Subgroups — C₂²×#16840159 #30549601

Order 15616.

2398 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	C₂²×#14519776	C₂	1	2	yes	maximal
2	C₂×#16840159	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14519709	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14519708	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₂	—	2	2	no	—
8	C₂³×#92418	C₂²	1	1	yes	characteristic
9	C₂×#14519776	—	2	12	no	—
10	C₂×#14519776	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂²×#11758196	—	2	2	no	—
13	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#16840159	C₂²	1	64	yes	direct product factor
15	C₂×#14519709	C₂²	1	12	yes	—
16	C₂×#14519708	C₂²	1	24	yes	—
17	C₂×C₈×Dic₆₁	C₂²	1	12	yes	—
18	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂×#106×C₆₁	C₂²	1	12	yes	—
21	C₂²×Dic₁₂₂	—	2	24	no	—
22	C₂²×Dic₁₂₂	—	4	16	no	—
23	C₂³×Dic₆₁	—	2	2	no	—
24	C₂³×Dic₆₁	D₄	1	2	yes	—
25	C₂²×#92418	Q₈	1	8	yes	complemented
26	C₂²×#92418	C₂³	1	6	yes	—
27	#14519776	—	2	48	no	—
28	#14519776	C₂³	1	32	yes	—
29	C₂×C₄⋊Dic₆₁	—	2	6	no	—
30	C₂×C₄⋊Dic₆₁	C₂³	1	12	yes	—
31	C₂×#11758196	—	2	24	no	—
32	C₂×C₄×Dic₆₁	D₄	1	6	yes	—
33	C₂×C₄×Dic₆₁	C₂³	1	6	yes	—
34	#14519709	C₂³	1	16	yes	—
35	#14519708	C₂³	1	32	yes	—
36	C₈×Dic₆₁	C₂³	1	16	yes	—
37	C₂³×C₄×C₆₁	—	2	2	no	—
38	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	Q₈	1	8	yes	complemented
40	C₂²×C₈×C₆₁	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₆₁	—	2	6	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂³	1	12	yes	—
43	#106×C₆₁	C₂³	1	16	yes	—
44	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	—	2	12	no	—
46	C₂²×C₄×C₆₁	—	4	8	no	—
47	C₂²×C₄×C₆₁	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₆₁	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	3	yes	—
51	C₂²×Dic₆₁	—	2	12	no	—
52	C₂²×Dic₆₁	—	4	8	no	—
53	C₂²×Dic₆₁	SD₁₆	1	16	yes	—
54	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
55	C₂×Dic₁₂₂	—	2	32	no	—
56	C₂×Dic₁₂₂	—	4	96	no	—
57	C₂×C₈×C₆₁	C₂×Q₈	1	24	yes	complemented
58	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
59	C₄⋊C₄×C₆₁	—	2	12	no	—
60	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
61	C₄⋊Dic₆₁	—	2	12	no	—
62	C₄⋊Dic₆₁	C₂⁴	1	8	yes	—
63	#11758196	—	2	32	no	—
64	C₄×Dic₆₁	C₂×D₄	1	12	yes	—
65	C₄×Dic₆₁	C₂⁴	1	4	yes	—
66	C₂×#92418	C₂×Q₈	1	24	yes	complemented
67	C₂×#92418	C₂⁴	1	4	yes	—
68	C₂³×C₆₁	C₄⋊Q₈	1	4	yes	—
69	C₂³×C₆₁	C₂×SD₁₆	1	8	yes	—
70	C₂³×C₆₁	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₆₁	—	2	8	no	—
