Subgroups — C₂³×#14519824 #30593902

Order 15616.

59806 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519824	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#11758222	C₂	1	3	yes	maximal, complemented
3	C₂³×#11758224	C₂	1	6	yes	maximal, complemented
4	C₂⁴×Dic₁₂₂	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×D₆₁	C₂	1	2	yes	maximal, complemented
6	C₂³×#41×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519824	C₂²	1	1792	yes	direct product factor
8	C₂²×#11758222	C₂²	1	168	yes	complemented
9	C₂²×#11758224	C₂²	1	336	yes	complemented
10	C₂³×Dic₁₂₂	C₂²	1	6	yes	—
11	C₂³×Dic₁₂₂	C₂²	1	3	yes	—
12	C₂³×Dic₁₂₂	C₂²	1	168	yes	complemented
13	C₂²×Q₈×D₆₁	C₂²	1	112	yes	complemented
14	C₂³×D₂₄₄	C₂²	1	1	yes	characteristic
15	C₂³×C₂²⋊D₆₁	C₂²	1	6	yes	—
16	C₂⁴×Dic₆₁	C₂²	1	6	yes	—
17	C₂³×C₄×D₆₁	C₂²	1	6	yes	—
18	C₂²×#41×C₆₁	C₂²	1	56	yes	complemented
19	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂³×D₄×C₆₁	C₂²	1	3	yes	—
21	C₂⁴×C₄×C₆₁	C₂²	1	3	yes	—
22	C₂⁴×D₆₁	C₂³	1	2	yes	—
23	#14519824	C₂³	1	4096	yes	direct product factor
24	C₂×#11758222	C₂³	1	1344	yes	complemented
25	C₂×#11758224	C₂³	1	2688	yes	complemented
26	C₂²×Dic₁₂₂	C₂³	1	168	yes	—
27	C₂²×Dic₁₂₂	C₂³	1	84	yes	—
28	C₂²×Dic₁₂₂	C₂³	1	1344	yes	complemented
29	C₂×Q₈×D₆₁	C₂³	1	896	yes	complemented
30	C₂²×D₂₄₄	C₂³	1	28	yes	—
31	C₂²×C₂²⋊D₆₁	C₂³	1	168	yes	—
32	C₂³×Dic₆₁	C₂³	1	6	yes	—
33	C₂³×Dic₆₁	C₂³	1	168	yes	—
34	C₂²×C₄×D₆₁	C₂³	1	168	yes	—
35	C₂⁵×C₆₁	C₂³	1	3	yes	—
36	C₂×#41×C₆₁	C₂³	1	448	yes	complemented
37	C₂²×Q₈×C₆₁	C₂³	1	28	yes	—
38	C₂²×D₄×C₆₁	C₂³	1	84	yes	—
39	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
41	C₂³×C₄×C₆₁	C₂³	1	84	yes	—
42	C₂⁴×C₆₁	—	2	24	no	—
43	C₂⁴×C₆₁	C₂⁴	1	42	yes	—
44	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
45	C₂³×D₆₁	—	2	16	no	—
46	C₂³×D₆₁	C₂⁴	1	28	yes	—
47	#41×C₆₁	C₂⁴	1	512	yes	complemented
48	C₂×Q₈×C₆₁	C₂⁴	1	112	yes	—
49	C₂×D₄×C₆₁	C₂⁴	1	336	yes	—
50	C₂²×C₄×C₆₁	C₂⁴	1	42	yes	—
51	C₂²×C₄×C₆₁	C₂⁴	1	14	yes	—
52	C₂²×C₄×C₆₁	C₂⁴	1	336	yes	—
53	C₂×C₂²⋊D₆₁	C₂⁴	1	672	yes	—
54	C₂²×Dic₆₁	C₂⁴	1	84	yes	—
55	C₂²×Dic₆₁	C₂⁴	1	672	yes	—
56	Q₈×D₆₁	C₂⁴	1	1024	yes	complemented
57	#11758224	C₂⁴	1	3072	yes	complemented
58	#11758222	C₂⁴	1	1536	yes	complemented
59	C₂×D₂₄₄	C₂⁴	1	112	yes	—
60	C₂×C₄×D₆₁	C₂⁴	1	672	yes	—
61	C₂×Dic₁₂₂	C₂⁴	1	672	yes	—
62	C₂×Dic₁₂₂	C₂⁴	1	336	yes	—
63	C₂×Dic₁₂₂	C₂⁴	1	1536	yes	complemented
64	C₂³×C₆₁	—	2	168	no	—
65	C₂³×C₆₁	2₋¹⁺⁴	1	8	yes	complemented
66	C₂³×C₆₁	C₂⁵	1	84	yes	—
67	C₂³×C₆₁	C₂⁵	1	7	yes	—
68	C₂²×D₆₁	—	2	112	no	—
