Subgroups — C₄×#16837224 #28504074

Order 15104.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837224	C₂	1	2	yes	maximal
2	C₄×#14273667	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273663	C₂	1	1	yes	maximal, characteristic
4	#21422194	C₂	1	2	yes	maximal
5	C₄×#106×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273667	C₂²	1	1	yes	characteristic
7	C₂×#14273667	C₂²	1	1	yes	characteristic
8	C₂×#14273663	C₂²	1	2	yes	—
9	C₄×#11745664	—	2	1	no	—
10	#16837224	C₄	1	8	yes	direct product factor
11	#16837178	C₄	1	2	yes	complemented
12	C₄×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
13	#16837167	—	2	1	no	—
14	C₂×#106×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
17	C₅₉×#340	—	2	1	no	—
18	C₂×#11745664	—	2	2	no	—
19	#14273667	—	2	2	no	—
20	#14273667	C₂×C₄	1	4	yes	—
21	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
23	#14273663	—	2	2	no	—
24	#14273663	C₂×C₄	1	4	yes	—
25	#14273638	—	4	1	no	—
26	C₄×C₅₉⋊C₈	—	4	1	no	—
27	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
30	C₂×C₄²×C₅₉	—	4	1	no	—
31	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
32	#106×C₅₉	—	2	2	no	—
33	#106×C₅₉	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₅₉	Q₈	1	1	yes	characteristic
38	#95×C₅₉	—	4	1	no	—
39	C₂²×C₄×C₅₉	—	4	2	no	—
40	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₉	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₉	—	2	2	no	—
44	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₉	#41	1	2	yes	—
50	C₄⋊C₄×C₅₉	—	2	4	no	—
51	C₄⋊C₄×C₅₉	—	4	2	no	—
52	C₄²×C₅₉	—	8	2	no	—
53	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₅₉	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
56	C₄²×C₅₉	SD₁₆	1	1	yes	characteristic
57	#11745664	—	2	4	no	—
58	#11745664	—	4	2	no	—
59	C₂×C₅₉⋊C₈	—	4	2	no	—
60	C₅₉⋊C₁₆	—	2	2	no	—
61	C₅₉⋊C₁₆	C₄²	1	4	yes	complemented
62	C₂³×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₉	—	2	2	no	—
64	C₂×C₄×C₅₉	—	4	1	no	—
65	C₂×C₄×C₅₉	—	4	4	no	—
66	C₂×C₄×C₅₉	—	8	2	no	—
67	C₂×C₄×C₅₉	—	8	2	no	—
68	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
69	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
70	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
71	C₂×C₄×C₅₉	#106	1	1	yes	characteristic
72	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₉	#135	1	2	yes	—
76	C₈×C₅₉	—	2	1	no	—
77	C₈×C₅₉	—	2	1	no	—
78	C₈×C₅₉	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₅₉	#117	1	1	yes	characteristic
80	C₈×C₅₉	C₄×Q₈	1	2	yes	—
81	C₅₉⋊C₈	—	4	4	no	—
82	C₄×#365	—	59	1	no	maximal, Sylow
83	C₂²×C₅₉	—	2	1	no	—
84	C₂²×C₅₉	—	2	1	no	—
85	C₂²×C₅₉	C₂×#106	1	1	yes	characteristic
86	C₂²×C₅₉	C₄×SD₁₆	1	2	yes	—
87	C₄×C₅₉	—	2	1	no	—
88	C₄×C₅₉	—	2	1	no	—
89	C₄×C₅₉	—	2	1	no	—
90	C₄×C₅₉	—	8	4	no	—
91	C₄×C₅₉	#365	1	2	yes	complemented
92	C₄×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₅₉	#427	1	1	yes	characteristic
94	C₄×C₅₉	C₄×SD₁₆	1	2	yes	—
95	C₂×#365	—	59	2	no	—
96	C₄×M₃₂	—	59	1	no	—
97	C₄×#108	—	59	1	no	—
98	C₄×#106	D₅₉	1	1	yes	characteristic
99	#1383	—	59	2	no	—
100	C₂×C₅₉	—	2	2	no	—
101	C₂×C₅₉	C₄×#106	1	1	yes	characteristic
102	C₂×C₅₉	C₂×#365	1	1	yes	characteristic
103	C₂×C₅₉	#2155	1	1	yes	characteristic
104	C₂×M₃₂	—	59	1	no	—
105	C₂×M₃₂	—	59	1	no	—
106	C₂×#108	—	59	2	no	—
107	C₂×#106	C₂×D₅₉	1	2	yes	—
108	C₄×M₁₆	—	118	1	no	—
