Subgroups — C₂³×#13133133 #28247573

Order 13568.

7134 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133133	C₂	1	28	yes	maximal, direct product factor
4	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₅₃	C₄	1	2	yes	—
6	C₂³×D₂₁₂	—	2	1	no	—
7	C₂³×D₂₁₂	—	2	7	no	—
8	C₂³×D₂₁₂	—	2	1	no	—
9	C₂³×D₂₁₂	—	2	7	no	—
10	C₂³×D₂₁₂	C₄	1	14	yes	—
11	C₂³×D₂₁₂	C₂²	1	14	yes	—
12	C₂²×#11719917	—	2	7	no	—
13	C₂²×#11719917	C₂²	1	14	yes	complemented
14	C₂³×#88322	—	2	1	no	—
15	C₂×#13133133	C₂²	1	112	yes	direct product factor
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₅₃	—	2	7	no	—
18	C₂²×M₁₆×C₅₃	C₂²	1	14	yes	complemented
19	C₂³×C₈×C₅₃	—	2	1	no	—
20	C₂⁴×D₅₃	—	2	14	no	—
21	C₂⁴×D₅₃	—	2	2	no	—
22	C₂⁴×D₅₃	—	4	16	no	—
23	C₂⁴×D₅₃	C₂×C₄	1	28	yes	—
24	C₂²×D₂₁₂	—	2	84	no	—
25	C₂²×D₂₁₂	—	2	14	no	—
26	C₂²×D₂₁₂	—	2	14	no	—
27	C₂²×D₂₁₂	—	2	14	no	—
28	C₂²×D₂₁₂	—	2	42	no	—
29	C₂²×D₂₁₂	—	2	14	no	—
30	C₂²×D₂₁₂	—	2	42	no	—
31	C₂²×D₂₁₂	C₂×C₄	1	84	yes	—
32	C₂²×D₂₁₂	C₂³	1	28	yes	—
33	C₂×#11719917	—	2	42	no	—
34	C₂×#11719917	C₂³	1	28	yes	complemented
35	C₂²×#88322	—	2	14	no	—
36	#13133133	C₂³	1	64	yes	direct product factor
37	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
39	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	D₄	1	7	yes	—
41	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	D₄	1	7	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
44	C₂×M₁₆×C₅₃	—	2	42	no	—
45	C₂×M₁₆×C₅₃	C₂³	1	28	yes	complemented
46	C₂²×C₈×C₅₃	—	2	14	no	—
47	C₂⁴×C₅₃	—	2	8	no	—
48	C₂⁴×C₅₃	C₂²⋊C₄	1	7	yes	—
49	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
51	C₂³×D₅₃	—	2	84	no	—
52	C₂³×D₅₃	—	2	28	no	—
53	C₂³×D₅₃	—	4	112	no	—
54	C₂³×D₅₃	—	4	16	no	—
55	C₂³×D₅₃	—	4	112	no	—
56	C₂³×D₅₃	C₂²×C₄	1	56	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	42	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	7	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	7	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	21	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
66	C₂×D₂₁₂	—	2	168	no	—
67	C₂×D₂₁₂	—	2	84	no	—
68	C₂×D₂₁₂	—	2	84	no	—
69	C₂×D₂₁₂	—	2	28	no	—
70	C₂×D₂₁₂	—	2	28	no	—
71	C₂×D₂₁₂	—	2	28	no	—
72	C₂×D₂₁₂	—	2	28	no	—
73	C₂×D₂₁₂	C₂²×C₄	1	56	yes	—
74	C₂×D₂₁₂	C₂⁴	1	8	yes	—
75	M₁₆×C₅₃	—	2	28	no	—
76	M₁₆×C₅₃	C₂⁴	1	8	yes	complemented
77	C₂×C₈×C₅₃	—	2	28	no	—
78	#11719917	—	2	28	no	—
79	#11719917	C₂⁴	1	8	yes	complemented
80	C₂×#88322	—	2	28	no	—
81	C₂³×C₅₃	—	2	28	no	—
82	C₂³×C₅₃	—	2	28	no	—
83	C₂³×C₅₃	#100	1	8	yes	complemented
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	21	yes	—
85	C₂³×C₅₃	C₂×C₂²⋊C₄	1	7	yes	—
86	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
87	C₂²×D₅₃	—	2	56	no	—
88	C₂²×D₅₃	—	2	56	no	—
89	C₂²×D₅₃	—	4	336	no	—
90	C₂²×D₅₃	—	4	112	no	—
91	C₂²×D₅₃	—	4	112	no	—
92	C₂²×D₅₃	C₂³×C₄	1	16	yes	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	42	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	21	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	21	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
