Subgroups — #27403094

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	3	yes	maximal
4	#20275488	C₂	1	3	yes	maximal
5	#20274852	C₂	1	3	yes	maximal
6	C₄×#13133152	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16810754	—	2	3	no	—
9	Q₈×#88322	—	2	1	no	—
10	Q₈×#88322	C₄	1	6	yes	—
11	Q₈×#88322	C₄	1	2	yes	—
12	C₂×#13133152	C₂²	1	3	yes	—
13	C₂×#13133152	C₂²	1	3	yes	—
14	#16810647	C₂²	1	3	yes	—
15	C₂×C₄⋊#88322	C₂²	1	3	yes	—
16	C₄×#11719917	C₂²	1	3	yes	—
17	C₂×C₄×#88322	C₂²	1	3	yes	—
18	#16810387	C₂²	1	6	yes	—
19	#16810387	C₂²	1	3	yes	—
20	#16810387	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#446	—	2	3	no	—
23	C₈×Q₈×C₅₃	—	2	1	no	—
24	C₅₃×#423	C₂²	1	3	yes	—
25	C₄×M₁₆×C₅₃	C₂²	1	3	yes	—
26	C₂×#11719917	C₂³	1	3	yes	—
27	C₂×#11719917	C₂³	1	1	yes	characteristic
28	C₂²×#88322	C₂³	1	1	yes	characteristic
29	C₂²×#88322	C₂³	1	3	yes	—
30	#13133152	—	2	6	no	—
31	#13133152	—	2	6	no	—
32	C₄⋊#88322	—	2	3	no	—
33	C₄⋊#88322	—	2	3	no	—
34	C₄⋊#88322	C₂×C₄	1	6	yes	—
35	C₄⋊#88322	C₂×C₄	1	6	yes	—
36	#13133128	—	2	3	no	—
37	C₄×#88322	—	2	3	no	—
38	C₄×#88322	C₂×C₄	1	6	yes	—
39	C₄×#88322	C₂×C₄	1	6	yes	—
40	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
41	C₂×M₁₆×C₅₃	C₂³	1	3	yes	—
42	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₅₃	C₂×C₄	1	3	yes	—
44	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	D₄	1	3	yes	—
46	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	Q₈	1	3	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
49	C₄⋊C₈×C₅₃	—	2	3	no	—
50	C₄⋊C₈×C₅₃	—	2	3	no	—
51	#97×C₅₃	—	2	3	no	—
52	C₄×C₈×C₅₃	—	2	3	no	—
53	C₂×Q₈×C₅₃	—	2	4	no	—
54	C₂×Q₈×C₅₃	C₄⋊C₄	1	3	yes	—
55	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
58	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
59	M₁₆×C₅₃	—	2	1	no	—
60	M₁₆×C₅₃	—	2	3	no	—
61	M₁₆×C₅₃	—	2	3	no	—
62	M₁₆×C₅₃	#41	1	2	yes	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	3	no	—
65	C₄⋊C₄×C₅₃	C₄⋊C₄	1	6	yes	—
66	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
67	C₄⋊C₄×C₅₃	C₄⋊C₄	1	3	yes	—
68	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
70	C₄²×C₅₃	C₂×D₄	1	3	yes	—
71	C₄²×C₅₃	C₂×D₄	1	3	yes	—
72	#11719917	—	2	1	no	—
73	#11719917	—	2	3	no	—
74	#11719917	—	2	3	no	—
75	#11719917	#41	1	2	yes	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	3	no	—
78	C₂×#88322	—	2	3	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	—	2	3	no	—
82	C₂×#88322	C₂²×C₄	1	2	yes	—
83	C₂×#88322	C₂²×C₄	1	6	yes	—
84	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	6	no	—
86	Q₈×C₅₃	#108	1	4	yes	complemented
87	C₂×C₄×C₅₃	—	2	6	no	—
88	C₂×C₄×C₅₃	—	4	3	no	—
89	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
98	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	4	3	no	—
101	#88322	—	2	6	no	—
102	#88322	—	2	2	no	—
103	#88322	—	2	2	no	—
104	#88322	—	4	3	no	—
105	#22746	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₅₃	#557	1	1	yes	characteristic
110	C₂²×C₅₃	#557	1	1	yes	characteristic
111	C₄×C₅₃	—	2	3	no	—
112	C₄×C₅₃	—	2	3	no	—
113	C₄×C₅₃	—	2	3	no	—
114	C₄×C₅₃	C₂×#108	1	6	yes	—
115	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
116	C₄×C₅₃	#519	1	1	yes	characteristic
117	C₄×C₅₃	#548	1	1	yes	characteristic
118	C₄×C₅₃	#548	1	1	yes	characteristic
119	Q₈×M₁₆	—	53	1	no	—
120	Q₈×M₁₆	D₅₃	1	1	yes	characteristic
121	C₂×C₈×Q₈	—	53	1	no	—
122	#1950	—	53	3	no	—
123	#1950	—	53	3	no	—
124	#1837	—	53	3	no	—
125	C₄×#108	—	53	3	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	#2351	1	1	yes	characteristic
129	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
130	C₂×C₅₃	#2916	1	1	yes	characteristic
131	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
