Subgroups — C₂×#21460842 #30605743

Order 15616.

286 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	2	yes	maximal
2	#21460842	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758187	—	2	2	no	—
5	C₄⋊#11758187	—	2	2	no	—
6	C₄⋊#11758187	C₂²	1	4	yes	—
7	C₂×C₄×#92418	C₄	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₈×#92418	C₄	1	2	yes	—
10	C₈×#92418	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758187	—	2	4	no	—
13	C₂×#11758187	—	4	4	no	—
14	C₂²×#92418	C₈	1	1	yes	characteristic
15	C₂²×#92418	C₈	1	1	yes	characteristic
16	C₂²×#92418	D₄	1	1	yes	characteristic
17	C₂²×#92418	Q₈	1	1	yes	characteristic
18	C₄×#92418	C₈	1	2	yes	—
19	C₄×#92418	C₂×C₄	1	1	yes	characteristic
20	C₄×#92418	C₂×C₄	1	2	yes	—
21	C₄×#92418	D₄	1	1	yes	characteristic
22	C₄×#92418	Q₈	1	1	yes	characteristic
23	C₄×#92418	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	—	2	1	no	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	—	2	1	no	—
27	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₆₁	—	2	4	no	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
36	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	C₁₆	1	4	yes	complemented
40	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#92418	C₂×C₈	1	2	yes	—
43	C₂×#92418	C₂×C₈	1	2	yes	—
44	C₂×#92418	SD₁₆	1	4	yes	—
45	C₂×#92418	C₂×D₄	1	1	yes	characteristic
46	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
47	#11758187	—	4	4	no	—
48	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	#106	1	2	yes	—
54	C₂×C₄×C₆₁	#108	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₆₁	—	2	4	no	—
61	#92418	—	2	1	no	—
62	#92418	—	2	1	no	—
63	#92418	—	2	2	no	—
64	#92418	#106	1	2	yes	—
65	#92418	C₂×C₁₆	1	4	yes	complemented
66	#92418	C₂×SD₁₆	1	2	yes	—
67	C₂×#1367	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	#334	1	2	yes	—
69	C₂²×C₆₁	C₄⋊C₁₆	1	2	yes	—
70	C₂²×C₆₁	#364	1	2	yes	—
71	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₆₁	—	2	1	no	—
73	C₄×C₆₁	—	2	1	no	—
74	C₄×C₆₁	#334	1	2	yes	—
75	C₄×C₆₁	C₄⋊C₁₆	1	2	yes	—
76	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
77	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
78	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
79	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
80	C₂×C₄⋊C₁₆	—	61	2	no	—
81	C₂×C₈²	—	61	1	no	—
82	#1367	—	61	4	no	—
83	C₂×C₆₁	#1367	1	4	yes	complemented
84	C₂×C₆₁	C₂×#334	1	1	yes	characteristic
85	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
86	C₂×C₆₁	C₂×#364	1	1	yes	characteristic
87	C₂²×C₁₆	—	122	2	no	—
88	C₂×C₄×C₈	—	61	1	no	—
89	C₂×C₄×C₈	—	61	1	no	—
90	C₂×C₄×C₈	#6923	1	1	yes	characteristic
91	C₄⋊C₁₆	—	61	4	no	—
92	C₄⋊C₁₆	—	122	2	no	—
93	C₈²	—	61	2	no	—
94	C₈²	—	61	2	no	—
95	C₆₁	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	61	1	no	—
98	C₂²×C₈	—	61	1	no	—
99	C₂²×C₈	—	61	1	no	—
100	C₂²×C₈	—	61	1	no	—
101	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
102	C₂×C₁₆	—	122	4	no	—
103	C₂×C₁₆	—	244	4	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	61	2	no	—
106	C₄×C₈	—	61	1	no	—
107	C₄×C₈	—	61	1	no	—
108	C₄×C₈	—	61	1	no	—
109	C₄×C₈	—	61	1	no	—
110	C₄×C₈	—	61	2	no	—
111	C₄×C₈	C₂×#6923	1	2	yes	—
112	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
114	C₂²×C₄	#11758217	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	61	4	no	—
118	C₂×C₈	—	61	4	no	—
119	C₂×C₈	—	61	1	no	—
120	C₂×C₈	—	61	1	no	—
121	C₂×C₈	—	61	2	no	—
122	C₂×C₈	—	61	1	no	—
123	C₂×C₈	—	61	1	no	—
124	C₂×C₈	—	61	2	no	—
125	C₂×C₈	—	122	2	no	—
126	C₂×C₈	—	122	2	no	—
127	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
128	C₄²	C₄⋊#6923	1	1	yes	characteristic
129	C₄²	#11758217	1	1	yes	characteristic
130	C₄²	C₂²×#6923	1	1	yes	characteristic
131	C₁₆	—	244	4	no	—
132	C₂³	#14519844	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	#14519844	1	1	yes	characteristic
135	C₂×C₄	#14519844	1	1	yes	characteristic
136	C₂×C₄	#14519844	1	1	yes	characteristic
137	C₂×C₄	#14519845	1	2	yes	—
138	C₂×C₄	#14519848	1	2	yes	—
139	C₂×C₄	D₆₁⋊C₁₆	1	2	yes	—
140	C₂×C₄	#14519852	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
142	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
143	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	61	2	no	—
146	C₈	—	61	2	no	—
147	C₈	—	61	4	no	—
148	C₈	—	122	1	no	—
149	C₈	—	122	1	no	—
150	C₈	—	122	2	no	—
151	C₂²	#16840624	1	2	yes	—
152	C₂²	#16840683	1	2	yes	—
153	C₂²	#16840685	1	2	yes	—
154	C₂²	C₂×#14519844	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#16840624	1	2	yes	—
158	C₄	#16840683	1	2	yes	—
159	C₄	C₂×#14519845	1	1	yes	characteristic
160	C₄	C₂×#14519848	1	1	yes	characteristic
161	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
162	C₄	C₂×#14519852	1	1	yes	characteristic
163	C₂	#21460842	1	4	yes	direct product factor
164	C₂	C₂×#16840624	1	1	yes	characteristic
165	C₂	C₂×#16840683	1	1	yes	characteristic
166	C₂	C₂×#16840685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₄²	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₄⋊C₈
cosocle	C₄×#92418	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	C₂×#1367
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #21460842	C₂	direct (N × H)
39. C₂×#92418	C₁₆	semidirect
65. #92418	C₂×C₁₆	semidirect
83. C₂×C₆₁	#1367	semidirect
95. C₆₁	C₂×#1367	semidirect
163. C₂	#21460842	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×C₄⋊#11758187	2
#21460842	2
C₂×C₈×#92418	2
C₂×#1367	61

Subgroups — C2×#21460842 #30605743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21460842 #30605743