109	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	2	1	no	—
111	#365	—	59	8	no	—
112	#346	—	59	2	no	—
113	C₄×C₁₆	—	59	2	no	—
114	#341	—	118	1	no	—
115	#340	—	2	1	no	—
116	C₅₉	C₄×#365	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	118	2	no	—
118	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
123	M₃₂	—	59	4	no	—
124	M₃₂	—	118	2	no	—
125	C₂×C₁₆	—	59	2	no	—
126	C₂×C₁₆	—	59	2	no	—
127	#108	—	59	4	no	—
128	#108	—	118	2	no	—
129	#106	—	2	2	no	—
130	#106	C₄×D₅₉	1	4	yes	—
131	#97	—	236	1	no	—
132	C₄×C₈	—	236	1	no	—
133	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
134	C₄×C₈	Dic₁₁₈	1	1	yes	characteristic
135	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
136	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
137	#95	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	#11745670	1	1	yes	characteristic
140	C₂²×C₄	#11745670	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	M₁₆	—	118	4	no	—
143	M₁₆	—	236	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	236	2	no	—
146	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
147	C₂×C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
151	C₂×C₈	#11745685	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	8	2	no	—
155	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
156	C₄²	Dic₅₉⋊C₄	1	1	yes	characteristic
157	C₄²	#11745671	1	1	yes	characteristic
158	C₄²	#11745671	1	1	yes	characteristic
159	C₁₆	—	59	4	no	—
160	C₁₆	—	118	2	no	—
161	C₂³	C₂×#11745670	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	#14273658	1	1	yes	characteristic
168	C₂×C₄	#14273658	1	1	yes	characteristic
169	C₂×C₄	#14273658	1	1	yes	characteristic
170	C₂×C₄	#14273658	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
172	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
173	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
174	C₂×C₄	#14273747	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	236	4	no	—
178	C₈	C₄²×D₅₉	1	1	yes	characteristic
179	C₈	#14273684	1	1	yes	characteristic
180	C₈	C₄×Dic₁₁₈	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#14273658	1	1	yes	characteristic
184	C₂²	C₄×#11745670	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	#16837224	1	2	yes	direct product factor
190	C₄	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
191	C₄	#16837438	1	1	yes	characteristic
192	C₄	C₄×#11745671	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#14273658	1	1	yes	characteristic
195	C₂	C₂×#16837224	1	1	yes	characteristic
196	C₂	#21425868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273658
commutator	C₈×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₄×#106×C₅₉	C₂
socle	C₂²×C₅₉	C₂×#106
prime radical	C₂³×C₅₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₄×#106	D₅₉
hypocenter	C₅₉	C₄×#365
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. #16837224	C₄	direct (N × H)
11. #16837178	C₄	semidirect
12. C₄×C₅₉⋊C₁₆	C₄	semidirect
61. C₅₉⋊C₁₆	C₄²	semidirect
91. C₄×C₅₉	#365	semidirect
116. C₅₉	C₄×#365	semidirect
189. C₄	#16837224	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×#16837224	2
C₄×#14273667	2
C₄×#14273663	2
#21422194	2
C₄×#106×C₅₉	2
C₄×#365	59

Subgroups — C4×#16837224 #28504074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16837224 #28504074