Subgroups — #28328428

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291129	C₂	1	4	yes	maximal, complemented
3	#20291052	C₂	1	1	yes	maximal, characteristic
4	#20291050	C₂	1	1	yes	maximal, characteristic
5	#20291049	C₂	1	1	yes	maximal, characteristic
6	#20290926	C₂	1	1	yes	maximal, characteristic
7	#20290924	C₂	1	1	yes	maximal, characteristic
8	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
9	#20283648	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133291	C₂²	1	1	yes	characteristic
11	C₂×#13133291	C₂²	1	1	yes	characteristic
12	C₂²×#11719910	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	1	yes	characteristic
14	#16811544	C₂²	1	4	yes	—
15	#16811544	C₂²	1	4	yes	—
16	C₄⋊#11719910	C₂²	1	4	yes	—
17	#16811512	C₂²	1	2	yes	—
18	#16811511	C₂²	1	2	yes	—
19	#16811509	C₂²	1	2	yes	—
20	#16811509	C₂²	1	2	yes	—
21	#16811509	C₂²	1	2	yes	—
22	C₄×#11719910	C₂²	1	2	yes	—
23	#16811176	C₄	1	1	yes	characteristic
24	#16810752	C₄	1	4	yes	—
25	#16810747	C₂²	1	4	yes	—
26	#16810649	C₄	1	1	yes	characteristic
27	#16810646	C₄	1	1	yes	characteristic
28	#16810581	C₂²	1	1	yes	characteristic
29	#16810579	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#424	C₄	1	1	yes	characteristic
32	#13133291	—	2	2	no	—
33	#13133291	—	2	2	no	—
34	C₂×#11719910	—	2	2	no	—
35	C₂×#11719910	Q₈	1	4	yes	complemented
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	1	yes	characteristic
40	C₂×#11719910	C₂³	1	1	yes	characteristic
41	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	#13133216	C₈	1	2	yes	—
44	#13133208	C₈	1	2	yes	—
45	#13133194	C₈	1	4	yes	—
46	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
47	#13133176	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	#13133146	C₂×C₄	1	4	yes	—
50	#13133146	C₂×C₄	1	4	yes	—
51	#13133128	C₂×C₄	1	2	yes	—
52	#13133127	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂³	1	2	yes	—
54	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₃	C₂³	1	2	yes	—
56	C₈×Dic₅₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
63	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
64	M₁₆×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
68	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
69	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
72	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
74	#11719919	C₂×C₈	1	4	yes	—
75	#11719919	C₂×C₈	1	4	yes	—
76	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
78	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
80	#11719917	—	2	2	no	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
83	C₂×#88322	C₂×Q₈	1	2	yes	—
84	C₂×#88322	#41	1	2	yes	—
85	C₂×#88322	C₂⁴	1	1	yes	characteristic
86	#11719910	—	2	2	no	—
87	#11719910	—	2	2	no	—
88	#11719910	—	2	2	no	—
89	#11719910	C₂×Q₈	1	2	yes	—
90	#11719910	C₂×Q₈	1	2	yes	—
91	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#131	1	2	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₅₃	—	2	4	no	—
98	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
99	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
100	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
101	Dic₁₀₆	—	2	8	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	#88322	—	2	2	no	—
105	#88322	C₄×Q₈	1	2	yes	—
106	#88322	C₂²×Q₈	1	1	yes	characteristic
107	#88322	C₂×#41	1	1	yes	characteristic
108	#33474	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
111	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
112	C₄×C₅₃	—	2	4	no	—
113	C₄×C₅₃	C₈×Q₈	1	2	yes	—
114	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
116	Dic₅₃	—	2	4	no	—
117	Dic₅₃	—	2	2	no	—
118	Dic₅₃	#504	1	4	yes	—
119	#2960	—	53	1	no	—
120	#2183	—	53	4	no	—
121	Q₈×C₁₆	—	53	4	no	—
