Subgroups — C₂²×#16810426 #27536967

Order 13568.

1214 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	C₂³×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810426	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₆	—	2	1	no	—
6	C₂²×Dic₂₁₂	—	2	4	no	—
7	C₂²×Dic₂₁₂	—	2	3	no	—
8	C₂²×Dic₂₁₂	C₄	1	8	yes	complemented
9	C₂²×Dic₂₁₂	C₂²	1	6	yes	—
10	C₂²×#11719908	—	2	4	no	—
11	C₂²×#11719908	C₂²	1	6	yes	—
12	#16810426	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×#107×C₅₃	—	2	3	no	—
16	C₂×#107×C₅₃	C₂²	1	6	yes	—
17	C₂²×Dic₁₀₆	—	2	8	no	—
18	C₂²×Dic₁₀₆	—	2	6	no	—
19	C₂²×Dic₁₀₆	—	4	3	no	—
20	C₂²×Dic₁₀₆	—	4	4	no	—
21	C₂³×Dic₅₃	—	4	1	no	—
22	C₂×Dic₂₁₂	—	2	24	no	—
23	C₂×Dic₂₁₂	—	2	12	no	—
24	C₂×Dic₂₁₂	—	2	6	no	—
25	C₂×Dic₂₁₂	C₂×C₄	1	24	yes	complemented
26	C₂×Dic₂₁₂	C₂³	1	4	yes	—
27	C₂×#11719908	—	2	12	no	—
28	C₂×#11719908	C₂³	1	4	yes	—
29	C₂³×C₄×C₅₃	—	4	1	no	—
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	D₄	1	4	yes	—
33	C₂²×C₈×C₅₃	D₄	1	3	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
35	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	3	no	—
37	#107×C₅₃	—	2	6	no	—
38	#107×C₅₃	C₂³	1	4	yes	—
39	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	4	6	no	—
41	C₂²×C₄×C₅₃	—	8	4	no	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₄×C₅₃	D₈	1	3	yes	—
44	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
46	C₂²×Dic₅₃	—	4	8	no	—
47	C₂²×Dic₅₃	—	4	6	no	—
48	C₂×Dic₁₀₆	—	2	24	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	24	no	—
51	C₂×Dic₁₀₆	—	4	6	no	—
52	C₂×Dic₁₀₆	—	4	12	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	12	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
57	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	4	6	no	—
60	Dic₂₁₂	—	2	24	no	—
61	Dic₂₁₂	—	2	4	no	—
62	Dic₂₁₂	C₂²×C₄	1	8	yes	complemented
63	#11719908	—	2	4	no	—
64	C₂³×C₅₃	#102	1	4	yes	—
65	C₂³×C₅₃	SD₃₂	1	4	yes	—
66	C₂³×C₅₃	Q₃₂	1	4	yes	—
67	C₂³×C₅₃	C₂×D₈	1	3	yes	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	—	8	12	no	—
70	C₂×C₄×C₅₃	#102	1	12	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
72	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×SD₁₆	1	6	yes	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
75	C₂×Dic₅₃	—	4	4	no	—
76	C₂×Dic₅₃	—	4	24	no	—
77	Dic₁₀₆	—	2	8	no	—
78	Dic₁₀₆	—	4	4	no	—
79	Dic₁₀₆	—	4	24	no	—
80	C₈×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
81	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂²×#358	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	#358	1	16	yes	complemented
85	C₂²×C₅₃	C₂×#102	1	6	yes	—
86	C₂²×C₅₃	C₂×SD₃₂	1	6	yes	—
87	C₂²×C₅₃	C₂×Q₃₂	1	6	yes	—
88	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	C₂×#102	1	6	yes	—
91	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
93	Dic₅₃	—	4	8	no	—
94	C₂³×Q₁₆	—	53	1	no	—
95	C₂³×C₁₆	—	53	1	no	—
96	C₂²×#107	D₅₃	1	1	yes	characteristic
97	C₂×#358	—	53	12	no	—
98	C₂×C₅₃	C₂×#358	1	12	yes	complemented
99	C₂×C₅₃	C₂²×#102	1	1	yes	characteristic
100	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
101	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
102	C₂³×Q₈	—	106	1	no	—
103	C₂²×Q₁₆	—	53	8	no	—
104	C₂²×Q₁₆	—	53	6	no	—
105	C₂²×Q₁₆	—	106	4	no	—
106	C₂²×Q₁₆	—	106	3	no	—
107	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	—	2	1	no	—
109	C₂²×C₁₆	—	53	6	no	—
110	C₂²×C₁₆	—	106	4	no	—
