Subgroups — C₈×#14273646 #28479252

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273646	C₂	1	1	yes	maximal, characteristic
2	#21422420	C₂	1	1	yes	maximal, characteristic
3	C₈×#11745664	C₂	1	2	yes	maximal
4	#21421946	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837455	C₂²	1	2	yes	—
7	C₂×#14273646	C₄	1	2	yes	—
8	C₄×#11745664	C₂²	1	2	yes	—
9	#16837171	C₄	1	2	yes	—
10	#16837128	C₄	1	4	yes	—
11	#16837124	—	2	2	no	—
12	C₈×C₅₉⋊C₈	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₅₉	—	2	2	no	—
15	C₈×Q₈×C₅₉	—	2	2	no	—
16	C₂×C₄⋊C₈×C₅₉	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₉	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×#11745664	C₂×C₄	1	4	yes	—
21	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
22	#14273646	C₈	1	8	yes	direct product factor
23	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
24	#14273638	—	2	2	no	—
25	C₄×C₅₉⋊C₈	—	2	2	no	—
26	C₄×C₅₉⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₅₉	—	2	2	no	—
32	C₄×Q₈×C₅₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₉	—	2	4	no	—
38	C₄⋊C₈×C₅₉	—	2	2	no	—
39	C₄×C₈×C₅₉	—	2	2	no	—
40	C₄×C₈×C₅₉	—	2	2	no	—
41	C₄×C₈×C₅₉	D₄	1	2	yes	—
42	C₄×C₈×C₅₉	D₄	1	2	yes	—
43	C₂×Q₈×C₅₉	—	2	2	no	—
44	C₂×Q₈×C₅₉	—	2	2	no	—
45	C₂×Q₈×C₅₉	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
47	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
48	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₉	—	2	2	no	—
52	C₂×C₈×C₅₉	—	2	2	no	—
53	C₂×C₈×C₅₉	—	4	1	no	—
54	C₂×C₈×C₅₉	—	4	2	no	—
55	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₉	—	2	4	no	—
59	C₄⋊C₄×C₅₉	—	2	2	no	—
60	C₄²×C₅₉	—	2	2	no	—
61	C₄²×C₅₉	—	2	2	no	—
62	C₄²×C₅₉	C₂×D₄	1	2	yes	—
63	C₄²×C₅₉	#41	1	2	yes	—
64	#11745664	—	2	4	no	—
65	#11745664	C₂×C₈	1	8	yes	—
66	C₂×C₅₉⋊C₈	—	2	4	no	—
67	C₂×C₅₉⋊C₈	—	2	4	no	—
68	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
69	Q₈×C₅₉	—	2	8	no	—
70	C₂×C₄×C₅₉	—	2	2	no	—
71	C₂×C₄×C₅₉	—	2	2	no	—
72	C₂×C₄×C₅₉	—	2	1	no	—
73	C₂×C₄×C₅₉	—	2	1	no	—
74	C₂×C₄×C₅₉	—	2	2	no	—
75	C₂×C₄×C₅₉	—	4	1	no	—
76	C₂×C₄×C₅₉	—	4	2	no	—
77	C₂×C₄×C₅₉	C₄×C₈	1	4	yes	—
78	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
80	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₅₉	#131	1	2	yes	—
84	C₈×C₅₉	—	2	1	no	—
85	C₈×C₅₉	—	2	2	no	—
86	C₈×C₅₉	—	4	2	no	—
87	C₈×C₅₉	#101	1	2	yes	complemented
88	C₅₉⋊C₈	—	2	8	no	—
89	C₅₉⋊C₈	—	4	4	no	—
90	C₈×#101	—	59	1	no	maximal, Sylow
91	C₂²×C₅₉	—	2	1	no	—
92	C₂²×C₅₉	—	2	1	no	—
93	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
95	C₂²×C₅₉	#405	1	1	yes	characteristic
96	C₄×C₅₉	—	2	4	no	—
97	C₄×C₅₉	—	2	1	no	—
98	C₄×C₅₉	—	2	2	no	—
99	C₄×C₅₉	—	4	2	no	—
100	C₄×C₅₉	C₂×#101	1	1	yes	characteristic
101	C₄×C₅₉	#413	1	1	yes	characteristic
102	C₄×C₅₉	C₈×D₄	1	4	yes	—
103	C₂×C₈×Q₈	D₅₉	1	1	yes	characteristic
104	C₄×#101	—	59	1	no	—
105	#1461	—	59	1	no	—
106	C₈×M₁₆	—	59	2	no	—
107	#1283	—	59	2	no	—
108	C₂×C₅₉	—	2	1	no	—
109	C₂×C₅₉	—	2	1	no	—
110	C₂×C₅₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₅₉	C₄×#101	1	1	yes	characteristic
112	C₂×C₅₉	#1757	1	1	yes	characteristic
113	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
114	C₈×Q₈	—	2	2	no	—
115	C₈×Q₈	—	2	2	no	—
116	#431	—	59	2	no	—
117	C₂×C₄⋊C₈	Dic₅₉	1	2	yes	—
118	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
119	C₂×#101	—	59	2	no	—
120	C₄×M₁₆	—	59	2	no	—
