Subgroups — C₂×#20287721 #28265021

Order 13568.

1894 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811352	C₂	1	2	yes	maximal
2	#20287721	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811003	C₂	1	3	yes	maximal
4	C₂×#16810834	C₂	1	6	yes	maximal
5	C₂×C₄×Dic₂₁₂	C₂	1	3	yes	maximal
6	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₁₂	C₂²	1	3	yes	—
8	C₂²×Dic₂₁₂	C₂²	1	6	yes	—
9	#16811352	—	2	8	no	—
10	#16811352	C₂²	1	16	yes	—
11	C₂×#13133217	C₂²	1	6	yes	—
12	C₂×#13133208	—	2	6	no	—
13	C₂×Q₈×Dic₅₃	—	2	2	no	—
14	C₂×C₄×Dic₁₀₆	C₂²	1	6	yes	—
15	#16811003	C₂²	1	24	yes	—
16	#16810834	C₂²	1	48	yes	—
17	C₄×Dic₂₁₂	C₂²	1	24	yes	—
18	C₂×#13133151	C₂²	1	1	yes	characteristic
19	C₂×#13133142	C₂²	1	6	yes	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₂²	1	8	yes	—
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
24	C₂²×Dic₁₀₆	C₂³	1	6	yes	—
25	C₂×Dic₂₁₂	—	2	12	no	—
26	C₂×Dic₂₁₂	—	4	12	no	—
27	C₂×Dic₂₁₂	C₂³	1	12	yes	—
28	C₂×Dic₂₁₂	C₂³	1	24	yes	—
29	#13133217	—	2	12	no	—
30	#13133217	C₂³	1	24	yes	—
31	#13133208	—	2	48	no	—
32	Q₈×Dic₅₃	—	2	16	no	—
33	C₄×Dic₁₀₆	—	2	12	no	—
34	C₄×Dic₁₀₆	C₂³	1	24	yes	—
35	C₂×C₄⋊Dic₅₃	—	2	6	no	—
36	C₂×C₄⋊Dic₅₃	—	2	6	no	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
38	C₂×#11719919	—	2	6	no	—
39	C₂×C₄×Dic₅₃	—	2	6	no	—
40	#13133151	C₂³	1	4	yes	—
41	#13133142	C₂³	1	24	yes	—
42	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	D₄	1	4	yes	—
46	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	3	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
49	C₄⋊C₈×C₅₃	C₂³	1	12	yes	—
50	C₄×C₈×C₅₃	C₂³	1	12	yes	—
51	C₂×Q₈×C₅₃	Q₁₆	1	8	yes	complemented
52	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	2	2	no	—
57	C₂²×Dic₅₃	—	2	6	no	—
58	C₂×Dic₁₀₆	—	2	12	no	—
59	C₂×Dic₁₀₆	—	2	12	no	—
60	C₂×Dic₁₀₆	—	2	6	no	—
61	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
62	C₂×C₈×C₅₃	—	2	6	no	—
63	C₂×C₈×C₅₃	#41	1	4	yes	—
64	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
65	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
66	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
67	C₄²×C₅₃	C₂×D₄	1	6	yes	—
68	C₄²×C₅₃	C₂⁴	1	6	yes	—
69	C₄⋊Dic₅₃	—	2	24	no	—
70	C₄⋊Dic₅₃	—	2	24	no	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
73	#11719919	—	2	24	no	—
74	C₄×Dic₅₃	—	2	24	no	—
75	Dic₂₁₂	—	2	24	no	—
76	Dic₂₁₂	—	4	24	no	—
77	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₅₃	C₂×Q₁₆	1	16	yes	complemented
79	C₂×C₄×C₅₃	—	2	6	no	—
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	12	yes	—
81	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
84	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
85	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
86	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	12	no	—
88	C₂×Dic₅₃	—	2	4	no	—
89	C₂×Dic₅₃	—	4	12	no	—
90	C₂×Dic₅₃	—	4	4	no	—
91	Dic₁₀₆	—	2	12	no	—
92	Dic₁₀₆	—	2	12	no	—
93	Dic₁₀₆	—	4	12	no	—
94	C₈×C₅₃	—	2	6	no	—
95	C₈×C₅₃	C₂×#41	1	4	yes	—
96	C₂×#3395	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	#550	1	2	yes	—
98	C₂²×C₅₃	C₂²×Q₁₆	1	2	yes	—
99	C₂²×C₅₃	#578	1	2	yes	—
100	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
101	C₄×C₅₃	—	2	6	no	—
102	C₄×C₅₃	#550	1	2	yes	—
103	C₄×C₅₃	C₂²×Q₁₆	1	12	yes	—
104	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
105	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
106	Dic₅₃	—	4	12	no	—
107	Dic₅₃	—	4	4	no	—
108	C₂×#557	—	53	2	no	—
109	#3395	—	53	16	no	—
110	C₂×#494	—	53	3	no	—
111	C₂×#462	—	53	6	no	—
112	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
113	C₂×C₄×Q₁₆	—	53	3	no	—
114	C₂×C₅₃	#3395	1	4	yes	complemented
115	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
116	C₂×C₅₃	C₂³×Q₁₆	1	1	yes	characteristic
117	C₂×C₅₃	C₂×#578	1	1	yes	characteristic
118	C₂²×Q₁₆	—	53	3	no	—
119	C₂²×Q₁₆	—	53	6	no	—
