Subgroups — #27299314

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810426	C₂	1	2	yes	maximal
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	C₄×Dic₂₁₂	—	2	2	no	—
10	C₄×Dic₂₁₂	C₄	1	4	yes	complemented
11	C₂×#13133151	C₂²	1	1	yes	characteristic
12	C₂×#13133142	—	2	1	no	—
13	#16810426	C₄	1	8	yes	complemented
14	#16810399	—	2	2	no	—
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₅₃×#340	—	2	1	no	—
19	C₂²×Dic₁₀₆	—	2	1	no	—
20	C₂×Dic₂₁₂	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₀₆	—	2	4	no	—
25	C₄×Dic₁₀₆	—	4	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×#11719919	—	4	1	no	—
28	C₂×C₄×Dic₅₃	—	4	1	no	—
29	C₂×#11719908	—	2	2	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	C₂×C₄	1	4	yes	—
32	#13133151	—	2	1	no	—
33	#13133151	C₂×C₄	1	2	yes	—
34	#13133142	—	2	4	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄²×C₅₃	—	4	1	no	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#107×C₅₃	—	2	2	no	—
41	#107×C₅₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	#95×C₅₃	—	4	1	no	—
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	1	no	—
50	C₂²×Dic₅₃	—	4	1	no	—
51	C₂×Dic₁₀₆	—	2	2	no	—
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×Dic₁₀₆	—	4	1	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	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	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	—	8	2	no	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	SD₁₆	1	2	yes	—
66	C₄⋊Dic₅₃	—	2	2	no	—
67	C₄⋊Dic₅₃	—	4	1	no	—
68	#11719919	—	4	4	no	—
69	C₄×Dic₅₃	—	4	4	no	—
70	Dic₂₁₂	—	2	2	no	—
71	Dic₂₁₂	—	2	4	no	—
72	Dic₂₁₂	C₄²	1	4	yes	complemented
73	#11719908	—	2	4	no	—
74	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	4	4	no	—
79	C₂×C₄×C₅₃	—	8	2	no	—
80	C₂×C₄×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	#102	1	2	yes	—
82	C₂×C₄×C₅₃	#102	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#135	1	2	yes	—
87	C₂×Dic₅₃	—	4	2	no	—
88	C₂×Dic₅₃	—	4	4	no	—
89	C₂×Dic₅₃	—	4	2	no	—
90	C₂×Dic₅₃	—	8	1	no	—
91	C₂×Dic₅₃	—	8	1	no	—
92	Dic₁₀₆	—	2	4	no	—
93	Dic₁₀₆	—	4	2	no	—
94	Dic₁₀₆	—	4	4	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₅₃	#117	1	1	yes	characteristic
98	C₈×C₅₃	C₄×D₄	1	2	yes	—
99	#12638	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
101	C₂²×C₅₃	C₄×D₈	1	2	yes	—
102	C₂²×C₅₃	#509	1	1	yes	characteristic
103	C₂²×C₅₃	#510	1	2	yes	—
104	C₂²×C₅₃	#510	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	8	4	no	—
108	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₅₃	#416	1	1	yes	characteristic
110	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
111	Dic₅₃	—	4	4	no	—
112	Dic₅₃	—	8	2	no	—
113	C₂×C₄×Q₁₆	—	53	1	no	—
114	C₂×#358	—	53	2	no	—
115	C₂×#346	—	53	1	no	—
116	C₄×#107	D₅₃	1	1	yes	characteristic
117	#1380	—	53	2	no	—
118	C₂×C₅₃	C₄×#102	1	1	yes	characteristic
119	C₂×C₅₃	#2139	1	1	yes	characteristic
120	C₂×C₅₃	#2141	1	1	yes	characteristic
121	C₂×C₅₃	#2175	1	2	yes	—
122	C₂×C₅₃	#2176	1	2	yes	—
123	C₂²×Q₁₆	—	53	1	no	—
124	C₂×C₄×Q₈	—	106	1	no	—
125	C₂²×C₁₆	—	53	2	no	—
126	C₄×Q₁₆	—	53	4	no	—
127	C₄×Q₁₆	—	106	2	no	—
128	C₂×#107	—	53	1	no	—
129	C₂×#107	C₂×D₅₃	1	2	yes	—
