Subgroups — C₄×#16839585 #29595397

Order 15616.

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×#16839585	C₂	1	2	yes	maximal
2	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
3	C₄×#14519709	C₂	1	1	yes	maximal, characteristic
4	#21443918	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519713	C₂²	1	2	yes	—
7	C₂×#14519709	C₂²	1	2	yes	—
8	C₄×#11758194	—	2	1	no	—
9	C₄×C₄⋊Dic₆₁	—	2	1	no	—
10	#16839585	C₄	1	8	yes	direct product factor
11	#16839527	—	2	1	no	—
12	#16839525	—	2	1	no	—
13	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
18	C₂×#11758194	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄×Dic₆₁	—	4	1	no	—
21	#14519713	—	2	2	no	—
22	#14519713	C₂×C₄	1	4	yes	—
23	#14519709	—	2	2	no	—
24	#14519709	C₂×C₄	1	4	yes	—
25	#14519688	—	4	1	no	—
26	C₄×#92418	—	4	1	no	—
27	#14519684	—	4	1	no	—
28	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
31	M₃₂×C₆₁	—	2	2	no	—
32	M₃₂×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	—	4	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₆₁	#41	1	2	yes	—
50	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
53	C₄²×C₆₁	SD₁₆	1	1	yes	characteristic
54	C₄⋊Dic₆₁	—	2	4	no	—
55	C₄⋊Dic₆₁	—	4	2	no	—
56	C₄×Dic₆₁	—	8	2	no	—
57	#11758194	—	2	4	no	—
58	#11758194	—	4	2	no	—
59	C₂×#92418	—	4	2	no	—
60	C₁₆×C₆₁	—	2	2	no	—
61	C₁₆×C₆₁	C₄²	1	4	yes	complemented
62	C₂³×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₆₁	—	2	2	no	—
64	C₂×C₄×C₆₁	—	4	1	no	—
65	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
66	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
67	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#135	1	2	yes	—
73	C₂×Dic₆₁	—	4	4	no	—
74	C₂×Dic₆₁	—	8	2	no	—
75	C₂×Dic₆₁	—	8	2	no	—
76	C₈×C₆₁	—	2	1	no	—
77	C₈×C₆₁	—	2	1	no	—
78	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₆₁	#117	1	1	yes	characteristic
80	C₈×C₆₁	C₄×Q₈	1	2	yes	—
81	#92418	—	4	4	no	—
82	C₄×#365	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	1	no	—
84	C₂²×C₆₁	—	2	1	no	—
85	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
86	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
87	C₄×C₆₁	—	2	1	no	—
88	C₄×C₆₁	—	2	1	no	—
89	C₄×C₆₁	—	2	1	no	—
90	C₄×C₆₁	#365	1	2	yes	complemented
91	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₆₁	#427	1	1	yes	characteristic
93	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
94	Dic₆₁	—	8	4	no	—
95	C₂×#365	—	61	2	no	—
96	C₄×M₃₂	D₆₁	1	1	yes	characteristic
97	C₄×#108	—	61	1	no	—
98	C₄×#106	—	61	1	no	—
99	#1383	—	61	2	no	—
100	C₂×C₆₁	—	2	2	no	—
101	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
102	C₂×C₆₁	C₂×#365	1	1	yes	characteristic
103	C₂×C₆₁	#2155	1	1	yes	characteristic
104	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
105	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
106	C₂×#108	—	61	2	no	—
107	C₂×#106	—	61	2	no	—
108	C₄×M₁₆	—	122	1	no	—
109	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	122	1	no	—
111	#365	—	61	8	no	—
112	#346	Dic₆₁	1	2	yes	complemented
113	C₄×C₁₆	Dic₆₁	1	2	yes	complemented
114	#341	—	122	1	no	—
115	#340	—	122	1	no	—
116	C₆₁	C₄×#365	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	122	2	no	—
118	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	122	2	no	—
121	C₂×C₄²	—	244	1	no	—
122	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	C₄×D₆₁	1	4	yes	—
125	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
127	#108	—	61	4	no	—
128	#108	—	122	2	no	—
129	#106	—	61	4	no	—
130	#106	—	122	2	no	—
131	#97	—	244	1	no	—
132	C₄×C₈	—	244	1	no	—
133	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
134	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
137	#95	—	244	1	no	—
138	C₂²×C₄	—	244	2	no	—
139	C₂²×C₄	#11758190	1	1	yes	characteristic
140	C₂²×C₄	#11758190	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
142	M₁₆	—	122	4	no	—
143	M₁₆	—	244	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	244	2	no	—
146	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₆₁	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
150	C₂×C₈	#11758222	1	2	yes	—
151	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
152	C₄⋊C₄	—	122	4	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄²	—	488	2	no	—
155	C₄²	#11758190	1	1	yes	characteristic
156	C₄²	#11758190	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
158	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	C₄×Dic₆₁	1	4	yes	complemented
161	C₂³	C₂×#11758190	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	488	2	no	—
165	C₂×C₄	—	488	2	no	—
166	C₂×C₄	—	244	4	no	—
167	C₂×C₄	#14519709	1	1	yes	characteristic
168	C₂×C₄	#14519709	1	1	yes	characteristic
169	C₂×C₄	#14519709	1	1	yes	characteristic
170	C₂×C₄	#14519709	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
172	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
173	C₂×C₄	C₂×#11758190	1	1	yes	characteristic
174	C₂×C₄	#14519799	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	244	4	no	—
178	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
179	C₈	#14519736	1	1	yes	characteristic
180	C₈	C₄×Dic₁₂₂	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#14519709	1	1	yes	characteristic
184	C₂²	C₄×#11758190	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	488	4	no	—
189	C₄	#16839585	1	2	yes	direct product factor
190	C₄	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
191	C₄	#16839800	1	1	yes	characteristic
192	C₄	C₄×#11758190	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#14519709	1	1	yes	characteristic
195	C₂	C₂×#16839585	1	1	yes	characteristic
196	C₂	#21447592	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14519709
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂³×C₆₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	C₄×#365
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16839585	C₄	direct (N × H)
16. C₆₁×#346	C₄	semidirect
17. C₄×C₁₆×C₆₁	C₄	semidirect
61. C₁₆×C₆₁	C₄²	semidirect
90. C₄×C₆₁	#365	semidirect
112. #346	Dic₆₁	semidirect
113. C₄×C₁₆	Dic₆₁	semidirect
116. C₆₁	C₄×#365	semidirect
160. C₁₆	C₄×Dic₆₁	semidirect
189. C₄	#16839585	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×#16839585	2
C₄×#14519713	2
C₄×#14519709	2
#21443918	2
C₄×M₃₂×C₆₁	2
C₄×#365	61

Subgroups — C4×#16839585 #29595397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16839585 #29595397