69	C₂²×D₆₁	C₂⁵	1	56	yes	—
70	Q₈×C₆₁	C₂⁵	1	64	yes	—
71	D₄×C₆₁	C₂⁵	1	192	yes	—
72	C₂×C₄×C₆₁	C₂⁵	1	28	yes	—
73	C₂×C₄×C₆₁	C₂⁵	1	84	yes	—
74	C₂×C₄×C₆₁	C₂⁵	1	192	yes	—
75	C₂²⋊D₆₁	C₂⁵	1	384	yes	—
76	C₂×Dic₆₁	C₂⁵	1	168	yes	—
77	C₂×Dic₆₁	C₂⁵	1	384	yes	—
78	D₂₄₄	C₂⁵	1	64	yes	—
79	C₄×D₆₁	C₂⁵	1	384	yes	—
80	Dic₁₂₂	C₂⁵	1	192	yes	—
81	Dic₁₂₂	C₂⁵	1	384	yes	—
82	C₂³×2₋¹⁺⁴	—	61	1	no	maximal, Sylow
83	C₂²×C₆₁	—	2	168	no	—
84	C₂²×C₆₁	C₂×2₋¹⁺⁴	1	28	yes	complemented
85	C₂²×C₆₁	C₂⁶	1	24	yes	—
86	C₂²×C₆₁	C₂⁶	1	7	yes	—
87	C₂×D₆₁	—	2	112	no	—
88	C₂×D₆₁	C₂⁶	1	16	yes	—
89	C₄×C₆₁	C₂⁶	1	8	yes	—
90	C₄×C₆₁	C₂⁶	1	24	yes	—
91	Dic₆₁	C₂⁶	1	48	yes	—
92	C₂²×2₋¹⁺⁴	—	61	112	no	—
93	C₂³×#41	—	61	6	no	—
94	C₂³×#41	—	61	3	no	—
95	C₂³×#41	D₆₁	1	1	yes	characteristic, complemented
96	C₂⁴×Q₈	—	61	2	no	—
97	C₂⁴×Q₈	—	61	3	no	—
98	C₂×C₆₁	—	2	24	no	—
99	C₂×C₆₁	C₂²×2₋¹⁺⁴	1	14	yes	complemented
100	C₂×C₆₁	C₂⁷	1	1	yes	characteristic
101	D₆₁	—	2	16	no	—
102	C₂×2₋¹⁺⁴	—	61	1792	no	—
103	C₂²×#41	—	61	336	no	—
104	C₂²×#41	—	61	168	no	—
105	C₂²×#41	C₂×D₆₁	1	56	yes	complemented
106	C₂³×Q₈	—	61	112	no	—
107	C₂³×Q₈	—	61	6	no	—
108	C₂³×Q₈	—	61	3	no	—
109	C₂³×Q₈	—	61	168	no	—
110	C₂³×Q₈	C₂×D₆₁	1	1	yes	characteristic
111	C₂³×D₄	—	61	6	no	—
112	C₂³×D₄	—	61	1	no	—
113	C₂³×D₄	C₂×D₆₁	1	3	yes	—
114	C₂⁴×C₄	—	61	6	no	—
115	C₂⁴×C₄	—	61	6	no	—
116	C₂⁴×C₄	C₂×D₆₁	1	3	yes	—
117	C₆₁	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
118	C₂⁵	—	61	2	no	—
119	C₂⁵	C₂²×D₆₁	1	3	yes	—
120	2₋¹⁺⁴	—	61	4096	no	—
121	C₂×#41	—	61	2688	no	—
122	C₂×#41	—	61	1344	no	—
123	C₂×#41	C₂²×D₆₁	1	448	yes	complemented
124	C₂²×Q₈	—	61	896	no	—
125	C₂²×Q₈	—	61	168	no	—
126	C₂²×Q₈	—	61	84	no	—
127	C₂²×Q₈	—	61	1344	no	—
128	C₂²×Q₈	C₂²×D₆₁	1	28	yes	—
129	C₂²×D₄	—	61	168	no	—
130	C₂²×D₄	—	61	28	no	—
131	C₂²×D₄	C₂²×D₆₁	1	84	yes	—
132	C₂³×C₄	—	61	168	no	—
133	C₂³×C₄	—	61	6	no	—
134	C₂³×C₄	—	61	168	no	—
135	C₂³×C₄	C₂²×D₆₁	1	84	yes	—
136	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
137	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
138	C₂⁴	—	2	24	no	—
139	C₂⁴	—	61	28	no	—
140	C₂⁴	—	122	16	no	—
141	C₂⁴	C₂³×D₆₁	1	42	yes	—
142	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
143	#41	—	61	3072	no	—
144	#41	—	61	1536	no	—
145	#41	C₂³×D₆₁	1	512	yes	complemented
146	C₂×Q₈	—	61	1024	no	—
147	C₂×Q₈	—	61	672	no	—
148	C₂×Q₈	—	61	336	no	—
149	C₂×Q₈	—	61	1536	no	—
150	C₂×Q₈	C₂³×D₆₁	1	112	yes	—
151	C₂×D₄	—	61	672	no	—
152	C₂×D₄	—	61	112	no	—
153	C₂×D₄	C₂³×D₆₁	1	336	yes	—
154	C₂²×C₄	—	61	84	no	—
