Subgroups — C₂²×#16811180 #28290798

Order 13568.

8238 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811180	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133217	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133209	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133196	C₂	1	4	yes	maximal
5	C₂²×C₄×Dic₁₀₆	C₂	1	4	yes	maximal, complemented
6	C₂²×#13133176	C₂	1	2	yes	maximal
7	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₁₀₆	C₂²	1	4	yes	—
9	#16811180	C₂²	1	256	yes	direct product factor
10	C₂×#13133217	C₂²	1	48	yes	complemented
11	C₂×#13133209	C₂²	1	48	yes	complemented
12	C₂×#13133196	C₂²	1	96	yes	—
13	C₂×C₄×Dic₁₀₆	C₂²	1	96	yes	complemented
14	C₂×#13133176	C₂²	1	48	yes	—
15	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
16	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
17	C₂²×#11719919	C₂²	1	8	yes	—
18	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
19	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
20	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
21	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	24	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₀₆	—	2	32	no	—
27	C₂²×Dic₁₀₆	C₂³	1	48	yes	—
28	C₂³×Dic₅₃	C₂³	1	8	yes	—
29	#13133217	C₂³	1	128	yes	complemented
30	#13133209	C₂³	1	128	yes	complemented
31	#13133196	C₂³	1	256	yes	—
32	C₄×Dic₁₀₆	C₂³	1	256	yes	complemented
33	#13133176	C₂³	1	128	yes	—
34	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
35	C₂×C₄⋊Dic₅₃	C₂³	1	48	yes	—
36	C₂×#11719919	C₂³	1	96	yes	—
37	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	48	yes	—
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	—	2	16	no	—
41	C₂³×C₄×C₅₃	Q₈	1	16	yes	complemented
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
50	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	64	yes	—
52	C₂⁴×C₅₃	—	2	8	no	—
53	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
54	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
55	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	8	no	—
57	C₂²×C₄×C₅₃	—	2	96	no	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
59	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
60	C₂²×C₄×C₅₃	C₂×Q₈	1	96	yes	complemented
61	C₂²×C₄×C₅₃	#41	1	16	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
68	C₂²×Dic₅₃	—	2	32	no	—
69	C₂²×Dic₅₃	C₂⁴	1	48	yes	—
70	C₂×Dic₁₀₆	—	2	192	no	—
71	C₂×Dic₁₀₆	C₂⁴	1	64	yes	—
72	C₄²×C₅₃	C₂⁴	1	32	yes	—
73	C₄²×C₅₃	C₂⁴	1	16	yes	—
74	C₄²×C₅₃	C₂⁴	1	16	yes	—
75	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
76	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
77	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
78	#11719919	C₂⁴	1	128	yes	—
79	C₄×Dic₅₃	C₂⁴	1	64	yes	—
80	C₂³×C₅₃	—	2	32	no	—
81	C₂³×C₅₃	—	2	24	no	—
82	C₂³×C₅₃	C₂²×Q₈	1	24	yes	—
83	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
84	C₂³×C₅₃	C₂×#41	1	8	yes	—
85	C₂³×C₅₃	C₂⁵	1	4	yes	—
86	C₂³×C₅₃	C₂⁵	1	3	yes	—
87	C₂×C₄×C₅₃	—	2	24	no	—
88	C₂×C₄×C₅₃	—	2	64	no	—
89	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
90	C₂×C₄×C₅₃	C₂²×Q₈	1	24	yes	—