72	C₂×C₄×C₆₁	—	4	24	no	—
73	C₂×C₄×C₆₁	C₄⋊Q₈	1	12	yes	—
74	C₂×C₄×C₆₁	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₆₁	C₂²×Q₈	1	6	yes	—
76	C₂×C₄×C₆₁	C₂²×Q₈	1	6	yes	—
77	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₆₁	—	2	8	no	—
79	C₂×Dic₆₁	—	4	24	no	—
80	C₂×Dic₆₁	C₂×SD₁₆	1	48	yes	—
81	C₂×Dic₆₁	C₂²×D₄	1	8	yes	—
82	Dic₁₂₂	—	4	64	no	—
83	C₈×C₆₁	C₂²×Q₈	1	8	yes	complemented
84	#92418	C₂²×Q₈	1	8	yes	complemented
85	C₂²×#498	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	#498	1	16	yes	complemented
87	C₂²×C₆₁	C₂×C₄⋊Q₈	1	6	yes	—
88	C₂²×C₆₁	C₂²×SD₁₆	1	12	yes	—
89	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
90	C₄×C₆₁	—	4	8	no	—
91	C₄×C₆₁	C₂×C₄⋊Q₈	1	6	yes	—
92	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
93	C₄×C₆₁	C₂³×Q₈	1	1	yes	characteristic
94	Dic₆₁	—	4	8	no	—
95	Dic₆₁	C₂²×SD₁₆	1	16	yes	—
96	C₂²×C₄⋊Q₈	—	61	2	no	—
97	C₂×#498	—	61	24	no	—
98	C₂²×#106	—	61	1	no	—
99	C₂²×#106	—	61	2	no	—
100	C₂²×#106	D₆₁	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	61	1	no	—
102	C₂×C₆₁	C₂×#498	1	12	yes	complemented
103	C₂×C₆₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₆₁	C₂³×SD₁₆	1	2	yes	—
105	C₂³×Q₈	—	122	2	no	—
106	C₂³×C₈	—	61	1	no	—
107	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	61	24	no	—
109	C₂×C₄⋊Q₈	—	122	12	no	—
110	C₂²×C₄⋊C₄	—	61	2	no	—
111	C₂²×C₄⋊C₄	—	122	2	no	—
112	C₂²×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
113	C₂²×C₄²	—	61	1	no	—
114	#498	—	61	64	no	—
115	C₂×#106	—	61	12	no	—
116	C₂×#106	—	61	24	no	—
117	C₂×#106	C₂×D₆₁	1	12	yes	—
118	C₂×C₄×C₈	—	61	12	no	—
119	C₆₁	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	122	24	no	—
121	C₂²×Q₈	—	244	16	no	—
122	C₂³×C₄	—	2	2	no	—
123	C₂³×C₄	—	61	2	no	—
124	C₂³×C₄	—	122	2	no	—
125	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
126	C₂²×C₈	—	61	8	no	—
127	C₂²×C₈	—	61	6	no	—
128	C₂²×C₈	Dic₁₂₂	1	8	yes	complemented
129	C₂²×C₈	C₂²×D₆₁	1	6	yes	—
130	C₄⋊Q₈	—	61	32	no	—
131	C₄⋊Q₈	—	122	48	no	—
132	C₂×C₄⋊C₄	—	2	6	no	—
133	C₂×C₄⋊C₄	—	61	12	no	—
134	C₂×C₄⋊C₄	—	122	24	no	—
135	C₂×C₄⋊C₄	—	122	6	no	—
136	C₂×C₄⋊C₄	C₂²×D₆₁	1	12	yes	—
137	C₂×C₄²	—	61	6	no	—
138	C₂×C₄²	—	61	6	no	—
139	#106	—	61	16	no	—
140	#106	—	61	32	no	—
141	#106	C₂²×D₆₁	1	16	yes	—
142	C₄×C₈	—	61	16	no	—
143	C₂⁴	D₄×D₆₁	1	1	yes	characteristic
144	C₂×Q₈	—	122	32	no	—
145	C₂×Q₈	—	244	96	no	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	4	8	no	—
148	C₂²×C₄	—	61	12	no	—
