Subgroups — C₂³×C₄×Dic₁₂₂ #30586582

Order 15616.

12446 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₂₂	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#11758196	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₆₁	C₂	1	2	yes	maximal
6	C₂³×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₂	C₄	1	32	yes	direct product factor
8	C₂³×Dic₁₂₂	C₂²	1	28	yes	—
9	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
10	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₂₂	C₂²	1	448	yes	direct product factor
12	C₂²×C₄⋊Dic₆₁	C₂²	1	28	yes	—
13	C₂²×#11758196	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₆₁	C₂²	1	56	yes	—
15	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₁	C₂²	1	28	yes	—
19	C₂²×Dic₁₂₂	C₂×C₄	1	448	yes	direct product factor
20	C₂²×Dic₁₂₂	C₂³	1	112	yes	—
21	C₂³×Dic₆₁	—	2	16	no	—
22	C₂³×Dic₆₁	C₂×C₄	1	32	yes	—
23	C₂³×Dic₆₁	C₂³	1	28	yes	—
24	C₂³×Dic₆₁	C₂³	1	28	yes	—
25	C₄×Dic₁₂₂	C₂³	1	512	yes	direct product factor
26	C₂×C₄⋊Dic₆₁	C₂³	1	112	yes	—
27	C₂×#11758196	C₂³	1	224	yes	—
28	C₂×C₄×Dic₆₁	C₂³	1	224	yes	—
29	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₆₁	—	2	8	no	—
31	C₂³×C₄×C₆₁	C₂×C₄	1	16	yes	—
32	C₂³×C₄×C₆₁	Q₈	1	16	yes	complemented
33	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
34	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
35	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
36	C₂×C₄²×C₆₁	C₂³	1	112	yes	—
37	C₂⁴×C₆₁	C₂²×C₄	1	8	yes	—
38	C₂⁴×C₆₁	C₂×Q₈	1	8	yes	—
39	C₂⁴×C₆₁	#41	1	8	yes	—
40	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
41	C₂²×C₄×C₆₁	—	2	56	no	—
42	C₂²×C₄×C₆₁	C₂²×C₄	1	112	yes	—
43	C₂²×C₄×C₆₁	C₂×Q₈	1	112	yes	complemented
44	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
45	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
46	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
47	C₂²×Dic₆₁	—	2	112	no	—
48	C₂²×Dic₆₁	C₂²×C₄	1	224	yes	—
49	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
50	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
51	C₂×Dic₁₂₂	C₂²×C₄	1	896	yes	direct product factor
52	C₂×Dic₁₂₂	C₂⁴	1	64	yes	—
53	C₄²×C₆₁	C₂⁴	1	64	yes	—
54	C₄⋊Dic₆₁	C₂⁴	1	64	yes	—
55	#11758196	C₂⁴	1	128	yes	—
56	C₄×Dic₆₁	C₂⁴	1	128	yes	—
57	C₂³×C₆₁	C₄×Q₈	1	64	yes	complemented
58	C₂³×C₆₁	C₂³×C₄	1	28	yes	—
59	C₂³×C₆₁	C₂²×Q₈	1	28	yes	—
60	C₂³×C₆₁	C₂×#41	1	28	yes	—
61	C₂³×C₆₁	C₂⁵	1	7	yes	—
62	C₂×C₄×C₆₁	—	2	56	no	—
63	C₂×C₄×C₆₁	C₂³×C₄	1	112	yes	—
64	C₂×C₄×C₆₁	C₂²×Q₈	1	112	yes	complemented
65	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
