Subgroups — C₂×#20276510 #27537148

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	2	yes	maximal
2	#20276510	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810453	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	2	no	—
6	C₂×#13133153	—	2	4	no	—
7	C₂×#13133153	—	2	2	no	—
8	C₂×#13133153	C₂²	1	4	yes	—
9	C₂×#13133153	C₂²	1	8	yes	—
10	#16810453	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	C₄	1	4	yes	—
13	C₂×M₃₂×C₅₃	C₄	1	4	yes	—
14	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	8	no	—
18	C₂×#11719917	—	4	2	no	—
19	C₂×#11719917	—	4	4	no	—
20	C₂²×#88322	—	4	2	no	—
21	#13133153	—	2	4	no	—
22	#13133153	—	2	8	no	—
23	#13133153	C₂³	1	8	yes	—
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₅₃	D₄	1	2	yes	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
33	M₃₂×C₅₃	C₂×C₄	1	16	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
37	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	D₈	1	2	yes	—
41	C₂²×C₄×C₅₃	D₈	1	2	yes	—
42	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
59	#11719917	—	2	8	no	—
60	#11719917	—	4	8	no	—
61	#11719917	—	4	4	no	—
62	C₂×#88322	—	4	8	no	—
63	C₂×#88322	—	4	4	no	—
64	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂³×C₅₃	—	4	2	no	—
66	C₂³×C₅₃	#107	1	2	yes	—
67	C₂³×C₅₃	#107	1	2	yes	—
68	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂³×C₅₃	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	#107	1	4	yes	—
71	C₂×C₄×C₅₃	#107	1	2	yes	—
72	C₂×C₄×C₅₃	#107	1	4	yes	—
73	C₂×C₄×C₅₃	#107	1	2	yes	—
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
88	#88322	—	4	8	no	—
89	C₂×#2161	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	4	no	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	4	2	no	—
93	C₂²×C₅₃	—	4	2	no	—
94	C₂²×C₅₃	C₂×#107	1	4	yes	—
95	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
96	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
97	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#107	1	4	yes	—
99	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
100	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
103	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
104	C₂²×#108	—	53	2	no	—
105	#2161	—	53	8	no	—
106	C₂×#368	—	53	4	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	—	2	4	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	#2161	1	2	yes	complemented
111	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
112	C₂²×M₁₆	—	106	2	no	—
113	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
114	C₂×M₃₂	Dic₅₃	1	4	yes	—
115	C₂×M₃₂	Dic₅₃	1	4	yes	—
116	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
117	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
118	C₂×#108	—	53	4	no	—
119	C₂×#108	—	53	8	no	—
120	C₂×#108	—	106	4	no	—
121	C₂×#108	—	106	2	no	—
122	#368	—	53	16	no	—
123	C₅₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
125	C₂×M₁₆	—	106	8	no	—
126	C₂×M₁₆	—	106	4	no	—
127	C₂×M₁₆	—	212	4	no	—
128	C₂×M₁₆	—	212	2	no	—
129	C₂²×C₈	—	212	2	no	—
130	C₂²×C₈	Dic₁₀₆	1	2	yes	—
131	C₂²×C₈	Dic₁₀₆	1	2	yes	—
132	C₂²×C₈	D₂₁₂	1	2	yes	—
133	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
134	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
135	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
138	M₃₂	C₂×Dic₅₃	1	16	yes	—
139	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
140	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
141	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
142	#108	—	53	8	no	—
143	#108	—	106	4	no	—
144	#108	—	106	8	no	—
145	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
146	C₂²×C₄	D₄₂₄	1	2	yes	—
147	C₂²×C₄	D₄₂₄	1	2	yes	—
148	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
149	C₂²×C₄	Dic₂₁₂	1	2	yes	—
150	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
151	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
152	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
153	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
154	M₁₆	—	106	8	no	—
155	M₁₆	—	212	8	no	—
156	M₁₆	—	212	4	no	—
157	C₂×C₈	—	212	8	no	—
158	C₂×C₈	—	212	4	no	—
159	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
160	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
161	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
163	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
166	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
167	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
168	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
171	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
172	C₁₆	C₂²×Dic₅₃	1	8	yes	—
173	C₂³	—	4	2	no	—
174	C₂³	#13133151	1	2	yes	—
175	C₂³	#13133151	1	2	yes	—
176	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
177	C₂³	C₂×Dic₂₁₂	1	2	yes	—
178	C₂×C₄	#13133151	1	2	yes	—
179	C₂×C₄	#13133151	1	4	yes	—
180	C₂×C₄	#13133151	1	2	yes	—
181	C₂×C₄	#13133151	1	4	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
186	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
189	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
191	C₈	—	212	8	no	—
192	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
194	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
195	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
196	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×#13133151	1	4	yes	—
202	C₂²	C₂×#13133151	1	1	yes	characteristic
203	C₂²	C₂×#13133151	1	1	yes	characteristic
204	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
205	C₄	C₂×#13133151	1	4	yes	—
206	C₄	C₂×#13133151	1	1	yes	characteristic
207	C₄	C₂×#13133151	1	1	yes	characteristic
208	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
209	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	4	no	—
212	C₂	—	2	1	no	—
213	C₂	#20276510	1	2	yes	direct product factor
214	C₂	C₂²×#13133151	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20276510	C₂	direct (N × H)
3. C₂×#16810453	C₂	semidirect
10. #16810453	C₂²	semidirect
110. C₂×C₅₃	#2161	semidirect
123. C₅₃	C₂×#2161	semidirect
213. C₂	#20276510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
#20276510	2
C₂×#16810453	2
C₂²×M₃₂×C₅₃	2
C₂×#2161	53

Subgroups — C2×#20276510 #27537148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276510 #27537148