Subgroups — #28328434

Order 13568.

498 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	#20291132	C₂	1	4	yes	maximal, complemented
2	#20291131	C₂	1	2	yes	maximal, complemented
3	#20291129	C₂	1	2	yes	maximal, complemented
4	#20291052	C₂	1	2	yes	maximal
5	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic
6	#20290926	C₂	1	2	yes	maximal
7	C₄×#13133291	C₂	1	1	yes	maximal, characteristic
8	#20283645	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	C₂²	1	1	yes	characteristic
10	C₂×#13133291	C₂²	1	1	yes	characteristic
11	C₂²×#11719910	C₂²	1	2	yes	—
12	#16811544	C₂²	1	4	yes	—
13	#16811544	C₂²	1	4	yes	—
14	C₄⋊#11719910	C₂²	1	1	yes	characteristic
15	C₄⋊#11719910	C₂²	1	1	yes	characteristic
16	C₄⋊#11719910	C₂²	1	2	yes	—
17	#16811512	C₂²	1	4	yes	—
18	#16811511	C₂²	1	2	yes	—
19	#16811509	C₂²	1	4	yes	—
20	C₄×#11719910	C₂²	1	2	yes	—
21	#16811180	C₄	1	1	yes	characteristic
22	#16810750	C₄	1	2	yes	—
23	#16810750	C₄	1	2	yes	—
24	#16810747	C₂²	1	2	yes	—
25	#16810747	C₂²	1	2	yes	—
26	#16810651	C₄	1	2	yes	—
27	#16810581	C₂²	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₅₃×#423	C₄	1	1	yes	characteristic
30	#13133291	—	2	2	no	—
31	#13133291	Q₈	1	4	yes	complemented
32	C₂×#11719910	—	2	4	no	—
33	C₂×#11719910	C₂³	1	2	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂×#11719910	C₂³	1	2	yes	—
37	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	1	yes	characteristic
39	#13133217	C₈	1	1	yes	characteristic
40	#13133217	C₈	1	1	yes	characteristic
41	#13133209	C₈	1	2	yes	—
42	#13133196	C₈	1	2	yes	—
43	#13133196	C₈	1	2	yes	—
44	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
46	#13133176	C₂×C₄	1	2	yes	—
47	#13133146	C₂×C₄	1	4	yes	—
48	#13133146	C₂×C₄	1	4	yes	—
49	#13133127	C₂×C₄	1	4	yes	—
50	C₄×#88322	C₂×C₄	1	1	yes	characteristic
51	C₄×#88322	C₂×C₄	1	1	yes	characteristic
52	C₄×#88322	C₂³	1	1	yes	characteristic
53	C₄×#88322	C₂³	1	1	yes	characteristic
54	C₈×Dic₅₃	C₂³	1	4	yes	—
55	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
63	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
64	M₁₆×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
66	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
71	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
72	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
73	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
74	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
75	#11719919	C₂×C₈	1	4	yes	—
76	#11719919	C₂×C₈	1	4	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
80	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
81	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
82	C₂×#88322	C₂×Q₈	1	2	yes	—
83	C₂×#88322	#41	1	2	yes	—
84	C₂×#88322	C₂⁴	1	1	yes	characteristic
85	#11719910	—	2	2	no	—
86	#11719910	—	2	2	no	—
87	#11719910	—	2	2	no	—
88	#11719910	C₂×Q₈	1	4	yes	—
89	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	2	yes	—
99	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
101	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	4	2	no	—
104	C₈×C₅₃	—	2	4	no	—
105	#88322	—	2	1	no	—
106	#88322	—	2	1	no	—
107	#88322	C₄×Q₈	1	2	yes	—
108	#88322	C₂²×Q₈	1	1	yes	characteristic
109	#88322	C₂×#41	1	1	yes	characteristic
110	#33476	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
114	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	C₈×Q₈	1	2	yes	—
119	C₄×C₅₃	#504	1	2	yes	—
120	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
122	Dic₅₃	—	2	4	no	—
123	Dic₅₃	—	2	4	no	—
124	#2960	—	53	1	no	—
125	#2183	—	53	2	no	—
126	#2183	—	53	4	no	—
127	Q₈×C₁₆	—	53	2	no	—
128	#2151	—	53	2	no	—
129	C₂×C₄⋊C₁₆	—	53	1	no	—
130	#2108	—	53	2	no	—
