Subgroups — C₂²×#16840624 #30611995

Order 15616.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#92420	C₂	1	2	yes	maximal
2	C₂×#16840624	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#92420	—	2	2	no	—
5	C₂×C₄⋊#92420	—	2	6	no	—
6	C₂×C₄⋊#92420	C₂²	1	12	yes	—
7	#16840624	C₂²	1	16	yes	direct product factor
8	C₂³×#92418	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	6	yes	—
11	C₂×C₈×Dic₆₁	C₂²	1	6	yes	—
12	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
13	C₂²×#92420	—	2	12	no	—
14	C₂²×#92420	—	4	8	no	—
15	C₄⋊#92420	—	2	12	no	—
16	C₄⋊#92420	C₂³	1	8	yes	—
17	C₂³×Dic₆₁	D₄	1	1	yes	characteristic
18	C₂³×Dic₆₁	Q₈	1	1	yes	characteristic
19	C₂²×#92418	C₈	1	8	yes	complemented
20	C₂²×#92418	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
22	C₂×C₄×Dic₆₁	D₄	1	3	yes	—
23	C₂×C₄×Dic₆₁	Q₈	1	3	yes	—
24	C₂×C₄×Dic₆₁	C₂³	1	3	yes	—
25	C₈×Dic₆₁	C₂×C₄	1	12	yes	—
26	C₈×Dic₆₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	—	2	4	no	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
30	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₄⋊C₄	1	3	yes	—
32	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
33	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
35	C₂²×Dic₆₁	—	2	4	no	—
36	C₂²×Dic₆₁	C₄⋊C₄	1	3	yes	—
37	C₂²×Dic₆₁	C₄⋊C₄	1	3	yes	—
38	C₂²×Dic₆₁	SD₁₆	1	8	yes	—
39	C₂²×Dic₆₁	C₂×D₄	1	3	yes	—
40	C₂²×Dic₆₁	C₂×Q₈	1	3	yes	—
41	C₂×#92420	—	2	8	no	—
42	C₂×#92420	—	4	24	no	—
43	C₂×C₈×C₆₁	—	2	12	no	—
44	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
45	C₄×Dic₆₁	C₄⋊C₄	1	6	yes	—
46	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
47	C₄×Dic₆₁	C₂×D₄	1	3	yes	—
48	C₄×Dic₆₁	C₂×Q₈	1	3	yes	—
49	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×#92418	C₂×C₈	1	24	yes	complemented
51	C₂×#92418	C₂²×C₄	1	4	yes	—
52	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
53	C₂³×C₆₁	#106	1	4	yes	—
54	C₂³×C₆₁	#108	1	4	yes	—
55	C₂³×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₆₁	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
59	C₂×C₄×C₆₁	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₆₁	—	2	6	no	—
62	C₂×Dic₆₁	—	2	6	no	—
63	C₂×Dic₆₁	#106	1	12	yes	—
64	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×Dic₆₁	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×Dic₆₁	C₂×SD₁₆	1	12	yes	—
67	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
68	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
69	#92420	—	4	8	no	—
70	C₈×C₆₁	—	2	4	no	—
71	#92418	C₂²×C₈	1	8	yes	complemented
72	C₂²×#334	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	#334	1	16	yes	complemented
74	C₂²×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
75	C₂²×C₆₁	C₂×#106	1	6	yes	—
76	C₂²×C₆₁	C₂×#108	1	6	yes	—
77	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₆₁	C₂×C₄⋊C₈	1	6	yes	—
79	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
81	Dic₆₁	—	2	3	no	—
82	Dic₆₁	—	2	1	no	—
83	Dic₆₁	C₂×#106	1	6	yes	—
84	Dic₆₁	C₂²×SD₁₆	1	2	yes	—
85	C₂²×C₄⋊C₈	—	61	2	no	—
86	C₂²×C₄×C₈	—	61	1	no	—
87	C₂×#334	—	61	12	no	—
88	C₂×C₆₁	C₂×#334	1	12	yes	complemented
89	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
90	C₂×C₆₁	C₂²×#106	1	1	yes	characteristic