111	C₂×#107	—	2	3	no	—
112	C₂×#107	C₂×D₅₃	1	6	yes	—
113	#358	—	53	16	no	—
114	C₅₃	C₂²×#358	1	1	yes	characteristic, complemented, Sylow
115	C₂²×Q₈	—	106	6	no	—
116	C₂²×Q₈	—	106	8	no	—
117	C₂²×Q₈	—	212	3	no	—
118	C₂²×Q₈	—	212	4	no	—
119	C₂³×C₄	—	4	1	no	—
120	C₂³×C₄	—	212	1	no	—
121	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×Q₁₆	—	53	24	no	—
123	C₂×Q₁₆	—	53	4	no	—
124	C₂×Q₁₆	—	106	12	no	—
125	C₂×Q₁₆	—	106	6	no	—
126	C₂×Q₁₆	—	106	24	no	—
127	C₂²×C₈	C₄×D₅₃	1	4	yes	—
128	C₂²×C₈	D₂₁₂	1	4	yes	—
129	C₂²×C₈	C₂²⋊D₅₃	1	3	yes	—
130	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
131	C₂×C₄⋊C₄	—	2	6	no	—
132	C₂×C₄⋊C₄	—	4	3	no	—
133	C₂×C₁₆	—	53	4	no	—
134	C₂×C₁₆	—	106	12	no	—
135	#107	—	2	6	no	—
136	#107	C₂²×D₅₃	1	4	yes	—
137	C₂⁴	#11719922	1	1	yes	characteristic
138	C₂×Q₈	—	106	24	no	—
139	C₂×Q₈	—	106	4	no	—
140	C₂×Q₈	—	212	24	no	—
141	C₂×Q₈	—	212	6	no	—
142	C₂×Q₈	—	212	12	no	—
143	C₂²×C₄	—	4	6	no	—
144	C₂²×C₄	—	8	4	no	—
145	C₂²×C₄	—	212	8	no	—
146	C₂²×C₄	—	212	6	no	—
147	C₂²×C₄	#11719921	1	4	yes	—
148	C₂²×C₄	#11719922	1	3	yes	—
149	C₂²×C₄	#11719924	1	4	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
151	Q₁₆	—	53	8	no	—
152	Q₁₆	—	106	24	no	—
153	Q₁₆	—	106	4	no	—
154	C₂×C₈	#11719921	1	12	yes	—
155	C₂×C₈	C₂×C₄×D₅₃	1	6	yes	—
156	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
157	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
158	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	6	no	—
161	C₁₆	—	106	4	no	—
162	C₂³	#13133137	1	4	yes	—
163	C₂³	#13133161	1	4	yes	—
164	C₂³	#13133163	1	4	yes	—
165	C₂³	C₂×#11719922	1	3	yes	—
166	Q₈	—	106	8	no	—
167	Q₈	—	212	24	no	—
168	Q₈	—	212	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	8	12	no	—
171	C₂×C₄	—	212	24	no	—
172	C₂×C₄	—	212	4	no	—
173	C₂×C₄	#13133137	1	12	yes	—
174	C₂×C₄	C₂×#11719921	1	6	yes	—
175	C₂×C₄	C₂×#11719922	1	3	yes	—
176	C₂×C₄	C₂×#11719924	1	6	yes	—
177	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₈	C₂×#11719921	1	6	yes	—
179	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
180	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
181	C₂²	#16810426	1	16	yes	direct product factor
182	C₂²	C₂×#13133137	1	6	yes	—
183	C₂²	C₂×#13133161	1	6	yes	—
184	C₂²	C₂×#13133163	1	6	yes	—
185	C₂²	C₂²×#11719922	1	1	yes	characteristic
186	C₄	—	8	4	no	—
187	C₄	—	212	8	no	—
188	C₄	C₂×#13133137	1	6	yes	—
189	C₄	C₂²×#11719921	1	1	yes	characteristic
190	C₄	C₂²×#11719924	1	1	yes	characteristic
191	C₂	C₂×#16810426	1	12	yes	direct product factor
192	C₂	C₂²×#13133137	1	1	yes	characteristic
193	C₂	C₂²×#13133161	1	1	yes	characteristic
194	C₂	C₂²×#13133163	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#107×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#107	D₅₃
hypocenter	C₅₃	C₂²×#358
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. C₂×#16810426	C₂	direct (N × H)
8. C₂²×Dic₂₁₂	C₄	semidirect
12. #16810426	C₂²	direct (N × H)
25. C₂×Dic₂₁₂	C₂×C₄	semidirect
62. Dic₂₁₂	C₂²×C₄	semidirect
84. C₂²×C₅₃	#358	semidirect
98. C₂×C₅₃	C₂×#358	semidirect
114. C₅₃	C₂²×#358	semidirect
181. C₂²	#16810426	direct (N × H)
191. C₂	C₂×#16810426	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₁₂	2
C₂³×#11719908	2
C₂×#16810426	2
C₂²×#107×C₅₃	2
C₂²×#358	53

Subgroups — C22×#16810426 #27536967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810426 #27536967