Subgroups — #27402998

Order 13568.

876 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×Q₈×#88322	C₂	1	2	yes	maximal
2	#20275434	C₂	1	6	yes	maximal
3	#20274832	C₂	1	3	yes	maximal
4	C₄×C₄⋊#88322	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88322	—	2	8	no	—
7	C₂×C₄⋊#88322	C₂²	1	6	yes	—
8	C₂×C₄⋊#88322	C₂²	1	6	yes	—
9	C₂×C₄×#88322	C₂²	1	6	yes	—
10	#16810378	C₂²	1	1	yes	characteristic
11	#16810378	C₂²	1	3	yes	—
12	#16810378	C₂²	1	6	yes	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
17	C₄×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
19	C₄³×C₅₃	C₂²	1	3	yes	—
20	C₂²×#88322	C₂³	1	6	yes	—
21	C₂²×#88322	C₂³	1	2	yes	—
22	C₄⋊#88322	—	2	12	no	—
23	C₄⋊#88322	—	2	12	no	—
24	C₄×#88322	—	2	12	no	—
25	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₅₃	—	2	4	no	—
27	C₄×Q₈×C₅₃	C₈	1	8	yes	complemented
28	C₄×Q₈×C₅₃	D₄	1	4	yes	—
29	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
40	#95×C₅₃	C₂×C₄	1	6	yes	—
41	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
42	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
43	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	—	2	6	no	—
52	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
53	C₄⋊C₄×C₅₃	C₂×C₈	1	12	yes	—
54	C₄²×C₅₃	—	2	6	no	—
55	C₄²×C₅₃	—	2	6	no	—
56	C₄²×C₅₃	—	2	6	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	4	3	no	—
59	C₄²×C₅₃	C₂×C₈	1	12	yes	—
60	C₄²×C₅₃	C₂×D₄	1	6	yes	—
61	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	6	no	—
65	C₂×#88322	—	2	6	no	—
66	C₂×#88322	—	2	6	no	—
67	C₂×#88322	#41	1	4	yes	—
68	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	Q₈×C₅₃	C₄⋊C₈	1	16	yes	complemented
70	C₂×C₄×C₅₃	—	2	3	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	3	no	—
75	C₂×C₄×C₅₃	—	2	3	no	—
76	C₂×C₄×C₅₃	—	2	3	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	3	no	—
79	C₂×C₄×C₅₃	—	2	3	no	—
80	C₂×C₄×C₅₃	—	2	3	no	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
91	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
92	#88322	—	2	4	no	—
93	#88322	—	4	6	no	—
94	#22734	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
97	C₂²×C₅₃	#519	1	1	yes	characteristic
98	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
101	C₂²×C₅₃	#568	1	1	yes	characteristic
102	C₄×C₅₃	—	2	6	no	—
103	C₄×C₅₃	—	2	6	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	4	3	no	—
107	C₄×C₅₃	C₂×C₄⋊C₈	1	12	yes	—
108	C₄×C₅₃	#548	1	2	yes	—
109	C₄×C₅₃	#557	1	2	yes	—
110	C₂×C₈×Q₈	—	53	2	no	—
111	C₄²×Q₈	D₅₃	1	1	yes	characteristic
112	#1939	—	53	6	no	—
113	#1831	—	53	3	no	—
114	C₄×C₄⋊C₈	—	53	3	no	—
115	C₂×C₅₃	#2351	1	1	yes	characteristic
116	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₅₃	#2906	1	1	yes	characteristic
118	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
119	C₂×C₅₃	#2971	1	1	yes	characteristic
120	C₂×C₅₃	#2982	1	1	yes	characteristic
121	C₂×C₅₃	#2984	1	1	yes	characteristic
122	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
123	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
125	C₈×Q₈	—	106	8	no	—
126	C₂×C₄⋊C₈	—	53	6	no	—
127	C₂×C₄⋊C₈	—	53	6	no	—
128	C₂×C₄×C₈	—	53	6	no	—
129	C₄×C₄⋊C₄	Dic₅₃	1	3	yes	—
130	C₄×C₄⋊C₄	Dic₅₃	1	3	yes	—
131	C₄×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
132	C₄³	C₂×D₅₃	1	3	yes	—