91	C₂×C₄×C₅₃	C₂²×Q₈	1	64	yes	complemented
92	C₂×C₄×C₅₃	C₂×#41	1	48	yes	—
93	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
94	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
95	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
96	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
97	C₂×Dic₅₃	—	2	96	no	—
98	C₂×Dic₅₃	C₂⁵	1	32	yes	—
99	Dic₁₀₆	—	2	128	no	—
100	C₂²×#533	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	8	no	—
102	C₂²×C₅₃	—	2	48	no	—
103	C₂²×C₅₃	#533	1	16	yes	complemented
104	C₂²×C₅₃	C₂³×Q₈	1	8	yes	—
105	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
106	C₂²×C₅₃	C₂²×#41	1	12	yes	—
107	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
108	C₄×C₅₃	—	2	8	no	—
109	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
110	C₄×C₅₃	C₂³×Q₈	1	8	yes	—
111	C₄×C₅₃	C₂²×#41	1	16	yes	—
112	Dic₅₃	—	2	32	no	—
113	C₂×#533	—	53	48	no	—
114	C₂²×C₄⋊Q₈	—	53	2	no	—
115	C₂²×#125	—	53	2	no	—
116	C₂²×C₂²⋊Q₈	—	53	4	no	—
117	C₂²×C₄×Q₈	—	53	4	no	—
118	C₂²×#117	—	53	2	no	—
119	C₂³×C₄²	D₅₃	1	1	yes	characteristic
120	C₂×C₅₃	—	2	8	no	—
121	C₂×C₅₃	C₂×#533	1	12	yes	complemented
122	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
123	C₂×C₅₃	C₂³×#41	1	2	yes	—
124	C₂³×Q₈	—	53	4	no	—
125	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
126	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
127	#533	—	53	256	no	—
128	C₂×C₄⋊Q₈	—	53	48	no	—
129	C₂×#125	—	53	48	no	—
130	C₂×C₂²⋊Q₈	—	53	96	no	—
131	C₂×C₄×Q₈	—	53	96	no	—
132	C₂×#117	—	53	48	no	—
133	C₂²×C₄⋊C₄	—	53	8	no	—
134	C₂²×C₄⋊C₄	—	53	4	no	—
135	C₂²×C₄⋊C₄	—	53	4	no	—
136	C₂²×C₂²⋊C₄	—	53	4	no	—
137	C₂²×C₄²	—	53	4	no	—
138	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
139	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	C₂²×C₄²	C₂×D₅₃	1	24	yes	—
141	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
142	C₅₃	C₂²×#533	1	1	yes	characteristic, complemented, Sylow
143	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
144	C₂²×Q₈	—	53	48	no	—
145	C₂²×Q₈	—	106	32	no	—
146	C₂³×C₄	—	2	16	no	—
147	C₂³×C₄	—	53	8	no	—
148	C₂³×C₄	Dic₁₀₆	1	16	yes	complemented
149	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
151	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
152	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
153	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
154	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
155	C₄⋊Q₈	—	53	128	no	—
156	#125	—	53	128	no	—
157	C₂²⋊Q₈	—	53	256	no	—
158	C₄×Q₈	—	53	256	no	—
159	#117	—	53	128	no	—
160	C₂×C₄⋊C₄	—	53	48	no	—
161	C₂×C₄⋊C₄	—	53	96	no	—
162	C₂×C₄⋊C₄	—	53	48	no	—
163	C₂×C₂²⋊C₄	—	53	48	no	—
164	C₂×C₄²	—	53	48	no	—
165	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
166	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
167	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
168	C₂×C₄²	C₂²×D₅₃	1	64	yes	—
169	C₂⁴	—	2	8	no	—
170	C₂⁴	C₂×Dic₁₀₆	1	8	yes	—
171	C₂⁴	C₂³×D₅₃	1	6	yes	—
172	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
173	C₂×Q₈	—	53	64	no	—
174	C₂×Q₈	—	106	192	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	96	no	—
