Subgroups — C₈×#13133287 #28302158

Order 13568.

374 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133287	C₂	1	1	yes	maximal, characteristic
2	#20290263	C₂	1	1	yes	maximal, characteristic
3	C₈×#11719936	C₂	1	2	yes	maximal
4	#20289547	C₂	1	2	yes	maximal
5	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811642	C₂²	1	2	yes	—
7	C₂×#13133287	C₄	1	2	yes	—
8	C₄×#11719936	C₂²	1	2	yes	—
9	#16811499	C₄	1	2	yes	—
10	#16811487	C₄	1	2	yes	—
11	#16811484	C₄	1	2	yes	—
12	#88324⋊C₈	—	2	2	no	—
13	C₈×#88324	—	2	2	no	—
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₈×#88322	C₄	1	1	yes	characteristic
18	C₈×#88322	C₄	1	1	yes	characteristic
19	C₈×#88322	C₄	1	1	yes	characteristic
20	C₈²×C₅₃	C₄	1	1	yes	characteristic
21	C₂×#11719936	C₂×C₄	1	4	yes	—
22	#13133287	C₈	1	8	yes	direct product factor
23	C₄⋊#88324	C₂×C₄	1	4	yes	—
24	#13133274	—	2	2	no	—
25	#13133272	—	2	2	no	—
26	C₄×#88324	—	2	2	no	—
27	C₄×#88324	—	2	2	no	—
28	C₂×C₈×D₅₃	—	2	1	no	—
29	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
32	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
33	C₄²×D₅₃	C₂³	1	1	yes	characteristic
34	C₄×#88322	C₂×C₄	1	1	yes	characteristic
35	C₄×#88322	C₂×C₄	1	1	yes	characteristic
36	C₄×#88322	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅₃	—	2	1	no	—
38	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅₃	D₄	1	1	yes	characteristic
41	C₈×Dic₅₃	D₄	1	1	yes	characteristic
42	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	2	4	no	—
50	#11719936	—	2	4	no	—
51	#11719936	C₂×C₈	1	8	yes	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₅₃	#41	1	1	yes	characteristic
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	C₄²	1	1	yes	characteristic
63	C₂×#88322	C₄²	1	1	yes	characteristic
64	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
65	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
66	C₈×D₅₃	—	2	4	no	—
67	C₈×D₅₃	—	2	2	no	—
68	C₈×D₅₃	—	2	2	no	—
69	C₈×D₅₃	—	2	2	no	—
70	C₈×D₅₃	C₂×C₈	1	4	yes	—
71	C₂²×D₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
76	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
77	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
78	C₄×D₅₃	—	2	1	no	—
79	C₄×D₅₃	—	2	1	no	—
80	C₄×D₅₃	—	2	2	no	—
81	C₄×D₅₃	C₂²×C₈	1	2	yes	—
82	C₄×D₅₃	#131	1	2	yes	—
83	#88324	—	2	8	no	—
84	#88324	—	4	4	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	1	no	—
87	C₈×C₅₃	—	2	1	no	—
88	C₈×C₅₃	C₄×C₈	1	2	yes	—
89	C₈×C₅₃	#108	1	2	yes	complemented
90	#88322	—	2	2	no	—
91	#88322	—	2	1	no	—
92	#88322	—	2	1	no	—
93	#88322	C₄×C₈	1	2	yes	—
94	#88322	#108	1	2	yes	complemented
95	C₈×#108	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂×D₅₃	—	2	1	no	—
98	C₂×D₅₃	—	2	1	no	—
99	C₂×D₅₃	C₈×Q₈	1	2	yes	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
102	C₄×C₅₃	#405	1	1	yes	characteristic
103	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
104	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	C₈×D₄	1	2	yes	—
107	Dic₅₃	#443	1	2	yes	—
108	C₄×#108	—	53	1	no	—
109	#1564	—	53	1	no	—
110	C₈×M₁₆	—	53	2	no	—
111	C₂×C₈²	—	53	1	no	—
112	#1278	—	53	2	no	—
113	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₅₃	#1772	1	1	yes	characteristic
115	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
116	D₅₃	—	2	2	no	—