66	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
67	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
68	C₂×Dic₆₁	—	2	112	no	—
69	C₂×Dic₆₁	C₂³×C₄	1	224	yes	—
70	C₂×Dic₆₁	C₂⁵	1	16	yes	—
71	C₂×Dic₆₁	C₂⁵	1	16	yes	—
72	Dic₁₂₂	C₂³×C₄	1	256	yes	direct product factor
73	C₂³×C₄×Q₈	—	61	1	no	maximal, Sylow
74	C₂²×C₆₁	C₂×C₄×Q₈	1	112	yes	complemented
75	C₂²×C₆₁	C₂⁴×C₄	1	14	yes	—
76	C₂²×C₆₁	C₂³×Q₈	1	14	yes	—
77	C₂²×C₆₁	C₂²×#41	1	14	yes	—
78	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
79	C₄×C₆₁	—	2	8	no	—
80	C₄×C₆₁	C₂⁴×C₄	1	16	yes	—
81	C₄×C₆₁	C₂³×Q₈	1	16	yes	complemented
82	Dic₆₁	—	2	16	no	—
83	Dic₆₁	C₂⁴×C₄	1	32	yes	—
84	C₂⁴×Q₈	—	61	1	no	—
85	C₂²×C₄×Q₈	—	61	56	no	—
86	C₂³×C₄⋊C₄	—	61	2	no	—
87	C₂³×C₄⋊C₄	—	61	1	no	—
88	C₂³×C₄²	—	61	2	no	—
89	C₂³×C₄²	D₆₁	1	1	yes	characteristic
90	C₂×C₆₁	C₂²×C₄×Q₈	1	28	yes	complemented
91	C₂×C₆₁	C₂⁵×C₄	1	1	yes	characteristic
92	C₂×C₆₁	C₂⁴×Q₈	1	1	yes	characteristic
93	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
94	C₂³×Q₈	—	61	32	no	—
95	C₂³×Q₈	—	61	28	no	—
96	C₂⁴×C₄	—	61	2	no	—
97	C₂⁴×C₄	—	61	2	no	—
98	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
99	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
100	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
101	C₂×C₄×Q₈	—	61	448	no	—
102	C₂²×C₄⋊C₄	—	61	56	no	—
103	C₂²×C₄⋊C₄	—	61	28	no	—
104	C₂²×C₄²	—	61	56	no	—
105	C₂²×C₄²	C₂×D₆₁	1	28	yes	—
106	C₆₁	C₂³×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
107	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
108	C₂²×Q₈	—	61	448	no	—
109	C₂²×Q₈	—	61	112	no	—
110	C₂³×C₄	—	2	8	no	—
111	C₂³×C₄	—	61	32	no	—
112	C₂³×C₄	—	61	28	no	—
113	C₂³×C₄	—	61	28	no	—
114	C₂³×C₄	—	122	16	no	—
115	C₂³×C₄	Dic₁₂₂	1	16	yes	direct product factor
116	C₂³×C₄	C₄×D₆₁	1	16	yes	—
117	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
118	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
119	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
120	C₄×Q₈	—	61	512	no	—
121	C₂×C₄⋊C₄	—	61	224	no	—
122	C₂×C₄⋊C₄	—	61	112	no	—
123	C₂×C₄²	—	61	224	no	—
124	C₂×C₄²	C₂²×D₆₁	1	112	yes	—
125	C₂⁴	C₂×Dic₁₂₂	1	8	yes	—
126	C₂⁴	C₂×C₄×D₆₁	1	8	yes	—
127	C₂⁴	#11758222	1	8	yes	—
128	C₂⁴	C₂³×D₆₁	1	7	yes	—
129	C₂×Q₈	—	61	896	no	—
130	C₂×Q₈	—	61	64	no	—
131	C₂²×C₄	—	2	56	no	—
132	C₂²×C₄	—	61	224	no	—
133	C₂²×C₄	—	61	56	no	—
134	C₂²×C₄	—	61	56	no	—
135	C₂²×C₄	—	122	112	no	—
136	C₂²×C₄	C₂×Dic₁₂₂	1	112	yes	direct product factor
137	C₂²×C₄	C₂×C₄×D₆₁	1	112	yes	—
138	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
139	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
