Subgroups — C₄×#16837227 #28504089

Order 15104.

366 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837227	C₂	1	2	yes	maximal
2	C₂×C₄×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic
4	#21422187	C₂	1	2	yes	maximal
5	C₄×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#14273661	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₅₉	—	2	1	no	—
10	#16837227	C₄	1	8	yes	direct product factor
11	C₄×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
12	#16837165	—	2	1	no	—
13	C₂×#107×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
16	C₅₉×#340	—	2	1	no	—
17	C₂×C₄⋊Dic₅₉	—	2	2	no	—
18	C₂×C₄×Dic₅₉	—	4	1	no	—
19	C₂×C₅₉⋊C₁₆	—	2	2	no	—
20	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	—
23	#14273661	—	2	2	no	—
24	#14273661	C₂×C₄	1	4	yes	—
25	C₅₉⋊#95	—	4	1	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
29	C₂×C₄²×C₅₉	—	4	1	no	—
30	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
31	#107×C₅₉	—	2	2	no	—
32	#107×C₅₉	C₂×C₄	1	4	yes	—
33	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₉	Q₈	1	2	yes	—
35	#95×C₅₉	—	4	1	no	—
36	C₂²×C₄×C₅₉	—	4	2	no	—
37	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₉	—	4	2	no	—
41	C₂×C₈×C₅₉	—	2	2	no	—
42	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₉	#41	1	2	yes	—
47	C₄⋊C₄×C₅₉	—	2	4	no	—
48	C₄⋊C₄×C₅₉	—	4	2	no	—
49	C₄²×C₅₉	—	8	2	no	—
50	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₄²×C₅₉	Q₁₆	1	2	yes	—
52	C₄⋊Dic₅₉	—	2	4	no	—
53	C₄⋊Dic₅₉	—	4	2	no	—
54	C₄×Dic₅₉	—	8	2	no	—
55	C₅₉⋊C₁₆	—	2	2	no	—
56	C₅₉⋊C₁₆	C₄²	1	4	yes	complemented
57	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₉	—	2	2	no	—
59	C₂×C₄×C₅₉	—	4	4	no	—
60	C₂×C₄×C₅₉	—	8	2	no	—
61	C₂×C₄×C₅₉	—	8	2	no	—
62	C₂×C₄×C₅₉	#107	1	2	yes	—
63	C₂×C₄×C₅₉	#107	1	2	yes	—
64	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
65	C₂×C₄×C₅₉	Q₃₂	1	2	yes	—
66	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#135	1	2	yes	—
70	C₂×Dic₅₉	—	4	4	no	—
71	C₂×Dic₅₉	—	8	2	no	—
72	C₂×Dic₅₉	—	8	2	no	—
73	C₈×C₅₉	—	2	2	no	—
74	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₅₉	#117	1	1	yes	characteristic
76	C₈×C₅₉	C₄×Q₈	1	2	yes	—
77	C₄×#366	—	59	1	no	maximal, Sylow
78	C₂²×C₅₉	C₂×#107	1	1	yes	characteristic
79	C₂²×C₅₉	C₄×D₈	1	2	yes	—
80	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
81	C₂²×C₅₉	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₅₉	#508	1	2	yes	—
83	C₄×C₅₉	—	2	2	no	—
84	C₄×C₅₉	—	8	4	no	—
85	C₄×C₅₉	#366	1	4	yes	complemented
86	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₅₉	#427	1	1	yes	characteristic
88	C₄×C₅₉	C₄×Q₁₆	1	2	yes	—
89	Dic₅₉	—	8	4	no	—
90	C₂×#366	—	59	2	no	—
91	C₂×C₄×C₁₆	—	59	1	no	—
92	C₄×#107	—	59	1	no	—
93	C₄×#107	D₅₉	1	1	yes	characteristic
94	#1380	—	59	2	no	—
95	C₂×C₅₉	C₄×#107	1	1	yes	characteristic
96	C₂×C₅₉	C₂×#366	1	1	yes	characteristic
97	C₂×C₅₉	#2158	1	1	yes	characteristic
98	C₂×C₅₉	C₄×D₁₆	1	2	yes	—
99	C₂×C₅₉	C₄×Q₃₂	1	2	yes	—
100	C₂²×C₁₆	—	59	1	no	—
101	C₂²×C₁₆	—	59	1	no	—
102	C₂×#107	—	59	2	no	—
