Subgroups — C₂²×#16810650 #28247708

Order 13568.

3006 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	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810650	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄⋊#88322	C₂	1	4	yes	maximal, complemented
5	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	8	no	—
7	C₂²×#11719917	C₂²	1	12	yes	—
8	C₂³×#88322	—	2	8	no	—
9	C₂³×#88322	C₂²	1	2	yes	—
10	C₂³×#88322	C₂²	1	2	yes	—
11	C₂³×#88322	C₂²	1	12	yes	—
12	#16810650	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊#88322	C₂²	1	48	yes	complemented
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₃	C₄	1	12	yes	—
16	C₂×#117×C₅₃	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
18	C₂²×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
20	C₂×#11719917	—	2	48	no	—
21	C₂×#11719917	C₂³	1	16	yes	—
22	C₂²×#88322	—	2	8	no	—
23	C₂²×#88322	—	2	48	no	—
24	C₂²×#88322	—	2	8	no	—
25	C₂²×#88322	C₂³	1	12	yes	—
26	C₂²×#88322	C₂³	1	16	yes	—
27	C₂²×#88322	C₂³	1	12	yes	—
28	C₄⋊#88322	C₂³	1	64	yes	complemented
29	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
33	C₂³×C₄×C₅₃	D₄	1	8	yes	—
34	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	#117×C₅₃	C₂×C₄	1	48	yes	—
38	#117×C₅₃	C₂³	1	16	yes	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
40	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	24	yes	—
41	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
43	C₂⁴×C₅₃	—	2	8	no	—
44	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
45	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
46	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	16	no	—
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	48	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
56	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
60	C₄⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
61	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
63	C₄²×C₅₃	C₂⁴	1	8	yes	—
64	#11719917	—	2	32	no	—
65	C₂×#88322	—	2	24	no	—
66	C₂×#88322	—	2	32	no	—
67	C₂×#88322	—	2	24	no	—
68	C₂×#88322	C₂⁴	1	8	yes	—
69	C₂×#88322	C₂⁴	1	8	yes	—
70	C₂³×C₅₃	—	2	24	no	—
71	C₂³×C₅₃	—	2	32	no	—
72	C₂³×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
73	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
74	C₂³×C₅₃	#131	1	8	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
76	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
77	C₂×C₄×C₅₃	—	2	48	no	—
78	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
79	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
85	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
87	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
89	#88322	—	2	8	no	—
90	#88322	—	2	8	no	—
91	C₂²×#424	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	48	no	—
93	C₂²×C₅₃	—	2	8	no	—
94	C₂²×C₅₃	#424	1	16	yes	complemented
95	C₂²×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
96	C₂²×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
97	C₂²×C₅₃	C₂×#131	1	12	yes	—
98	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
99	C₄×C₅₃	—	2	16	no	—
100	C₄×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
101	C₄×C₅₃	C₂²×C₄⋊C₄	1	8	yes	—
102	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
103	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
104	C₂³×M₁₆	—	53	1	no	—
105	C₂⁴×C₈	—	53	1	no	—
106	C₂²×#117	D₅₃	1	1	yes	characteristic
107	C₂×#424	—	53	24	no	—
108	C₂²×C₄⋊C₈	—	53	4	no	—
109	C₂×C₅₃	—	2	8	no	—
110	C₂×C₅₃	C₂×#424	1	12	yes	complemented
111	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₅₃	C₂²×#131	1	2	yes	—
113	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
114	C₂²×M₁₆	—	53	12	no	—
115	C₂²×M₁₆	—	106	8	no	—
116	C₂³×C₈	—	53	12	no	—
117	C₂³×C₈	—	53	2	no	—
118	C₂³×C₈	—	53	2	no	—
119	C₂³×C₈	—	106	8	no	—
120	C₂×#117	Dic₅₃	1	12	yes	—
121	C₂×#117	C₂×D₅₃	1	12	yes	—
122	C₂²×C₄⋊C₄	Dic₅₃	1	2	yes	—
123	C₂²×C₂²⋊C₄	Dic₅₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
125	#424	—	53	64	no	—
126	C₂×C₄⋊C₈	—	53	48	no	—