120	#557	—	53	16	no	—
121	#557	—	106	8	no	—
122	C₂×C₄⋊Q₈	—	53	6	no	—
123	C₂×#125	—	106	6	no	—
124	C₂×C₄×Q₈	—	53	6	no	—
125	C₂×C₄×Q₈	—	106	2	no	—
126	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
127	#494	—	53	24	no	—
128	#462	—	53	48	no	—
129	C₈×Q₈	C₂×D₅₃	1	8	yes	—
130	C₄×Q₁₆	—	53	24	no	—
131	C₂×#107	—	53	1	no	—
132	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
133	C₂×#103	—	53	6	no	—
134	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
135	C₅₃	C₂×#3395	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₈	—	53	6	no	—
137	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
138	C₂×Q₁₆	—	53	12	no	—
139	C₂×Q₁₆	—	53	24	no	—
140	C₂×Q₁₆	—	106	12	no	—
141	C₂×Q₁₆	—	212	12	no	—
142	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₄⋊Q₈	—	53	24	no	—
145	C₄⋊Q₈	—	106	12	no	—
146	#125	—	106	48	no	—
147	C₄×Q₈	—	53	24	no	—
148	C₄×Q₈	—	106	16	no	—
149	C₄×Q₈	—	106	12	no	—
150	C₄×Q₈	D₂₁₂	1	4	yes	—
151	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
152	C₂×C₄⋊C₄	—	53	2	no	—
153	C₂×C₄⋊C₄	—	106	6	no	—
154	C₂×C₄⋊C₄	—	106	6	no	—
155	C₂×C₄⋊C₄	—	106	6	no	—
156	C₂×C₄⋊C₄	D₂₁₂	1	3	yes	—
157	C₂×C₄²	—	106	6	no	—
158	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
159	#107	—	53	4	no	—
160	C₄⋊C₈	C₂²×D₅₃	1	12	yes	—
161	#103	—	53	24	no	—
162	C₄×C₈	C₂²×D₅₃	1	12	yes	—
163	C₂×Q₈	—	53	12	no	—
164	C₂×Q₈	—	106	6	no	—
165	C₂×Q₈	—	106	12	no	—
166	C₂×Q₈	—	106	12	no	—
167	C₂×Q₈	Dic₂₁₂	1	8	yes	complemented
168	C₂×Q₈	C₂×D₂₁₂	1	4	yes	—
169	C₂²×C₄	—	106	2	no	—
170	C₂²×C₄	—	106	6	no	—
171	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
172	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
173	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
174	Q₁₆	—	106	24	no	—
175	Q₁₆	—	212	24	no	—
176	C₂×C₈	—	2	6	no	—
177	C₂×C₈	#11719949	1	4	yes	—
178	C₂×C₈	C₂³×D₅₃	1	6	yes	—
179	C₂×C₈	C₂³×D₅₃	1	2	yes	—
180	C₄⋊C₄	—	53	4	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₄⋊C₄	—	106	24	no	—
183	C₄⋊C₄	—	106	24	no	—
184	C₄⋊C₄	—	106	24	no	—
185	C₄⋊C₄	C₂×D₂₁₂	1	12	yes	—
186	C₄²	—	106	24	no	—
187	C₄²	C₂×D₂₁₂	1	6	yes	—
188	C₄²	C₂³×D₅₃	1	6	yes	—
189	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
190	Q₈	—	106	12	no	—
191	Q₈	—	106	12	no	—
192	Q₈	—	212	12	no	—
193	Q₈	C₂×Dic₂₁₂	1	16	yes	complemented
194	C₂×C₄	—	2	6	no	—
195	C₂×C₄	—	106	4	no	—
196	C₂×C₄	—	106	12	no	—
197	C₂×C₄	—	212	12	no	—
198	C₂×C₄	—	212	4	no	—
199	C₂×C₄	C₂×Dic₂₁₂	1	12	yes	—
200	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
201	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
203	C₂×C₄	C₂×#11719949	1	2	yes	—
204	C₂×C₄	#13133263	1	2	yes	—
205	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
206	C₈	—	2	6	no	—
207	C₈	C₂×#11719949	1	4	yes	—
208	C₂²	#16811295	1	2	yes	—
209	C₂²	C₂²×Dic₂₁₂	1	2	yes	—
210	C₂²	#16811440	1	2	yes	—
211	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
212	C₄	—	2	6	no	—
213	C₄	—	212	12	no	—
214	C₄	—	212	4	no	—
215	C₄	#16811295	1	2	yes	—
216	C₄	C₂²×Dic₂₁₂	1	12	yes	—
217	C₄	C₂²×#11719949	1	1	yes	characteristic
218	C₄	C₂×#13133263	1	1	yes	characteristic
219	C₂	#20287721	1	4	yes	direct product factor
220	C₂	C₂×#16811295	1	1	yes	characteristic
221	C₂	C₂³×Dic₂₁₂	1	1	yes	characteristic
222	C₂	C₂×#16811440	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₂×#3395
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #20287721	C₂	direct (N × H)
51. C₂×Q₈×C₅₃	Q₁₆	semidirect
78. Q₈×C₅₃	C₂×Q₁₆	semidirect
114. C₂×C₅₃	#3395	semidirect
135. C₅₃	C₂×#3395	semidirect
167. C₂×Q₈	Dic₂₁₂	semidirect
193. Q₈	C₂×Dic₂₁₂	semidirect
219. C₂	#20287721	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×#16811352	2
#20287721	2
C₂×#16811003	2
C₂×#16810834	2
C₂×C₄×Dic₂₁₂	2
C₂×C₈×Q₈×C₅₃	2
C₂×#3395	53

Subgroups — C2×#20287721 #28265021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20287721 #28265021