133	#336	—	53	3	no	—
134	#336	—	53	6	no	—
135	#336	—	53	1	no	—
136	C₅₃	#22734	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	—	53	6	no	—
140	C₄×Q₈	—	2	4	no	—
141	C₄×Q₈	#88322	1	8	yes	complemented
142	C₄×Q₈	Dic₁₀₆	1	4	yes	—
143	C₄×Q₈	D₂₁₂	1	4	yes	—
144	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
145	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
147	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
148	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
149	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
150	C₂×C₄²	C₂×Dic₅₃	1	3	yes	—
151	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
153	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
154	C₄⋊C₈	—	106	12	no	—
155	C₄⋊C₈	—	106	12	no	—
156	C₄×C₈	—	106	12	no	—
157	#95	C₂×Dic₅₃	1	6	yes	—
158	C₂×Q₈	C₂×#88322	1	4	yes	—
159	C₂×Q₈	#11719917	1	4	yes	—
160	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
161	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
162	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
163	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
164	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂²×Dic₅₃	1	3	yes	—
167	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	6	no	—
172	C₂×C₈	—	106	6	no	—
173	C₂×C₈	—	106	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	C₂×#88322	1	12	yes	—
176	C₄⋊C₄	C₄⋊Dic₅₃	1	12	yes	—
177	C₄²	—	2	6	no	—
178	C₄²	—	2	6	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	4	3	no	—
182	C₄²	C₂×#88322	1	12	yes	—
183	C₄²	C₂×Dic₁₀₆	1	6	yes	—
184	C₄²	C₂×D₂₁₂	1	6	yes	—
185	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
186	Q₈	C₄⋊#88322	1	16	yes	complemented
187	C₂×C₄	—	2	3	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	3	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
199	C₂×C₄	C₂×C₄⋊Dic₅₃	1	6	yes	—
200	C₂×C₄	C₂²×#88322	1	2	yes	—
201	C₂×C₄	C₂²×#88322	1	6	yes	—
202	C₂×C₄	C₂²×#88322	1	6	yes	—
203	C₂×C₄	C₂×#11719917	1	6	yes	—
204	C₂×C₄	#13133224	1	2	yes	—
205	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
207	C₂×C₄	#41×D₅₃	1	2	yes	—
208	C₂×C₄	#13133265	1	2	yes	—
209	C₈	—	106	4	no	—
210	C₈	—	212	6	no	—
211	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
212	C₂²	#41×Dic₅₃	1	1	yes	characteristic
213	C₂²	#16811101	1	1	yes	characteristic
214	C₂²	C₂³×#88322	1	1	yes	characteristic
215	C₂²	C₂²×#11719917	1	1	yes	characteristic
216	C₂²	C₂×#13133224	1	1	yes	characteristic
217	C₂²	#16811390	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	6	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊#88322	1	12	yes	—
224	C₄	#16811285	1	2	yes	—
225	C₄	#16811349	1	2	yes	—
226	C₂	#20277636	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
228	C₂	#20283283	1	1	yes	characteristic
229	C₂	#41×#88322	1	1	yes	characteristic
230	C₂	#20283735	1	1	yes	characteristic
231	C₂	#20283796	1	1	yes	characteristic
232	C₂	#20283804	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#22734
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
27. C₄×Q₈×C₅₃	C₈	semidirect
69. Q₈×C₅₃	C₄⋊C₈	semidirect
136. C₅₃	#22734	semidirect
141. C₄×Q₈	#88322	semidirect
186. Q₈	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
#20275434	2
#20274832	2
C₄×C₄⋊#88322	2
C₄²×Q₈×C₅₃	2
#22734	53