127	C₅₃	C₂²×#424	1	1	yes	characteristic, complemented, Sylow
128	C₂⁵	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂³×C₄	Dic₁₀₆	1	8	yes	—
130	C₂³×C₄	D₂₁₂	1	8	yes	—
131	C₂³×C₄	C₂×Dic₅₃	1	4	yes	—
132	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
134	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
135	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
136	C₂×M₁₆	—	53	16	no	—
137	C₂×M₁₆	—	106	48	no	—
138	C₂²×C₈	—	53	12	no	—
139	C₂²×C₈	—	53	16	no	—
140	C₂²×C₈	—	53	12	no	—
141	C₂²×C₈	—	106	8	no	—
142	C₂²×C₈	—	106	48	no	—
143	C₂²×C₈	—	106	8	no	—
144	#117	C₂×Dic₅₃	1	48	yes	—
145	#117	C₂²×D₅₃	1	16	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₅₃	1	24	yes	—
147	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	24	yes	—
148	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
149	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
150	C₄⋊C₈	—	53	64	no	—
151	C₂⁴	—	2	8	no	—
152	C₂⁴	C₄⋊Dic₅₃	1	8	yes	—
153	C₂⁴	C₂²×Dic₅₃	1	6	yes	—
154	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄	—	2	16	no	—
156	C₂²×C₄	C₄⋊Dic₅₃	1	48	yes	—
157	C₂²×C₄	C₄⋊Dic₅₃	1	8	yes	—
158	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
159	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
160	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
161	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
162	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
163	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
164	C₂²×C₄	C₂²×Dic₅₃	1	6	yes	—
165	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
166	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
167	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
168	M₁₆	—	106	32	no	—
169	C₂×C₈	—	53	8	no	—
170	C₂×C₈	—	53	8	no	—
171	C₂×C₈	—	106	24	no	—
172	C₂×C₈	—	106	32	no	—
173	C₂×C₈	—	106	24	no	—
174	C₄⋊C₄	C₂²×Dic₅₃	1	32	yes	—
175	C₂²⋊C₄	C₂²×Dic₅₃	1	32	yes	—
176	C₄²	C₂²×Dic₅₃	1	24	yes	—
177	C₄²	C₂³×D₅₃	1	8	yes	—
178	C₂³	—	2	24	no	—
179	C₂³	—	2	32	no	—
180	C₂³	C₂×C₄⋊Dic₅₃	1	24	yes	—
181	C₂³	C₂×C₄⋊Dic₅₃	1	4	yes	—
182	C₂³	#13133224	1	8	yes	—
183	C₂³	C₂³×Dic₅₃	1	4	yes	—
184	C₂³	C₂³×Dic₅₃	1	3	yes	—
185	C₂×C₄	—	2	48	no	—
186	C₂×C₄	C₂×C₄⋊Dic₅₃	1	48	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
188	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
189	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
190	C₂×C₄	C₂³×Dic₅₃	1	3	yes	—
191	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
192	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
193	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
194	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
195	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
196	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
197	C₈	—	106	8	no	—
198	C₈	—	106	8	no	—
199	C₂²	—	2	48	no	—
200	C₂²	—	2	8	no	—
201	C₂²	#16810650	1	16	yes	direct product factor
202	C₂²	C₂²×C₄⋊Dic₅₃	1	8	yes	—
203	C₂²	C₂²×C₄⋊Dic₅₃	1	6	yes	—
204	C₂²	C₂×#13133224	1	12	yes	—
205	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
206	C₄	—	2	16	no	—
207	C₄	C₂²×C₄⋊Dic₅₃	1	6	yes	—
208	C₄	C₂²×C₄⋊Dic₅₃	1	8	yes	—
209	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#16810650	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
214	C₂	C₂²×#13133224	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#117×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×#117	D₅₃
hypocenter	C₅₃	C₂²×#424
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#16810650	C₂	direct (N × H)
4. C₂²×C₄⋊#88322	C₂	semidirect
12. #16810650	C₂²	direct (N × H)
13. C₂×C₄⋊#88322	C₂²	semidirect
28. C₄⋊#88322	C₂³	semidirect
94. C₂²×C₅₃	#424	semidirect
110. C₂×C₅₃	C₂×#424	semidirect
127. C₅₃	C₂²×#424	semidirect
201. C₂²	#16810650	direct (N × H)
212. C₂	C₂×#16810650	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×#11719917	2
C₂⁴×#88322	2
C₂×#16810650	2
C₂²×C₄⋊#88322	2
C₂²×#117×C₅₃	2
C₂²×#424	53

Subgroups — C22×#16810650 #28247708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810650 #28247708