	4	no	—
13	C₂×#11719910	—	4	4	no	—
14	C₂²×#88322	C₈	1	1	yes	characteristic
15	C₂²×#88322	C₈	1	1	yes	characteristic
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	C₂²×#88322	Q₈	1	1	yes	characteristic
18	C₄×#88322	C₈	1	2	yes	—
19	C₄×#88322	C₂×C₄	1	1	yes	characteristic
20	C₄×#88322	C₂×C₄	1	2	yes	—
21	C₄×#88322	D₄	1	1	yes	characteristic
22	C₄×#88322	Q₈	1	1	yes	characteristic
23	C₄×#88322	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	—	2	1	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	4	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
36	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	C₁₆	1	4	yes	complemented
40	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#88322	C₂×C₈	1	2	yes	—
43	C₂×#88322	C₂×C₈	1	2	yes	—
44	C₂×#88322	SD₁₆	1	4	yes	—
45	C₂×#88322	C₂×D₄	1	1	yes	characteristic
46	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
47	#11719910	—	4	4	no	—
48	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#106	1	2	yes	—
54	C₂×C₄×C₅₃	#108	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₅₃	—	2	4	no	—
61	#88322	—	2	1	no	—
62	#88322	—	2	1	no	—
63	#88322	—	2	2	no	—
64	#88322	#106	1	2	yes	—
65	#88322	C₂×C₁₆	1	4	yes	complemented
66	#88322	C₂×SD₁₆	1	2	yes	—
67	C₂×#1367	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#334	1	2	yes	—
69	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
70	C₂²×C₅₃	#364	1	2	yes	—
71	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#334	1	2	yes	—
75	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
76	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
78	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
79	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	—	53	2	no	—
81	C₂×C₈²	—	53	1	no	—
82	#1367	—	53	4	no	—
83	C₂×C₅₃	#1367	1	4	yes	complemented
84	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
85	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
86	C₂×C₅₃	C₂×#364	1	1	yes	characteristic
87	C₂²×C₁₆	—	106	2	no	—
88	C₂×C₄×C₈	—	53	1	no	—
89	C₂×C₄×C₈	—	53	1	no	—
90	C₂×C₄×C₈	#6157	1	1	yes	characteristic
91	C₄⋊C₁₆	—	53	4	no	—
92	C₄⋊C₁₆	—	106	2	no	—
93	C₈²	—	53	2	no	—
94	C₈²	—	53	2	no	—
95	C₅₃	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	53	1	no	—
98	C₂²×C₈	—	53	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
102	C₂×C₁₆	—	106	4	no	—
103	C₂×C₁₆	—	212	4	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₄×C₈	—	53	1	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	1	no	—
110	C₄×C₈	—	53	2	no	—
111	C₄×C₈	C₂×#6157	1	2	yes	—
112	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
114	C₂²×C₄	#11719940	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	2	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	2	no	—
125	C₂×C₈	—	106	2	no	—
126	C₂×C₈	—	106	2	no	—
127	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
128	C₄²	C₄⋊#6157	1	1	yes	characteristic
129	C₄²	#11719940	1	1	yes	characteristic
130	C₄²	C₂²×#6157	1	1	yes	characteristic
131	C₁₆	—	212	4	no	—
132	C₂³	#13133284	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	#13133284	1	1	yes	characteristic
135	C₂×C₄	#13133284	1	1	yes	characteristic
136	C₂×C₄	#13133284	1	1	yes	characteristic
137	C₂×C₄	#13133285	1	2	yes	—
138	C₂×C₄	#13133288	1	2	yes	—
139	C₂×C₄	D₅₃⋊C₁₆	1	2	yes	—
140	C₂×C₄	#13133292	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
143	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	53	2	no	—
146	C₈	—	53	2	no	—
147	C₈	—	53	4	no	—
148	C₈	—	106	1	no	—
149	C₈	—	106	1	no	—
150	C₈	—	106	2	no	—
151	C₂²	#16811485	1	2	yes	—
152	C₂²	#16811544	1	2	yes	—
153	C₂²	#16811546	1	2	yes	—
154	C₂²	C₂×#13133284	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#16811485	1	2	yes	—
158	C₄	#16811544	1	2	yes	—
159	C₄	C₂×#13133285	1	1	yes	characteristic
160	C₄	C₂×#13133288	1	1	yes	characteristic
161	C₄	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
162	C₄	C₂×#13133292	1	1	yes	characteristic
163	C₂	#20289738	1	4	yes	direct product factor
164	C₂	C₂×#16811485	1	1	yes	characteristic
165	C₂	C₂×#16811544	1	1	yes	characteristic
166	C₂	C₂×#16811546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄⋊C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#1367
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289738	C₂	direct (N × H)
39. C₂×#88322	C₁₆	semidirect
65. #88322	C₂×C₁₆	semidirect
83. C₂×C₅₃	#1367	semidirect
95. C₅₃	C₂×#1367	semidirect
163. C₂	#20289738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719910	2
#20289738	2
C₂×C₈×#88322	2
C₂×#1367	53

Subgroups — C2×#20289738 #28309822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289738 #28309822