177	C₂²×C₄	—	53	48	no	—
178	C₂²×C₄	—	106	32	no	—
179	C₂²×C₄	C₂×Dic₁₀₆	1	8	yes	—
180	C₂²×C₄	C₂×Dic₁₀₆	1	8	yes	—
181	C₂²×C₄	C₂×Dic₁₀₆	1	96	yes	complemented
182	C₂²×C₄	#11719945	1	16	yes	—
183	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
184	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
185	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
186	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
187	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
188	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
189	C₄⋊C₄	—	53	64	no	—
190	C₄⋊C₄	—	53	128	no	—
191	C₄⋊C₄	—	53	64	no	—
192	C₂²⋊C₄	—	53	64	no	—
193	C₄²	—	53	64	no	—
194	C₄²	C₂³×D₅₃	1	32	yes	—
195	C₄²	C₂³×D₅₃	1	16	yes	—
196	C₄²	C₂³×D₅₃	1	16	yes	—
197	C₂³	—	2	32	no	—
198	C₂³	—	2	24	no	—
199	C₂³	C₂²×Dic₁₀₆	1	24	yes	—
200	C₂³	C₂²×Dic₁₀₆	1	4	yes	—
201	C₂³	C₂×#11719945	1	8	yes	—
202	C₂³	C₂⁴×D₅₃	1	4	yes	—
203	C₂³	C₂⁴×D₅₃	1	3	yes	—
204	Q₈	—	106	128	no	—
205	C₂×C₄	—	2	24	no	—
206	C₂×C₄	—	2	64	no	—
207	C₂×C₄	—	53	32	no	—
208	C₂×C₄	—	106	96	no	—
209	C₂×C₄	C₂²×Dic₁₀₆	1	24	yes	—
210	C₂×C₄	C₂²×Dic₁₀₆	1	24	yes	—
211	C₂×C₄	C₂²×Dic₁₀₆	1	64	yes	complemented
212	C₂×C₄	C₂×#11719945	1	48	yes	—
213	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
214	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
215	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
216	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
217	C₂²	—	2	48	no	—
218	C₂²	—	2	8	no	—
219	C₂²	#16811180	1	16	yes	direct product factor
220	C₂²	C₂³×Dic₁₀₆	1	8	yes	—
221	C₂²	C₂³×Dic₁₀₆	1	6	yes	—
222	C₂²	C₂²×#11719945	1	12	yes	—
223	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	106	32	no	—
226	C₄	C₂³×Dic₁₀₆	1	8	yes	—
227	C₄	C₂³×Dic₁₀₆	1	8	yes	—
228	C₄	C₂²×#11719945	1	16	yes	—
229	C₂	—	2	8	no	—
230	C₂	C₂×#16811180	1	12	yes	direct product factor
231	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
232	C₂	C₂³×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#533
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×#16811180	C₂	direct (N × H)
2. C₂²×#13133217	C₂	semidirect
3. C₂²×#13133209	C₂	semidirect
5. C₂²×C₄×Dic₁₀₆	C₂	semidirect
9. #16811180	C₂²	direct (N × H)
10. C₂×#13133217	C₂²	semidirect
11. C₂×#13133209	C₂²	semidirect
13. C₂×C₄×Dic₁₀₆	C₂²	semidirect
29. #13133217	C₂³	semidirect
30. #13133209	C₂³	semidirect
32. C₄×Dic₁₀₆	C₂³	semidirect
41. C₂³×C₄×C₅₃	Q₈	semidirect
60. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
91. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
103. C₂²×C₅₃	#533	semidirect
121. C₂×C₅₃	C₂×#533	semidirect
142. C₅₃	C₂²×#533	semidirect
148. C₂³×C₄	Dic₁₀₆	semidirect
181. C₂²×C₄	C₂×Dic₁₀₆	semidirect
211. C₂×C₄	C₂²×Dic₁₀₆	semidirect
219. C₂²	#16811180	direct (N × H)
230. C₂	C₂×#16811180	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811180	2
C₂²×#13133217	2
C₂²×#13133209	2
C₂²×#13133196	2
C₂²×C₄×Dic₁₀₆	2
C₂²×#13133176	2
C₂³×C₄²×C₅₃	2
C₂²×#533	53

Subgroups — C22×#16811180 #28290798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811180 #28290798