Subgroups — C₂³×#14519756 #30586656

Order 15616.

14078 conjugacy classes of subgroups.

The subgroup lattice has 208 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, complemented
2	C₂²×#14519756	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11758196	C₂	1	2	yes	maximal, complemented
5	C₂³×C₂²⋊Dic₆₁	C₂	1	2	yes	maximal
6	C₂⁵×C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₂	—	2	16	no	—
8	C₂³×Dic₁₂₂	C₂²	1	28	yes	complemented
9	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
10	C₂⁴×Dic₆₁	C₂²	1	2	yes	—
11	C₂×#14519756	C₂²	1	448	yes	direct product factor
12	C₂²×C₄⋊Dic₆₁	C₂²	1	28	yes	complemented
13	C₂²×#11758196	C₂²	1	56	yes	complemented
14	C₂²×C₂²⋊Dic₆₁	C₂²	1	56	yes	—
15	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	—	2	16	no	—
17	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄×C₆₁	C₂²	1	28	yes	—
19	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₂₂	—	2	224	no	—
21	C₂²×Dic₁₂₂	C₂³	1	112	yes	complemented
22	C₂³×Dic₆₁	—	2	16	no	—
23	C₂³×Dic₆₁	—	2	16	no	—
24	C₂³×Dic₆₁	C₂³	1	28	yes	—
25	C₂³×Dic₆₁	C₂³	1	28	yes	—
26	#14519756	C₂³	1	512	yes	direct product factor
27	C₂×C₄⋊Dic₆₁	C₂³	1	112	yes	complemented
28	C₂×#11758196	C₂³	1	224	yes	complemented
29	C₂×C₂²⋊Dic₆₁	C₂³	1	224	yes	—
30	C₂⁵×C₆₁	—	2	8	no	—
31	C₂⁵×C₆₁	—	2	8	no	—
32	C₂⁵×C₆₁	Q₈	1	16	yes	complemented
33	C₂⁵×C₆₁	C₂³	1	14	yes	—
34	C₂⁵×C₆₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₆₁	—	2	8	no	—
36	C₂³×C₄×C₆₁	—	2	224	no	—
37	C₂³×C₄×C₆₁	D₄	1	16	yes	—
38	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
39	C₂³×C₄×C₆₁	C₂³	1	112	yes	—
40	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
41	C₂⁴×C₆₁	—	2	128	no	—
42	C₂⁴×C₆₁	—	2	56	no	—
43	C₂⁴×C₆₁	—	2	56	no	—
44	C₂⁴×C₆₁	C₂×D₄	1	8	yes	—
45	C₂⁴×C₆₁	C₂×Q₈	1	8	yes	—
46	C₂⁴×C₆₁	C₂×Q₈	1	112	yes	complemented
47	C₂⁴×C₆₁	#41	1	8	yes	—
48	C₂⁴×C₆₁	C₂⁴	1	28	yes	—
49	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
50	C₂²×C₄×C₆₁	—	2	56	no	—
51	C₂²×C₄×C₆₁	—	2	448	no	—
52	C₂²×C₄×C₆₁	C₂×D₄	1	112	yes	—
53	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
54	C₂²×C₄×C₆₁	C₂⁴	1	64	yes	—
55	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
56	C₂²×Dic₆₁	—	2	112	no	—
57	C₂²×Dic₆₁	—	2	112	no	—
58	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
59	C₂²×Dic₆₁	C₂⁴	1	56	yes	—
60	C₂×Dic₁₂₂	—	2	448	no	—
61	C₂×Dic₁₂₂	C₂⁴	1	64	yes	complemented
62	C₂²⋊Dic₆₁	C₂⁴	1	128	yes	—
63	C₄⋊Dic₆₁	C₂⁴	1	64	yes	complemented
64	#11758196	C₂⁴	1	128	yes	complemented
65	C₂³×C₆₁	—	2	448	no	—
66	C₂³×C₆₁	—	2	56	no	—
67	C₂³×C₆₁	—	2	56	no	—
68	C₂³×C₆₁	C₂²⋊Q₈	1	64	yes	complemented
69	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
70	C₂³×C₆₁	C₂²×Q₈	1	28	yes	—
71	C₂³×C₆₁	C₂²×Q₈	1	112	yes	complemented
72	C₂³×C₆₁	C₂×#41	1	28	yes	—
73	C₂³×C₆₁	C₂⁵	1	8	yes	—
74	C₂³×C₆₁	C₂⁵	1	7	yes	—
75	C₂×C₄×C₆₁	—	2	56	no	—
76	C₂×C₄×C₆₁	—	2	128	no	—
77	C₂×C₄×C₆₁	C₂²×D₄	1	112	yes	—
78	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
79	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