103	C₂×#107	C₂×D₅₉	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₄⋊C₄	—	118	1	no	—
107	#366	—	59	8	no	—
108	C₄×C₁₆	—	59	4	no	—
109	#340	—	2	1	no	—
110	#340	—	118	1	no	—
111	C₅₉	C₄×#366	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	118	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	—	236	1	no	—
118	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₂×C₁₆	—	59	2	no	—
120	C₂×C₁₆	—	59	4	no	—
121	C₂×C₁₆	—	59	2	no	—
122	C₂×C₁₆	—	118	2	no	—
123	#107	—	2	2	no	—
124	#107	—	59	4	no	—
125	#107	—	118	2	no	—
126	#107	C₄×D₅₉	1	4	yes	—
127	C₄×C₈	Dic₁₁₈	1	2	yes	—
128	C₄×C₈	C₄×D₅₉	1	2	yes	—
129	#95	—	4	1	no	—
130	#95	—	236	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	236	2	no	—
133	C₂²×C₄	#11745669	1	1	yes	characteristic
134	C₂²×C₄	#11745669	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
141	C₂×C₈	#11745685	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	118	4	no	—
145	C₄⋊C₄	—	236	2	no	—
146	C₄²	—	8	2	no	—
147	C₄²	—	472	2	no	—
148	C₄²	Dic₅₉⋊C₄	1	2	yes	—
149	C₄²	#11745672	1	2	yes	—
150	C₁₆	—	59	4	no	—
151	C₁₆	—	118	2	no	—
152	C₂³	C₂×#11745669	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	472	2	no	—
158	C₂×C₄	—	472	2	no	—
159	C₂×C₄	—	236	4	no	—
160	C₂×C₄	#14273660	1	2	yes	—
161	C₂×C₄	#14273660	1	2	yes	—
162	C₂×C₄	#14273668	1	2	yes	—
163	C₂×C₄	#14273673	1	2	yes	—
164	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
166	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
167	C₂×C₄	#14273747	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₄²×D₅₉	1	1	yes	characteristic
170	C₈	#14273684	1	1	yes	characteristic
171	C₈	C₄×Dic₁₁₈	1	2	yes	—
172	C₂²	C₂×#14273660	1	1	yes	characteristic
173	C₂²	C₄×#11745669	1	2	yes	—
174	C₂²	C₂×#14273668	1	1	yes	characteristic
175	C₂²	C₂×#14273673	1	1	yes	characteristic
176	C₂²	#16837832	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	8	4	no	—
179	C₄	—	472	4	no	—
180	C₄	#16837227	1	4	yes	direct product factor
181	C₄	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
182	C₄	#16837438	1	1	yes	characteristic
183	C₄	C₄×#11745672	1	2	yes	—
184	C₂	C₄×#14273660	1	1	yes	characteristic
185	C₂	C₂×#16837227	1	1	yes	characteristic
186	C₂	#21425878	1	1	yes	characteristic
187	C₂	C₄×#14273668	1	2	yes	—
188	C₂	C₄×#14273673	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11745669
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#107×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×D₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#107	D₅₉
hypocenter	C₅₉	C₄×#366
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. #16837227	C₄	direct (N × H)
11. C₄×C₅₉⋊C₁₆	C₄	semidirect
56. C₅₉⋊C₁₆	C₄²	semidirect
85. C₄×C₅₉	#366	semidirect
111. C₅₉	C₄×#366	semidirect
180. C₄	#16837227	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16837227	2
C₂×C₄×C₅₉⋊C₁₆	2
C₄×#14273661	2
#21422187	2
C₄×#107×C₅₉	2
C₄×#366	59

Subgroups — C4×#16837227 #28504089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837227 #28504089