155	C₂²×C₄	—	61	672	no	—
156	C₂²×C₄	—	61	672	no	—
157	C₂²×C₄	C₂³×D₆₁	1	42	yes	—
158	C₂²×C₄	C₂³×D₆₁	1	14	yes	—
159	C₂²×C₄	C₂³×D₆₁	1	336	yes	—
160	C₂³	—	2	168	no	—
161	C₂³	—	61	56	no	—
162	C₂³	—	122	112	no	—
163	C₂³	#14519824	1	8	yes	direct product factor
164	C₂³	C₂⁴×D₆₁	1	84	yes	—
165	C₂³	C₂⁴×D₆₁	1	7	yes	—
166	Q₈	—	61	384	no	—
167	Q₈	—	61	192	no	—
168	Q₈	C₂⁴×D₆₁	1	64	yes	—
169	D₄	—	61	384	no	—
170	D₄	—	61	64	no	—
171	D₄	C₂⁴×D₆₁	1	192	yes	—
172	C₂×C₄	—	61	168	no	—
173	C₂×C₄	—	61	384	no	—
174	C₂×C₄	—	61	384	no	—
175	C₂×C₄	C₂⁴×D₆₁	1	84	yes	—
176	C₂×C₄	C₂⁴×D₆₁	1	28	yes	—
177	C₂×C₄	C₂⁴×D₆₁	1	192	yes	—
178	C₂²	—	2	168	no	—
179	C₂²	—	61	16	no	—
180	C₂²	—	122	112	no	—
181	C₂²	C₂×#14519824	1	28	yes	direct product factor
182	C₂²	C₂⁵×D₆₁	1	24	yes	—
183	C₂²	C₂⁵×D₆₁	1	7	yes	—
184	C₄	—	61	48	no	—
185	C₄	C₂⁵×D₆₁	1	24	yes	—
186	C₄	C₂⁵×D₆₁	1	8	yes	—
187	C₂	—	2	24	no	—
188	C₂	—	122	16	no	—
189	C₂	C₂²×#14519824	1	14	yes	direct product factor
190	C₂	C₂⁶×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂×C₆₁	C₂⁷
Frattini	C₂	C₂⁶×D₆₁
Fitting	C₂³×#41×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂³×#41	D₆₁
hypocenter	C₆₁	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#14519824	C₂	direct (N × H)
2. C₂³×#11758222	C₂	semidirect
3. C₂³×#11758224	C₂	semidirect
4. C₂⁴×Dic₁₂₂	C₂	semidirect
5. C₂³×Q₈×D₆₁	C₂	semidirect
6. C₂³×#41×C₆₁	C₂	semidirect
7. C₂×#14519824	C₂²	direct (N × H)
8. C₂²×#11758222	C₂²	semidirect
9. C₂²×#11758224	C₂²	semidirect
12. C₂³×Dic₁₂₂	C₂²	semidirect
13. C₂²×Q₈×D₆₁	C₂²	semidirect
18. C₂²×#41×C₆₁	C₂²	semidirect
23. #14519824	C₂³	direct (N × H)
24. C₂×#11758222	C₂³	semidirect
25. C₂×#11758224	C₂³	semidirect
28. C₂²×Dic₁₂₂	C₂³	semidirect
29. C₂×Q₈×D₆₁	C₂³	semidirect
36. C₂×#41×C₆₁	C₂³	semidirect
47. #41×C₆₁	C₂⁴	semidirect
56. Q₈×D₆₁	C₂⁴	semidirect
57. #11758224	C₂⁴	semidirect
58. #11758222	C₂⁴	semidirect
63. C₂×Dic₁₂₂	C₂⁴	semidirect
65. C₂³×C₆₁	2₋¹⁺⁴	semidirect
84. C₂²×C₆₁	C₂×2₋¹⁺⁴	semidirect
95. C₂³×#41	D₆₁	semidirect
99. C₂×C₆₁	C₂²×2₋¹⁺⁴	semidirect
105. C₂²×#41	C₂×D₆₁	semidirect
117. C₆₁	C₂³×2₋¹⁺⁴	semidirect
123. C₂×#41	C₂²×D₆₁	semidirect
145. #41	C₂³×D₆₁	semidirect
163. C₂³	#14519824	direct (N × H)
181. C₂²	C₂×#14519824	direct (N × H)
189. C₂	C₂²×#14519824	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×#14519824	2
C₂³×#11758222	2
C₂³×#11758224	2
C₂⁴×Dic₁₂₂	2
C₂³×Q₈×D₆₁	2
C₂³×#41×C₆₁	2
C₂³×2₋¹⁺⁴	61

Subgroups — C23×#14519824 #30593902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519824 #30593902