117	#431	—	53	2	no	—
118	C₂×#108	—	53	2	no	—
119	C₄×M₁₆	—	53	2	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	—	53	1	no	—
122	C₂×C₄×C₈	—	53	1	no	—
123	#341	—	53	2	no	—
124	#335	—	53	2	no	—
125	#334	—	53	2	no	—
126	#322	—	106	2	no	—
127	C₈²	—	53	1	no	—
128	C₈²	—	53	1	no	—
129	C₈²	—	53	1	no	—
130	C₈²	—	106	2	no	—
131	C₈²	#6157	1	1	yes	characteristic
132	C₅₃	C₈×#108	1	1	yes	characteristic, complemented, Sylow
133	C₂×M₁₆	—	53	4	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₂×C₄²	—	53	1	no	—
140	#108	—	53	8	no	—
141	C₄⋊C₈	—	53	4	no	—
142	C₂²⋊C₈	—	106	2	no	—
143	#97	—	106	2	no	—
144	C₄×C₈	—	53	1	no	—
145	C₄×C₈	—	53	1	no	—
146	C₄×C₈	—	53	1	no	—
147	C₄×C₈	—	53	1	no	—
148	C₄×C₈	—	53	1	no	—
149	C₄×C₈	—	53	1	no	—
150	C₄×C₈	—	53	1	no	—
151	C₄×C₈	—	106	1	no	—
152	C₄×C₈	—	106	2	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	C₂×#6157	1	1	yes	characteristic
155	C₄×C₈	C₂×#6157	1	1	yes	characteristic
156	C₄×C₈	C₂×#6157	1	1	yes	characteristic
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	M₁₆	—	53	8	no	—
161	M₁₆	—	106	4	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	53	1	no	—
164	C₂×C₈	—	53	1	no	—
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	1	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	2	no	—
170	C₂×C₈	—	106	4	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	2	no	—
174	C₂×C₈	—	106	4	no	—
175	C₂×C₈	C₄×#6157	1	1	yes	characteristic
176	C₂×C₈	C₄×#6157	1	1	yes	characteristic
177	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
178	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
179	C₄²	—	53	1	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	C₂²×#6157	1	1	yes	characteristic
183	C₂³	—	53	1	no	—
184	C₂×C₄	—	53	1	no	—
185	C₂×C₄	—	53	1	no	—
186	C₂×C₄	—	53	2	no	—
187	C₂×C₄	—	53	2	no	—
188	C₂×C₄	—	53	1	no	—
189	C₂×C₄	—	106	2	no	—
190	C₂×C₄	—	106	1	no	—
191	C₂×C₄	—	106	1	no	—
192	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
194	C₂×C₄	#13133296	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	53	2	no	—
199	C₈	—	53	2	no	—
200	C₈	—	106	2	no	—
201	C₈	—	106	8	no	—
202	C₈	—	106	1	no	—
203	C₈	—	106	1	no	—
204	C₈	—	212	4	no	—
205	C₈	C₈×#6157	1	2	yes	—
206	C₈	#13133287	1	2	yes	direct product factor
207	C₂²	—	53	2	no	—
208	C₂²	—	106	1	no	—
209	C₂²	—	106	1	no	—
210	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	53	2	no	—
213	C₄	—	53	2	no	—
214	C₄	—	106	1	no	—
215	C₄	C₂×C₈×#6157	1	1	yes	characteristic
216	C₄	#16811586	1	1	yes	characteristic
217	C₄	C₂×#13133287	1	1	yes	characteristic
218	C₄	D₅₃⋊#108	1	1	yes	characteristic
219	C₂	—	106	2	no	—
220	C₂	C₈×C₄⋊#6157	1	1	yes	characteristic
221	C₂	#20290517	1	1	yes	characteristic
222	C₂	C₄×#13133287	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₄⋊C₄
prime radical	C₂²×D₅₃	C₄×Q₈
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	C₈×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
22. #13133287	C₈	direct (N × H)
89. C₈×C₅₃	#108	semidirect
94. #88322	#108	semidirect
132. C₅₃	C₈×#108	semidirect
206. C₈	#13133287	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#13133287	2
#20290263	2
C₈×#11719936	2
#20289547	2
C₈²×D₅₃	2
C₈×#108	53

Subgroups — C8×#13133287 #28302158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133287 #28302158