Subgroups — C₂×#20276474 #27537075

Order 13568.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20276474	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11719908	C₂	1	4	yes	maximal, complemented
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	4	no	—
7	C₂×#13133157	C₂²	1	4	yes	—
8	C₂²×#11719908	—	2	4	no	—
9	C₂²×#11719908	C₂²	1	2	yes	—
10	C₂²×#11719908	C₂²	1	4	yes	—
11	C₂²×#11719908	C₂²	1	2	yes	—
12	C₄⋊#11719908	C₂²	1	16	yes	complemented
13	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	4	yes	—
17	C₅₃×#405	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
20	#13133157	—	2	8	no	—
21	C₂×#11719908	—	2	4	no	—
22	C₂×#11719908	—	2	8	no	—
23	C₂×#11719908	—	2	4	no	—
24	C₂×#11719908	C₂³	1	4	yes	—
25	C₂×#11719908	C₂³	1	4	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	—	2	4	no	—
28	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	4	yes	—
33	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	#117×C₅₃	C₈	1	4	yes	—
37	#117×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
39	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
41	#97×C₅₃	C₂×C₄	1	8	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₃	C₂³	1	4	yes	—
44	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
52	M₁₆×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
65	C₂²⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
66	C₄²×C₅₃	C₂×C₈	1	4	yes	—
67	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
68	#11719908	—	2	4	no	—
69	#11719908	—	2	4	no	—
70	C₂³×C₅₃	—	2	4	no	—
71	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
72	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
73	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
76	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
80	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
84	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₅₃	—	2	4	no	—
86	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
88	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×#2151	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	—	2	4	no	—
92	C₂²×C₅₃	—	2	8	no	—
93	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₅₃	#504	1	4	yes	—
96	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₅₃	—	2	8	no	—
98	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
99	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
100	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂²×M₃₂	—	53	1	no	—
103	C₂³×C₁₆	—	53	1	no	—
104	C₂×#405	D₅₃	1	1	yes	characteristic
105	#2151	—	53	8	no	—
106	C₂×C₄⋊C₁₆	—	53	4	no	—
107	C₂×C₅₃	—	2	4	no	—
108	C₂×C₅₃	#2151	1	4	yes	complemented
109	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₅₃	C₂×#504	1	2	yes	—
111	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
112	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
113	C₂×#117	Dic₅₃	1	1	yes	characteristic
114	C₂×M₃₂	—	53	4	no	—
115	C₂×M₃₂	—	106	4	no	—
116	C₂²×C₁₆	—	53	4	no	—
117	C₂²×C₁₆	—	53	2	no	—
118	C₂²×C₁₆	—	53	2	no	—
119	C₂²×C₁₆	—	106	4	no	—
120	#405	Dic₅₃	1	4	yes	—
121	#405	C₂×D₅₃	1	4	yes	—
122	C₂×#97	Dic₅₃	1	2	yes	—
123	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
124	C₄⋊C₁₆	—	53	16	no	—
125	C₅₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
129	C₂²×C₈	Dic₁₀₆	1	4	yes	—
130	C₂²×C₈	D₂₁₂	1	4	yes	—
131	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
134	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	#117	#88322	1	4	yes	—
137	#117	C₂×Dic₅₃	1	4	yes	—
138	C₂×C₄⋊C₄	#88322	1	2	yes	—
139	C₂×C₂²⋊C₄	#88322	1	2	yes	—
140	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
141	M₃₂	—	106	8	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₂×C₁₆	—	53	4	no	—
144	C₂×C₁₆	—	106	4	no	—
145	C₂×C₁₆	—	106	8	no	—
146	C₂×C₁₆	—	106	4	no	—
147	#97	C₂×Dic₅₃	1	8	yes	—
148	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
149	C₄×C₈	C₂²×D₅₃	1	4	yes	—
150	C₂⁴	C₂×#88322	1	1	yes	characteristic
151	C₂²×C₄	C₂×#88322	1	4	yes	—
152	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
153	C₂²×C₄	C₂×#88322	1	2	yes	—
154	C₂²×C₄	#11719917	1	4	yes	—
155	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
156	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
158	M₁₆	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
161	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
162	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
163	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
164	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
165	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
166	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
167	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
169	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
170	C₄⋊C₄	C₂×#88322	1	8	yes	—
171	C₂²⋊C₄	C₂×#88322	1	8	yes	—
172	C₄²	C₂×#88322	1	4	yes	—
173	C₄²	C₂²×Dic₅₃	1	4	yes	—
174	C₁₆	—	106	4	no	—
175	C₁₆	—	106	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄⋊#88322	1	4	yes	—
178	C₂³	C₂²×#88322	1	2	yes	—
179	C₂³	C₂²×#88322	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	C₄⋊#88322	1	8	yes	—
182	C₂×C₄	C₄⋊#88322	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
185	C₂×C₄	C₂²×#88322	1	8	yes	—
186	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
187	C₂×C₄	C₂²×#88322	1	2	yes	—
188	C₂×C₄	C₂×#11719917	1	2	yes	—
189	C₂×C₄	C₂×#11719917	1	4	yes	—
190	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
193	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
194	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
196	C₂²	—	2	4	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂×C₄⋊#88322	1	4	yes	—
199	C₂²	C₂×C₄⋊#88322	1	2	yes	—
200	C₂²	#16811040	1	4	yes	—
201	C₂²	C₂³×#88322	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₄⋊#88322	1	2	yes	—
204	C₄	C₂×C₄⋊#88322	1	4	yes	—
205	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
206	C₄	C₂²×#11719917	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#20276474	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
210	C₂	C₂×#16811040	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #20276474	C₂	direct (N × H)
4. C₂×C₄⋊#11719908	C₂	semidirect
12. C₄⋊#11719908	C₂²	semidirect
108. C₂×C₅₃	#2151	semidirect
125. C₅₃	C₂×#2151	semidirect
208. C₂	#20276474	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×#13133157	2
C₂³×#11719908	2
#20276474	2
C₂×C₄⋊#11719908	2
C₂×C₅₃×#405	2
C₂×#2151	53

Subgroups — C2×#20276474 #27537075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276474 #27537075