97	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
98	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
99	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
100	C₂×C₄×C₅₃	C₂²×D₄	1	7	yes	—
101	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
102	D₂₁₂	—	2	56	no	—
103	D₂₁₂	—	2	56	no	—
104	D₂₁₂	—	2	8	no	—
105	D₂₁₂	—	2	8	no	—
106	C₈×C₅₃	—	2	8	no	—
107	#88322	—	2	8	no	—
108	C₂³×#100	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	42	no	—
110	C₂²×C₅₃	—	2	14	no	—
111	C₂²×C₅₃	C₂×#100	1	28	yes	complemented
112	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	7	yes	—
113	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	7	yes	—
114	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
115	C₂×D₅₃	—	2	16	no	—
116	C₂×D₅₃	—	4	112	no	—
117	C₂×D₅₃	—	4	112	no	—
118	C₂×D₅₃	—	4	16	no	—
119	C₄×C₅₃	C₂²×C₂²⋊C₄	1	7	yes	—
120	C₄×C₅₃	C₂²×C₂²⋊C₄	1	7	yes	—
121	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
122	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
123	C₂⁴×D₄	—	53	1	no	—
124	C₂³×M₁₆	—	53	1	no	—
125	C₂³×M₁₆	D₅₃	1	1	yes	characteristic, complemented
126	C₂²×#100	—	53	28	no	—
127	C₂×C₅₃	—	2	7	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	C₂²×#100	1	14	yes	complemented
130	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
131	D₅₃	—	4	16	no	—
132	C₂⁶	—	53	2	no	—
133	C₂³×D₄	—	53	14	no	—
134	C₂³×D₄	—	53	14	no	—
135	C₂³×D₄	—	106	7	no	—
136	C₂³×D₄	—	106	1	no	—
137	C₂³×D₄	—	106	7	no	—
138	C₂³×D₄	—	106	1	no	—
139	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
140	C₂²×M₁₆	—	2	7	no	—
141	C₂²×M₁₆	—	53	14	no	—
142	C₂²×M₁₆	—	106	7	no	—
143	C₂²×M₁₆	C₂×D₅₃	1	14	yes	complemented
144	C₂³×C₈	—	2	1	no	—
145	C₂³×C₈	—	106	1	no	—
146	C₂×#100	—	53	112	no	—
147	C₅₃	C₂³×#100	1	1	yes	characteristic, complemented, Sylow
148	C₂⁵	—	53	28	no	—
149	C₂⁵	—	106	14	no	—
150	C₂⁵	—	106	2	no	—
151	C₂⁵	—	212	16	no	—
152	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
153	C₂²×D₄	—	53	84	no	—
154	C₂²×D₄	—	53	28	no	—
155	C₂²×D₄	—	106	14	no	—
156	C₂²×D₄	—	106	42	no	—
157	C₂²×D₄	—	106	14	no	—
158	C₂²×D₄	—	106	42	no	—
159	C₂²×D₄	—	106	84	no	—
160	C₂²×D₄	—	106	14	no	—
161	C₂²×D₄	—	106	14	no	—
162	C₂³×C₄	C₄×D₅₃	1	7	yes	—
163	C₂³×C₄	D₂₁₂	1	7	yes	—
164	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
165	C₂³×C₄	C₂²⋊D₅₃	1	7	yes	—
166	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
168	C₂×M₁₆	—	2	42	no	—
169	C₂×M₁₆	—	53	28	no	—
170	C₂×M₁₆	—	106	42	no	—
171	C₂×M₁₆	C₂²×D₅₃	1	28	yes	complemented
172	C₂²×C₈	—	2	14	no	—
173	C₂²×C₈	—	106	14	no	—
174	#100	—	53	64	no	—
175	C₂⁴	—	2	8	no	—
176	C₂⁴	—	53	56	no	—
177	C₂⁴	—	106	84	no	—
178	C₂⁴	—	106	28	no	—
179	C₂⁴	—	212	112	no	—
180	C₂⁴	—	212	112	no	—
181	C₂⁴	—	212	16	no	—
182	C₂⁴	#11719921	1	7	yes	—
183	C₂⁴	#11719921	1	1	yes	characteristic
184	C₂⁴	C₂×C₄×D₅₃	1	7	yes	—
185	C₂×D₄	—	53	56	no	—
186	C₂×D₄	—	53	8	no	—
187	C₂×D₄	—	106	168	no	—
188	C₂×D₄	—	106	84	no	—
189	C₂×D₄	—	106	84	no	—
190	C₂×D₄	—	106	28	no	—
191	C₂×D₄	—	106	28	no	—
192	C₂×D₄	—	106	28	no	—
193	C₂×D₄	—	106	28	no	—