91	C₂×C₆₁	C₂²×#108	1	1	yes	characteristic
92	C₂³×C₈	—	61	1	no	—
93	C₂³×C₈	—	122	2	no	—
94	C₂³×C₈	#6923	1	1	yes	characteristic
95	C₂²×C₄²	—	61	1	no	—
96	C₂×C₄⋊C₈	—	61	12	no	—
97	C₂×C₄⋊C₈	—	122	6	no	—
98	C₂×C₄×C₈	—	61	6	no	—
99	C₂×C₄×C₈	—	61	6	no	—
100	#334	—	61	16	no	—
101	C₆₁	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
102	C₂³×C₄	—	61	1	no	—
103	C₂³×C₄	—	61	1	no	—
104	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	61	8	no	—
107	C₂²×C₈	—	61	6	no	—
108	C₂²×C₈	—	122	12	no	—
109	C₂²×C₈	—	244	8	no	—
110	C₂²×C₈	C₂×#6923	1	6	yes	—
111	C₂×C₄²	—	61	3	no	—
112	C₂×C₄²	—	61	3	no	—
113	C₂×C₄²	—	61	3	no	—
114	C₂×C₄²	—	61	3	no	—
115	C₄⋊C₈	—	61	8	no	—
116	C₄⋊C₈	—	122	12	no	—
117	C₄×C₈	—	61	4	no	—
118	C₄×C₈	—	61	12	no	—
119	C₂⁴	C₄⋊#6923	1	1	yes	characteristic
120	C₂²×C₄	—	61	8	no	—
121	C₂²×C₄	—	61	3	no	—
122	C₂²×C₄	—	61	3	no	—
123	C₂²×C₄	—	61	3	no	—
124	C₂²×C₄	—	61	3	no	—
125	C₂²×C₄	—	122	4	no	—
126	C₂²×C₄	D₆₁⋊C₈	1	4	yes	—
127	C₂²×C₄	C₄⋊#6923	1	3	yes	—
128	C₂²×C₄	#11758217	1	4	yes	—
129	C₂²×C₄	C₂²×#6923	1	3	yes	—
130	C₂×C₈	—	2	12	no	—
131	C₂×C₈	—	61	24	no	—
132	C₂×C₈	—	61	4	no	—
133	C₂×C₈	—	122	8	no	—
134	C₂×C₈	—	244	24	no	—
135	C₂×C₈	C₂²×#6923	1	4	yes	—
136	C₄²	—	61	3	no	—
137	C₄²	—	61	6	no	—
138	C₄²	—	61	3	no	—
139	C₄²	—	61	1	no	—
140	C₄²	—	61	3	no	—
141	C₂³	#14519844	1	4	yes	—
142	C₂³	#14519845	1	4	yes	—
143	C₂³	#14519848	1	4	yes	—
144	C₂³	C₂×C₄⋊#6923	1	3	yes	—
145	C₂×C₄	—	61	12	no	—
146	C₂×C₄	—	61	12	no	—
147	C₂×C₄	—	61	3	no	—
148	C₂×C₄	—	61	1	no	—
149	C₂×C₄	—	61	3	no	—
150	C₂×C₄	—	61	1	no	—
151	C₂×C₄	—	122	6	no	—
152	C₂×C₄	—	122	6	no	—
153	C₂×C₄	#14519844	1	12	yes	—
154	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
155	C₂×C₄	C₂×D₆₁⋊C₈	1	6	yes	—
156	C₂×C₄	C₂×#11758217	1	6	yes	—
157	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	61	8	no	—
160	C₈	—	244	8	no	—
161	C₂²	#16840624	1	16	yes	direct product factor
162	C₂²	C₂×#14519844	1	6	yes	—
163	C₂²	C₂×#14519845	1	6	yes	—
164	C₂²	C₂×#14519848	1	6	yes	—
165	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
166	C₄	—	61	2	no	—
167	C₄	—	61	6	no	—
168	C₄	—	122	3	no	—
169	C₄	—	122	1	no	—
170	C₄	C₂×#14519844	1	6	yes	—
171	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
172	C₄	C₂²×#11758217	1	1	yes	characteristic
173	C₂	C₂×#16840624	1	12	yes	direct product factor
174	C₂	C₂²×#14519844	1	1	yes	characteristic
175	C₂	C₂²×#14519845	1	1	yes	characteristic
176	C₂	C₂²×#14519848	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₄⋊#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle, prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂²×#334
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16840624	C₂	direct (N × H)
7. #16840624	C₂²	direct (N × H)
19. C₂²×#92418	C₈	semidirect
50. C₂×#92418	C₂×C₈	semidirect
71. #92418	C₂²×C₈	semidirect
73. C₂²×C₆₁	#334	semidirect
88. C₂×C₆₁	C₂×#334	semidirect
101. C₆₁	C₂²×#334	semidirect
161. C₂²	#16840624	direct (N × H)
173. C₂	C₂×#16840624	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#92420	2
C₂×#16840624	2
C₂²×C₈×Dic₆₁	2
C₂²×#334	61

Subgroups — C22×#16840624 #30611995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840624 #30611995