130	C₂×#103	—	106	1	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	2	1	no	—
133	#358	—	53	8	no	—
134	#346	—	106	2	no	—
135	#340	—	2	1	no	—
136	C₅₃	#12638	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	—	106	1	no	—
138	C₂×Q₁₆	—	53	4	no	—
139	C₂×Q₁₆	—	53	2	no	—
140	C₂×Q₁₆	—	106	1	no	—
141	C₂×Q₁₆	—	106	2	no	—
142	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	C₄×Q₈	—	106	4	no	—
145	C₄×Q₈	—	212	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	106	1	no	—
148	C₂×C₄⋊C₄	—	212	1	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	212	1	no	—
151	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	—	53	4	no	—
153	C₂×C₁₆	—	106	2	no	—
154	C₂×C₁₆	—	106	2	no	—
155	#107	—	2	2	no	—
156	#107	—	53	2	no	—
157	#107	—	106	1	no	—
158	#107	C₄×D₅₃	1	4	yes	—
159	#103	—	106	4	no	—
160	C₄×C₈	C₄×D₅₃	1	2	yes	—
161	C₄×C₈	D₂₁₂	1	2	yes	—
162	#95	—	4	1	no	—
163	C₂×Q₈	—	106	2	no	—
164	C₂×Q₈	—	106	4	no	—
165	C₂×Q₈	—	212	1	no	—
166	C₂×Q₈	—	212	2	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	212	1	no	—
169	C₂²×C₄	—	212	1	no	—
170	C₂²×C₄	#11719922	1	1	yes	characteristic
171	C₂²×C₄	#11719922	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	Q₁₆	—	53	4	no	—
174	Q₁₆	—	106	2	no	—
175	Q₁₆	—	106	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	#11719921	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
180	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
181	C₂×C₈	#11719945	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	106	2	no	—
185	C₄⋊C₄	—	212	4	no	—
186	C₄⋊C₄	—	212	1	no	—
187	C₄²	—	8	2	no	—
188	C₄²	—	212	4	no	—
189	C₄²	#11719921	1	2	yes	—
190	C₄²	#11719924	1	2	yes	—
191	C₁₆	—	106	4	no	—
192	C₂³	C₂×#11719922	1	1	yes	characteristic
193	Q₈	—	106	4	no	—
194	Q₈	—	212	2	no	—
195	Q₈	—	212	4	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	—	424	1	no	—
203	C₂×C₄	—	424	1	no	—
204	C₂×C₄	—	212	2	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	—	212	4	no	—
207	C₂×C₄	#13133137	1	2	yes	—
208	C₂×C₄	#13133137	1	2	yes	—
209	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
211	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
212	C₂×C₄	#13133237	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄²×D₅₃	1	1	yes	characteristic
215	C₈	#13133174	1	1	yes	characteristic
216	C₈	C₄×D₂₁₂	1	2	yes	—
217	C₂²	C₂×#13133137	1	1	yes	characteristic
218	C₂²	C₄×#11719922	1	2	yes	—
219	C₂²	#16811059	1	1	yes	characteristic
220	C₂²	#16811063	1	2	yes	—
221	C₂²	#16811064	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	424	2	no	—
226	C₄	—	212	4	no	—
227	C₄	C₄×#11719921	1	1	yes	characteristic
228	C₄	#16810617	1	1	yes	characteristic
229	C₄	C₄×#11719924	1	2	yes	—
230	C₂	C₄×#13133137	1	1	yes	characteristic
231	C₂	#20276425	1	1	yes	characteristic
232	C₂	#20276434	1	1	yes	characteristic
233	C₂	#20276579	1	2	yes	—
234	C₂	#20276584	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#12638
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. C₄×Dic₂₁₂	C₄	semidirect
13. #16810426	C₄	semidirect
72. Dic₂₁₂	C₄²	semidirect
136. C₅₃	#12638	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₂×#16810426	2
C₂×#16810399	2
#20272806	2
C₄×#107×C₅₃	2
#12638	53