194	C₂²×C₄	#11719921	1	42	yes	—
195	C₂²×C₄	#11719921	1	7	yes	—
196	C₂²×C₄	#11719921	1	7	yes	—
197	C₂²×C₄	C₂×C₄×D₅₃	1	21	yes	—
198	C₂²×C₄	C₂×D₂₁₂	1	7	yes	—
199	C₂²×C₄	C₂×D₂₁₂	1	21	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₅₃	1	7	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
202	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
203	M₁₆	—	2	28	no	—
204	M₁₆	—	53	8	no	—
205	M₁₆	—	106	28	no	—
206	M₁₆	C₂³×D₅₃	1	8	yes	complemented
207	C₂×C₈	—	2	28	no	—
208	C₂×C₈	—	106	28	no	—
209	C₂³	—	2	28	no	—
210	C₂³	—	2	28	no	—
211	C₂³	—	53	16	no	—
212	C₂³	—	106	56	no	—
213	C₂³	—	106	56	no	—
214	C₂³	—	212	112	no	—
215	C₂³	—	212	336	no	—
216	C₂³	—	212	112	no	—
217	C₂³	#13133133	1	8	yes	direct product factor
218	C₂³	C₂×#11719921	1	21	yes	—
219	C₂³	C₂×#11719921	1	7	yes	—
220	C₂³	C₂²×C₄×D₅₃	1	7	yes	—
221	D₄	—	106	56	no	—
222	D₄	—	106	56	no	—
223	D₄	—	106	8	no	—
224	D₄	—	106	8	no	—
225	C₂×C₄	C₂×#11719921	1	42	yes	—
226	C₂×C₄	C₂×#11719921	1	21	yes	—
227	C₂×C₄	C₂×#11719921	1	21	yes	—
228	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
229	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
230	C₂×C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
231	C₂×C₄	C₂²×D₂₁₂	1	7	yes	—
232	C₂×C₄	C₂²×D₂₁₂	1	7	yes	—
233	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	106	8	no	—
236	C₂²	—	2	42	no	—
237	C₂²	—	2	14	no	—
238	C₂²	—	106	16	no	—
239	C₂²	—	212	16	no	—
240	C₂²	—	212	112	no	—
241	C₂²	—	212	112	no	—
242	C₂²	C₂×#13133133	1	28	yes	direct product factor
243	C₂²	C₂²×#11719921	1	7	yes	—
244	C₂²	C₂²×#11719921	1	7	yes	—
245	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
246	C₄	C₂²×#11719921	1	7	yes	—
247	C₄	C₂²×#11719921	1	7	yes	—
248	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
250	C₂	—	2	7	no	—
251	C₂	—	2	1	no	—
252	C₂	—	212	16	no	—
253	C₂	C₂²×#13133133	1	14	yes	direct product factor
254	C₂	C₂³×#11719921	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#11719921
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²⋊C₄
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂³×#100
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#11719917	C₂	semidirect
3. C₂²×#13133133	C₂	direct (N × H)
4. C₂³×M₁₆×C₅₃	C₂	semidirect
13. C₂²×#11719917	C₂²	semidirect
15. C₂×#13133133	C₂²	direct (N × H)
18. C₂²×M₁₆×C₅₃	C₂²	semidirect
34. C₂×#11719917	C₂³	semidirect
36. #13133133	C₂³	direct (N × H)
45. C₂×M₁₆×C₅₃	C₂³	semidirect
76. M₁₆×C₅₃	C₂⁴	semidirect
79. #11719917	C₂⁴	semidirect
83. C₂³×C₅₃	#100	semidirect
111. C₂²×C₅₃	C₂×#100	semidirect
125. C₂³×M₁₆	D₅₃	semidirect
129. C₂×C₅₃	C₂²×#100	semidirect
143. C₂²×M₁₆	C₂×D₅₃	semidirect
147. C₅₃	C₂³×#100	semidirect
171. C₂×M₁₆	C₂²×D₅₃	semidirect
206. M₁₆	C₂³×D₅₃	semidirect
217. C₂³	#13133133	direct (N × H)
242. C₂²	C₂×#13133133	direct (N × H)
253. C₂	C₂²×#13133133	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₁₂	2
C₂³×#11719917	2
C₂²×#13133133	2
C₂³×M₁₆×C₅₃	2
C₂³×#100	53

Subgroups — C23×#13133133 #28247573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133133 #28247573