122	#2151	—	53	1	no	—
123	#2151	—	53	1	no	—
124	#2150	—	53	1	no	—
125	#2108	—	53	1	no	—
126	#2108	—	53	1	no	—
127	C₂×C₄×C₁₆	—	53	1	no	—
128	C₂×C₅₃	—	2	2	no	—
129	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#504	1	2	yes	—
131	#533	—	53	1	no	—
132	C₂×M₃₂	—	53	1	no	—
133	C₂×M₃₂	—	53	1	no	—
134	C₂²×C₁₆	—	53	1	no	—
135	C₂²×C₁₆	—	53	1	no	—
136	#446	—	53	4	no	—
137	C₈×Q₈	—	53	4	no	—
138	#424	—	53	1	no	—
139	#424	#6157	1	1	yes	characteristic
140	#423	—	53	1	no	—
141	#405	—	53	1	no	—
142	#405	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₄⋊C₁₆	—	53	4	no	—
145	C₄⋊C₁₆	—	53	4	no	—
146	C₄⋊C₁₆	—	53	4	no	—
147	#346	—	53	2	no	—
148	#346	—	53	2	no	—
149	C₄×C₁₆	—	53	2	no	—
150	C₄×C₁₆	—	53	2	no	—
151	C₄×C₁₆	—	53	2	no	—
152	C₄×C₁₆	—	53	2	no	—
153	C₅₃	#33474	1	1	yes	characteristic, complemented, Sylow
154	C₂×M₁₆	—	53	1	no	—
155	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
156	C₂²×C₈	—	53	1	no	—
157	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
158	C₄⋊Q₈	—	53	2	no	—
159	#125	—	53	2	no	—
160	C₂²⋊Q₈	—	53	4	no	—
161	C₄×Q₈	—	53	4	no	—
162	#117	—	53	1	no	—
163	#117	C₂×#6157	1	1	yes	characteristic
164	C₂×C₄²	—	53	1	no	—
165	M₃₂	—	106	2	no	—
166	M₃₂	—	106	2	no	—
167	C₂×C₁₆	—	53	1	no	—
168	C₂×C₁₆	—	53	1	no	—
169	C₂×C₁₆	—	53	4	no	—
170	C₂×C₁₆	—	53	2	no	—
171	C₂×C₁₆	—	53	2	no	—
172	C₂×C₁₆	—	53	2	no	—
173	C₂×C₁₆	—	106	2	no	—
174	C₄⋊C₈	—	53	4	no	—
175	C₄⋊C₈	—	53	4	no	—
176	C₄⋊C₈	C₂×#6157	1	4	yes	—
177	#97	—	53	2	no	—
178	#97	—	53	2	no	—
179	C₄×C₈	—	53	2	no	—
180	C₄×C₈	—	53	2	no	—
181	C₄×C₈	—	53	2	no	—
182	C₄×C₈	—	53	2	no	—
183	C₂×Q₈	—	53	4	no	—
184	C₂²×C₄	—	53	2	no	—
185	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	106	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	53	2	no	—
190	C₂×C₈	—	53	2	no	—
191	C₂×C₈	—	53	1	no	—
192	C₂×C₈	—	53	1	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	C₂²×#6157	1	2	yes	—
195	C₂×C₈	C₂²×#6157	1	2	yes	—
196	C₄⋊C₄	—	53	2	no	—
197	C₄⋊C₄	—	53	4	no	—
198	C₄⋊C₄	—	53	4	no	—
199	C₄⋊C₄	—	53	4	no	—
200	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
201	C₂²⋊C₄	—	53	2	no	—
202	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
203	C₄²	—	53	1	no	—
204	C₄²	—	53	1	no	—
205	C₄²	—	53	2	no	—
206	C₄²	—	53	2	no	—
207	C₄²	C₂²×#6157	1	2	yes	—
208	C₁₆	—	53	2	no	—
209	C₁₆	—	53	2	no	—
210	C₁₆	—	106	2	no	—
211	C₁₆	—	106	2	no	—
212	C₁₆	—	106	2	no	—
213	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
214	Q₈	—	106	8	no	—
215	C₂×C₄	—	53	2	no	—
216	C₂×C₄	—	53	2	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	106	4	no	—
219	C₂×C₄	Q₈×#6157	1	2	yes	—
220	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
221	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
222	C₂×C₄	#13133304	1	2	yes	—
223	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	53	2	no	—
227	C₈	—	53	1	no	—
228	C₈	—	53	1	no	—
229	C₈	—	106	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	#16811672	1	2	yes	—
232	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	53	4	no	—
235	C₄	—	106	2	no	—
236	C₄	—	106	4	no	—
237	C₄	#16811672	1	2	yes	—
238	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
239	C₄	C₂×#13133304	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	C₂×#16811672	1	1	yes	characteristic
242	C₂	D₅₃⋊#504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#424	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#424	#6157
hypocenter	C₅₃	#33474
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291129	C₂	semidirect
35. C₂×#11719910	Q₈	semidirect
153. C₅₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291129	2
#20291052	2
#20291050	2
#20291049	2
#20290926	2
#20290924	2
C₂×#16811509	2
#20283648	2
#33474	53