140	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
141	C₄⋊C₄	—	61	128	no	—
142	C₄⋊C₄	—	61	64	no	—
143	C₄²	—	61	128	no	—
144	C₄²	C₂³×D₆₁	1	64	yes	—
145	C₂³	C₄×Dic₁₂₂	1	64	yes	direct product factor
146	C₂³	C₂²×C₄×D₆₁	1	28	yes	—
147	C₂³	C₂²×Dic₁₂₂	1	28	yes	—
148	C₂³	C₂×#11758222	1	28	yes	—
149	C₂³	C₂⁴×D₆₁	1	7	yes	—
150	Q₈	—	61	256	no	—
151	C₂×C₄	—	2	56	no	—
152	C₂×C₄	—	61	224	no	—
153	C₂×C₄	—	61	16	no	—
154	C₂×C₄	—	61	16	no	—
155	C₂×C₄	—	122	112	no	—
156	C₂×C₄	C₂²×C₄×D₆₁	1	112	yes	—
157	C₂×C₄	C₂²×Dic₁₂₂	1	112	yes	direct product factor
158	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
159	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
160	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
161	C₂²	C₂×C₄×Dic₁₂₂	1	112	yes	direct product factor
162	C₂²	C₂³×C₄×D₆₁	1	14	yes	—
163	C₂²	C₂³×Dic₁₂₂	1	14	yes	—
164	C₂²	C₂²×#11758222	1	14	yes	—
165	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
166	C₄	—	2	8	no	—
167	C₄	—	61	32	no	—
168	C₄	—	122	16	no	—
169	C₄	C₂³×C₄×D₆₁	1	16	yes	—
170	C₄	C₂³×Dic₁₂₂	1	16	yes	direct product factor
171	C₂	C₂²×C₄×Dic₁₂₂	1	28	yes	direct product factor
172	C₂	C₂⁴×C₄×D₆₁	1	1	yes	characteristic
173	C₂	C₂⁴×Dic₁₂₂	1	1	yes	characteristic
174	C₂	C₂³×#11758222	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₆₁
commutator	C₂×C₆₁	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₆₁
Fitting	C₂³×C₄²×C₆₁	C₂
socle, prime radical	C₂⁵×C₆₁	C₂³
cosocle	C₂²×C₆₁	C₂⁶
hypercenter	C₂³×C₄²	D₆₁
hypocenter	C₆₁	C₂³×C₄×Q₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₁₂₂	C₂	direct (N × H)
7. C₂³×Dic₁₂₂	C₄	direct (N × H)
11. C₂×C₄×Dic₁₂₂	C₂²	direct (N × H)
19. C₂²×Dic₁₂₂	C₂×C₄	direct (N × H)
25. C₄×Dic₁₂₂	C₂³	direct (N × H)
32. C₂³×C₄×C₆₁	Q₈	semidirect
43. C₂²×C₄×C₆₁	C₂×Q₈	semidirect
51. C₂×Dic₁₂₂	C₂²×C₄	direct (N × H)
57. C₂³×C₆₁	C₄×Q₈	semidirect
64. C₂×C₄×C₆₁	C₂²×Q₈	semidirect
72. Dic₁₂₂	C₂³×C₄	direct (N × H)
74. C₂²×C₆₁	C₂×C₄×Q₈	semidirect
81. C₄×C₆₁	C₂³×Q₈	semidirect
90. C₂×C₆₁	C₂²×C₄×Q₈	semidirect
106. C₆₁	C₂³×C₄×Q₈	semidirect
115. C₂³×C₄	Dic₁₂₂	direct (N × H)
136. C₂²×C₄	C₂×Dic₁₂₂	direct (N × H)
145. C₂³	C₄×Dic₁₂₂	direct (N × H)
157. C₂×C₄	C₂²×Dic₁₂₂	direct (N × H)
161. C₂²	C₂×C₄×Dic₁₂₂	direct (N × H)
170. C₄	C₂³×Dic₁₂₂	direct (N × H)
171. C₂	C₂²×C₄×Dic₁₂₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₂₂	2
C₂²×C₄×Dic₁₂₂	2
C₂³×C₄⋊Dic₆₁	2
C₂³×#11758196	2
C₂³×C₄×Dic₆₁	2
C₂³×C₄²×C₆₁	2
C₂³×C₄×Q₈	61

Subgroups — C23×C4×Dic122 #30586582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Dic₁₂₂ #30586582