131	C₄×M₃₂	—	53	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
134	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
135	C₂×C₅₃	#3407	1	1	yes	characteristic
136	#533	—	53	1	no	—
137	C₂×M₃₂	—	53	1	no	—
138	C₂×M₃₂	—	53	1	no	—
139	C₂²×C₁₆	—	53	2	no	—
140	#446	—	53	2	no	—
141	#446	—	53	2	no	—
142	C₈×Q₈	—	53	2	no	—
143	C₈×Q₈	—	53	2	no	—
144	#424	—	53	2	no	—
145	#423	#6157	1	1	yes	characteristic
146	#405	—	53	2	no	—
147	C₂×C₄×C₈	—	53	1	no	—
148	C₄⋊C₁₆	—	53	1	no	—
149	C₄⋊C₁₆	—	53	1	no	—
150	C₄⋊C₁₆	—	53	2	no	—
151	C₄⋊C₁₆	—	53	4	no	—
152	C₄⋊C₁₆	—	53	4	no	—
153	#346	—	53	2	no	—
154	#346	—	53	4	no	—
155	C₄×C₁₆	—	53	2	no	—
156	C₄×C₁₆	—	53	4	no	—
157	C₅₃	#33476	1	1	yes	characteristic, complemented, Sylow
158	C₂×M₁₆	C₂×#6157	1	2	yes	—
159	C₂²×C₈	—	53	1	no	—
160	C₂²×C₈	—	53	1	no	—
161	C₄⋊Q₈	—	53	1	no	—
162	C₄⋊Q₈	—	53	1	no	—
163	#125	—	53	2	no	—
164	C₂²⋊Q₈	—	53	2	no	—
165	C₂²⋊Q₈	—	53	2	no	—
166	C₄×Q₈	—	53	2	no	—
167	C₄×Q₈	—	53	2	no	—
168	#117	—	53	2	no	—
169	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
170	M₃₂	—	53	4	no	—
171	M₃₂	—	106	2	no	—
172	C₂×C₁₆	—	53	2	no	—
173	C₂×C₁₆	—	53	2	no	—
174	C₂×C₁₆	—	53	2	no	—
175	C₂×C₁₆	—	53	2	no	—
176	C₂×C₁₆	—	106	4	no	—
177	C₄⋊C₈	—	53	4	no	—
178	C₄⋊C₈	—	53	4	no	—
179	C₄⋊C₈	C₂×#6157	1	2	yes	—
180	C₄⋊C₈	C₂×#6157	1	2	yes	—
181	#97	—	53	4	no	—
182	C₄×C₈	—	53	4	no	—
183	C₄×C₈	—	53	1	no	—
184	C₄×C₈	—	53	1	no	—
185	C₄×C₈	—	53	1	no	—
186	C₄×C₈	—	53	1	no	—
187	C₂×Q₈	—	53	2	no	—
188	C₂×Q₈	—	53	2	no	—
189	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
190	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
191	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	—	53	1	no	—
196	C₂×C₈	—	53	1	no	—
197	C₂×C₈	—	53	1	no	—
198	C₂×C₈	—	53	1	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	C₂²×#6157	1	4	yes	—
201	C₄⋊C₄	—	53	2	no	—
202	C₄⋊C₄	—	53	2	no	—
203	C₄⋊C₄	—	53	4	no	—
204	C₄⋊C₄	—	53	4	no	—
205	C₄⋊C₄	—	53	2	no	—
206	C₄⋊C₄	—	53	2	no	—
207	C₂²⋊C₄	—	53	4	no	—
208	C₄²	—	53	4	no	—
209	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
210	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
211	C₄²	C₂²×#6157	1	1	yes	characteristic
212	C₄²	C₂²×#6157	1	1	yes	characteristic
213	C₁₆	—	53	4	no	—
214	C₁₆	—	106	2	no	—
215	C₁₆	—	106	2	no	—
216	C₁₆	—	106	2	no	—
217	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
218	Q₈	—	106	4	no	—
219	Q₈	—	212	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	Q₈×#6157	1	2	yes	—
225	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
226	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
227	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
230	C₂×C₄	#13133304	1	2	yes	—
231	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	53	2	no	—
234	C₈	—	53	1	no	—
235	C₈	—	53	1	no	—
236	C₈	—	106	1	no	—
237	C₈	—	106	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	#16811672	1	2	yes	—
241	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	106	4	no	—
246	C₄	—	106	4	no	—
247	C₄	#16811672	1	2	yes	—
248	C₄	#16811682	1	2	yes	—
249	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
250	C₄	C₂×#13133304	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×#16811672	1	1	yes	characteristic
253	C₂	C₂×#16811682	1	1	yes	characteristic
254	C₂	#20291684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#423	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#423	#6157
hypocenter	C₅₃	#33476
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291131	C₂	semidirect
3. #20291129	C₂	semidirect
31. #13133291	Q₈	semidirect
157. C₅₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291131	2
#20291129	2
#20291052	2
C₂×C₄⋊#11719910	2
#20290926	2
C₄×#13133291	2
#20283645	2
#33476	53