149	C₂²×C₄	—	61	16	no	—
150	C₂²×C₄	—	122	12	no	—
151	C₂²×C₄	—	244	8	no	—
152	C₂²×C₄	C₂×Dic₁₂₂	1	4	yes	—
153	C₂²×C₄	D₄×D₆₁	1	3	yes	—
154	C₂²×C₄	Q₈×D₆₁	1	4	yes	—
155	C₂²×C₄	C₂³×D₆₁	1	3	yes	—
156	C₂×C₈	—	61	24	no	—
157	C₂×C₈	—	61	4	no	—
158	C₂×C₈	C₂×Dic₁₂₂	1	24	yes	complemented
159	C₂×C₈	C₂³×D₆₁	1	4	yes	—
160	C₄⋊C₄	—	2	12	no	—
161	C₄⋊C₄	—	61	8	no	—
162	C₄⋊C₄	—	122	32	no	—
163	C₄⋊C₄	—	122	12	no	—
164	C₄⋊C₄	C₂³×D₆₁	1	8	yes	—
165	C₄²	—	61	12	no	—
166	C₄²	—	61	4	no	—
167	C₂³	#14519776	1	4	yes	—
168	C₂³	SD₁₆×D₆₁	1	8	yes	—
169	C₂³	C₂×D₄×D₆₁	1	3	yes	—
170	Q₈	—	244	64	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	24	no	—
173	C₂×C₄	—	61	48	no	—
174	C₂×C₄	—	61	8	no	—
175	C₂×C₄	—	122	8	no	—
176	C₂×C₄	—	244	24	no	—
177	C₂×C₄	#14519776	1	12	yes	—
178	C₂×C₄	C₂×D₄×D₆₁	1	3	yes	—
179	C₂×C₄	C₂×Q₈×D₆₁	1	6	yes	—
180	C₂×C₄	C₂²×Dic₁₂₂	1	6	yes	—
181	C₂×C₄	C₂⁴×D₆₁	1	1	yes	characteristic
182	C₈	—	61	8	no	—
183	C₈	C₂²×Dic₁₂₂	1	8	yes	complemented
184	C₂²	#16840159	1	16	yes	direct product factor
185	C₂²	C₂×#14519776	1	6	yes	—
186	C₂²	C₂×SD₁₆×D₆₁	1	12	yes	—
187	C₂²	C₂²×D₄×D₆₁	1	1	yes	characteristic
188	C₄	—	4	8	no	—
189	C₄	—	61	16	no	—
190	C₄	—	244	8	no	—
191	C₄	C₂×#14519776	1	6	yes	—
192	C₄	C₂²×Q₈×D₆₁	1	1	yes	characteristic
193	C₄	C₂³×Dic₁₂₂	1	1	yes	characteristic
194	C₂	C₂×#16840159	1	12	yes	direct product factor
195	C₂	C₂²×#14519776	1	1	yes	characteristic
196	C₂	C₂²×SD₁₆×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₆₁
commutator, cosocle	C₂×C₄×C₆₁	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₁
Fitting	C₂²×#106×C₆₁	C₂
socle, prime radical	C₂⁴×C₆₁	C₂×D₄
hypercenter	C₂²×#106	D₆₁
hypocenter	C₆₁	C₂²×#498
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#16840159	C₂	direct (N × H)
14. #16840159	C₂²	direct (N × H)
25. C₂²×#92418	Q₈	semidirect
39. C₂²×C₈×C₆₁	Q₈	semidirect
57. C₂×C₈×C₆₁	C₂×Q₈	semidirect
66. C₂×#92418	C₂×Q₈	semidirect
83. C₈×C₆₁	C₂²×Q₈	semidirect
84. #92418	C₂²×Q₈	semidirect
86. C₂²×C₆₁	#498	semidirect
102. C₂×C₆₁	C₂×#498	semidirect
119. C₆₁	C₂²×#498	semidirect
128. C₂²×C₈	Dic₁₂₂	semidirect
158. C₂×C₈	C₂×Dic₁₂₂	semidirect
183. C₈	C₂²×Dic₁₂₂	semidirect
184. C₂²	#16840159	direct (N × H)
194. C₂	C₂×#16840159	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×#14519776	2
C₂×#16840159	2
C₂²×#14519709	2
C₂²×#14519708	2
C₂²×C₈×Dic₆₁	2
C₂²×#106×C₆₁	2
C₂²×#498	61

Subgroups — C22×#16840159 #30549601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840159 #30549601