Subgroups — #27293899

Order 13568.

362 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	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719908	C₂	1	4	yes	maximal, complemented
4	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×#13133157	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	—
9	#16810399	—	2	1	no	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₈×Q₈×C₅₃	—	2	1	no	—
13	C₈×Q₈×C₅₃	—	2	1	no	—
14	C₈×Q₈×C₅₃	C₄	1	2	yes	—
15	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	2	4	no	—
21	#13133157	—	4	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×Q₈×C₅₃	—	2	1	no	—
29	C₄×Q₈×C₅₃	—	2	1	no	—
30	C₄×Q₈×C₅₃	C₈	1	2	yes	—
31	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	1	no	—
39	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	2	1	no	—
48	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
60	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	#11719908	—	2	4	no	—
74	#11719908	—	4	2	no	—
75	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
76	Q₈×C₅₃	—	2	4	no	—
77	Q₈×C₅₃	—	2	2	no	—
78	Q₈×C₅₃	M₃₂	1	4	yes	complemented
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#103	1	2	yes	—
89	C₂×C₄×C₅₃	#104	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₅₃	—	2	1	no	—
94	C₈×C₅₃	—	2	1	no	—
95	C₈×C₅₃	—	4	2	no	—
96	C₈×C₅₃	#103	1	2	yes	—
97	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
99	#11607	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
103	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
110	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
111	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
112	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
113	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
114	#2150	—	53	1	no	—
115	C₄×M₃₂	—	53	1	no	—
116	Q₈⋊C₁₆	—	53	4	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
119	C₂×C₅₃	#2112	1	1	yes	characteristic
120	C₂×C₅₃	#2117	1	1	yes	characteristic
121	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
122	C₂×M₃₂	—	106	1	no	—
123	C₂×M₃₂	—	106	1	no	—
124	C₈×Q₈	—	2	1	no	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	Dic₅₃	1	2	yes	—
127	C₈×Q₈	C₂×D₅₃	1	2	yes	—
128	C₂×C₄⋊C₈	—	2	1	no	—
129	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	53	2	no	—
133	C₄⋊C₁₆	—	106	1	no	—
134	#346	—	106	1	no	—
135	C₄×C₁₆	—	53	2	no	—
136	C₅₃	#11607	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₈	#88322	1	1	yes	characteristic
138	C₂²×C₈	—	2	1	no	—
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	#88322	1	2	yes	—
144	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
149	M₃₂	—	106	4	no	—
150	M₃₂	—	212	2	no	—
151	C₂×C₁₆	—	106	2	no	—
152	C₂×C₁₆	—	106	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂×#88322	1	2	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
169	C₂²×C₄	#11719917	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	#11719924	1	2	yes	—
175	C₂×C₈	#11719925	1	2	yes	—
176	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	C₂×#88322	1	2	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	C₂×#88322	1	1	yes	characteristic
187	C₄²	#11719917	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
190	C₁₆	—	106	4	no	—
191	C₁₆	—	212	2	no	—
192	C₂³	C₂²⋊#88322	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	2	2	no	—
195	Q₈	#13133157	1	4	yes	complemented
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
205	C₂×C₄	#13133141	1	2	yes	—
206	C₂×C₄	#13133144	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#13133141	1	2	yes	—
214	C₈	C₂×#11719924	1	1	yes	characteristic
215	C₈	C₂×#11719925	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	Q₈⋊#88322	1	2	yes	—
219	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	2	no	—
225	C₄	Q₈⋊#88322	1	2	yes	—
226	C₄	C₂×#13133141	1	1	yes	characteristic
227	C₄	C₂×#13133144	1	1	yes	characteristic
228	C₄	C₂×#13133157	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
231	C₂	#20276326	1	1	yes	characteristic
232	C₂	#20276343	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#11607
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. Q₈⋊#11719908	C₂	semidirect
78. Q₈×C₅₃	M₃₂	semidirect
136. C₅₃	#11607	semidirect
195. Q₈	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₄×#13133157	2
Q₈⋊#11719908	2
C₂×C₈×Q₈×C₅₃	2
#11607	53