132	#446	—	2	3	no	—
133	#446	—	106	3	no	—
134	C₈×Q₈	—	2	1	no	—
135	C₈×Q₈	—	53	6	no	—
136	C₈×Q₈	—	53	2	no	—
137	C₈×Q₈	—	106	1	no	—
138	C₂×#108	—	53	3	no	—
139	C₂×#108	—	53	3	no	—
140	#423	—	53	3	no	—
141	#423	C₂×D₅₃	1	3	yes	—
142	C₂×C₄⋊C₈	—	53	3	no	—
143	C₄×M₁₆	—	53	3	no	—
144	C₄×M₁₆	C₂×D₅₃	1	3	yes	—
145	C₂×C₄×C₈	—	53	3	no	—
146	#341	—	53	3	no	—
147	#341	—	53	6	no	—
148	#341	—	53	1	no	—
149	C₅₃	#22746	1	1	yes	characteristic, complemented, Sylow
150	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
151	C₂×M₁₆	—	53	1	no	—
152	C₂×M₁₆	—	53	3	no	—
153	C₂×M₁₆	C₂²×D₅₃	1	3	yes	—
154	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
155	C₂²×C₈	—	53	1	no	—
156	C₂²×C₈	—	53	3	no	—
157	C₄×Q₈	C₄×D₅₃	1	3	yes	—
158	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
159	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₅₃	1	3	yes	—
161	C₂×C₄⋊C₄	Dic₁₀₆	1	3	yes	—
162	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
163	#108	—	106	6	no	—
164	#108	—	106	6	no	—
165	C₄⋊C₈	—	2	3	no	—
166	C₄⋊C₈	—	2	3	no	—
167	C₄⋊C₈	—	53	6	no	—
168	C₄⋊C₈	—	53	6	no	—
169	C₄⋊C₈	—	106	3	no	—
170	C₄⋊C₈	—	106	3	no	—
171	#97	—	2	3	no	—
172	#97	—	106	3	no	—
173	C₄×C₈	—	2	3	no	—
174	C₄×C₈	—	53	6	no	—
175	C₄×C₈	—	53	6	no	—
176	C₄×C₈	—	106	3	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	#11719919	1	3	yes	—
179	C₂×Q₈	#11719919	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
181	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
182	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	3	no	—
185	M₁₆	—	2	3	no	—
186	M₁₆	—	53	2	no	—
187	M₁₆	—	106	1	no	—
188	M₁₆	—	106	3	no	—
189	M₁₆	—	106	3	no	—
190	M₁₆	#11719949	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	3	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	6	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	3	no	—
197	C₂×C₈	—	106	3	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	—	106	1	no	—
200	C₂×C₈	—	106	3	no	—
201	C₄⋊C₄	#11719919	1	6	yes	—
202	C₄⋊C₄	#11719919	1	3	yes	—
203	C₄⋊C₄	#11719919	1	3	yes	—
204	C₄²	C₂×C₄×D₅₃	1	3	yes	—
205	C₄²	C₂×C₄×D₅₃	1	3	yes	—
206	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
207	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
208	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	#13133152	1	4	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×#11719919	1	3	yes	—
214	C₂×C₄	C₂×#11719919	1	3	yes	—
215	C₂×C₄	C₂×#11719919	1	3	yes	—
216	C₂×C₄	C₂×#11719919	1	3	yes	—
217	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
221	C₂×C₄	#13133264	1	1	yes	characteristic
222	C₂×C₄	#13133265	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	3	no	—
225	C₈	—	106	2	no	—
226	C₈	—	106	6	no	—
227	C₈	—	106	2	no	—
228	C₈	—	212	3	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×#11719919	1	1	yes	characteristic
232	C₂²	#16811349	1	1	yes	characteristic
233	C₂²	#16811352	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#13133152	1	6	yes	—
238	C₄	C₄×#11719949	1	1	yes	characteristic
239	C₄	#16811099	1	1	yes	characteristic
240	C₄	#16811280	1	1	yes	characteristic
241	C₄	#16811284	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#20277633	1	1	yes	characteristic
245	C₂	C₂²×#13133152	1	1	yes	characteristic
246	C₂	#20283324	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	Q₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	Q₈×M₁₆	D₅₃
hypocenter	C₅₃	#22746
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
86. Q₈×C₅₃	#108	semidirect
149. C₅₃	#22746	semidirect
210. Q₈	#13133152	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11719917	2
C₂×Q₈×#88322	2
#20275490	2
#20275488	2
#20274852	2
C₄×#13133152	2
Q₈×M₁₆×C₅₃	2
#22746	53