121	C₂×C₄×C₈	Dic₅₉	1	2	yes	—
122	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
123	#343	—	59	2	no	—
124	#324	—	59	4	no	—
125	#322	—	118	2	no	—
126	C₈²	—	118	2	no	—
127	C₅₉	C₈×#101	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₈	C₄×D₅₉	1	1	yes	characteristic
129	C₂×M₁₆	—	59	4	no	—
130	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
132	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
133	C₄×Q₈	—	2	2	no	—
134	C₄×Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	C₄×D₅₉	1	2	yes	—
136	C₂×C₄⋊C₄	C₄×D₅₉	1	1	yes	characteristic
137	C₂×C₄²	C₂×Dic₅₉	1	2	yes	—
138	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
139	C₄⋊C₈	—	2	4	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	118	2	no	—
142	#101	—	59	8	no	—
143	C₂²⋊C₈	—	59	4	no	—
144	#97	—	118	2	no	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	—	118	2	no	—
148	C₄×C₈	—	118	2	no	—
149	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
150	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	C₈×D₅₉	1	4	yes	—
154	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
155	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
158	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
159	M₁₆	—	59	8	no	—
160	M₁₆	—	118	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	118	4	no	—
166	C₂×C₈	—	118	4	no	—
167	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
168	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
169	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	2	2	no	—
174	C₄²	#11745685	1	2	yes	—
175	C₄²	C₂×C₂²⋊D₅₉	1	2	yes	—
176	C₂³	C₂×C₄×Dic₅₉	1	1	yes	characteristic
177	Q₈	—	2	8	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	C₈×Dic₅₉	1	4	yes	—
186	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
187	C₂×C₄	#14273686	1	1	yes	characteristic
188	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
189	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
190	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
191	C₂×C₄	#14273733	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	118	8	no	—
196	C₈	—	236	4	no	—
197	C₈	#14273646	1	2	yes	direct product factor
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₄×C₂²⋊Dic₅₉	1	1	yes	characteristic
201	C₂²	C₂×C₈×Dic₅₉	1	1	yes	characteristic
202	C₂²	#16837357	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	4	no	—
206	C₄	—	4	2	no	—
207	C₄	C₂×#14273646	1	1	yes	characteristic
208	C₄	#16837386	1	1	yes	characteristic
209	C₄	C₈×C₂²⋊D₅₉	1	4	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	C₈×C₂²⋊Dic₅₉	1	1	yes	characteristic
213	C₂	C₄×#14273646	1	1	yes	characteristic
214	C₂	#21423896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₈×Q₈×C₅₉	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂³×C₅₉	C₂×C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×C₈×Q₈	D₅₉
hypocenter	C₅₉	C₈×#101
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
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
22. #14273646	C₈	direct (N × H)
87. C₈×C₅₉	#101	semidirect
127. C₅₉	C₈×#101	semidirect
197. C₈	#14273646	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×#14273646	2
#21422420	2
C₈×#11745664	2
#21421946	2
C₂×C₈×Q₈×C₅₉	2
C₈×#101	59

Subgroups — C8×#14273646 #28479252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14273646 #28479252