80	C₂×Dic₆₁	—	2	112	no	—
81	C₂×Dic₆₁	—	2	112	no	—
82	C₂×Dic₆₁	C₂⁵	1	16	yes	—
83	C₂×Dic₆₁	C₂⁵	1	16	yes	—
84	Dic₁₂₂	—	2	128	no	—
85	C₂³×C₂²⋊Q₈	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	—	2	8	no	—
87	C₂²×C₆₁	—	2	8	no	—
88	C₂²×C₆₁	—	2	224	no	—
89	C₂²×C₆₁	C₂×C₂²⋊Q₈	1	112	yes	complemented
90	C₂²×C₆₁	C₂³×D₄	1	14	yes	—
91	C₂²×C₆₁	C₂³×Q₈	1	14	yes	—
92	C₂²×C₆₁	C₂³×Q₈	1	16	yes	complemented
93	C₂²×C₆₁	C₂²×#41	1	14	yes	—
94	C₂²×C₆₁	C₂⁶	1	1	yes	characteristic
95	C₄×C₆₁	—	2	8	no	—
96	C₄×C₆₁	C₂³×D₄	1	16	yes	—
97	Dic₆₁	—	2	16	no	—
98	Dic₆₁	—	2	16	no	—
99	C₂⁴×Q₈	—	61	1	no	—
100	C₂⁵×C₄	D₆₁	1	1	yes	characteristic
101	C₂²×C₂²⋊Q₈	—	61	56	no	—
102	C₂³×C₄⋊C₄	—	61	2	no	—
103	C₂³×C₄⋊C₄	—	61	1	no	—
104	C₂³×C₂²⋊C₄	—	61	2	no	—
105	C₂×C₆₁	—	2	16	no	—
106	C₂×C₆₁	C₂²×C₂²⋊Q₈	1	28	yes	complemented
107	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
108	C₂×C₆₁	C₂⁴×Q₈	1	1	yes	characteristic
109	C₂×C₆₁	C₂³×#41	1	1	yes	characteristic
110	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
111	C₂³×Q₈	—	61	28	no	—
112	C₂³×Q₈	—	122	16	no	—
113	C₂⁴×C₄	—	2	16	no	—
114	C₂⁴×C₄	—	61	2	no	—
115	C₂⁴×C₄	—	61	2	no	—
116	C₂⁴×C₄	C₂×D₆₁	1	28	yes	—
117	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
118	C₂⁴×C₄	C₂×D₆₁	1	1	yes	characteristic
119	C₂×C₂²⋊Q₈	—	61	448	no	—
120	C₂²×C₄⋊C₄	—	61	56	no	—
121	C₂²×C₄⋊C₄	—	61	28	no	—
122	C₂²×C₂²⋊C₄	—	61	56	no	—
123	C₆₁	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
124	C₂⁵	—	2	8	no	—
125	C₂⁵	—	2	8	no	—
126	C₂⁵	Dic₁₂₂	1	16	yes	complemented
127	C₂⁵	C₂²×D₆₁	1	14	yes	—
128	C₂⁵	C₂²×D₆₁	1	1	yes	characteristic
129	C₂²×Q₈	—	61	112	no	—
130	C₂²×Q₈	—	122	224	no	—
131	C₂³×C₄	—	2	8	no	—
132	C₂³×C₄	—	2	224	no	—
133	C₂³×C₄	—	61	28	no	—
134	C₂³×C₄	—	61	28	no	—
135	C₂³×C₄	—	122	16	no	—
136	C₂³×C₄	—	122	16	no	—
137	C₂³×C₄	C₂²⋊D₆₁	1	16	yes	—
138	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
139	C₂³×C₄	C₂²×D₆₁	1	112	yes	—
140	C₂³×C₄	C₂²×D₆₁	1	14	yes	—
141	C₂²⋊Q₈	—	61	512	no	—
142	C₂×C₄⋊C₄	—	61	224	no	—
143	C₂×C₄⋊C₄	—	61	112	no	—
144	C₂×C₂²⋊C₄	—	61	224	no	—
145	C₂⁴	—	2	56	no	—
146	C₂⁴	—	2	56	no	—
147	C₂⁴	—	2	128	no	—
148	C₂⁴	C₂×Dic₁₂₂	1	8	yes	—
149	C₂⁴	C₂×Dic₁₂₂	1	112	yes	complemented
150	C₂⁴	#11758222	1	8	yes	—
151	C₂⁴	C₂×C₂²⋊D₆₁	1	8	yes	—
152	C₂⁴	C₂³×D₆₁	1	28	yes	—
153	C₂⁴	C₂³×D₆₁	1	7	yes	—
154	C₂×Q₈	—	61	64	no	—
155	C₂×Q₈	—	122	448	no	—
156	C₂²×C₄	—	2	56	no	—
157	C₂²×C₄	—	2	448	no	—
158	C₂²×C₄	—	61	56	no	—
159	C₂²×C₄	—	61	56	no	—
160	C₂²×C₄	—	122	112	no	—
161	C₂²×C₄	—	122	112	no	—
162	C₂²×C₄	C₂×C₂²⋊D₆₁	1	112	yes	—
163	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
164	C₂²×C₄	C₂³×D₆₁	1	64	yes	—
165	C₂²×C₄	C₂³×D₆₁	1	28	yes	—
166	C₄⋊C₄	—	61	128	no	—
167	C₄⋊C₄	—	61	64	no	—
168	C₂²⋊C₄	—	61	128	no	—
169	C₂³	—	2	448	no	—
170	C₂³	—	2	56	no	—
171	C₂³	—	2	56	no	—
172	C₂³	#14519756	1	64	yes	direct product factor
173	C₂³	C₂²×C₂²⋊D₆₁	1	28	yes	—
174	C₂³	C₂²×Dic₁₂₂	1	28	yes	—
175	C₂³	C₂²×Dic₁₂₂	1	112	yes	complemented
176	C₂³	C₂×#11758222	1	28	yes	—
177	C₂³	C₂⁴×D₆₁	1	8	yes	—
178	C₂³	C₂⁴×D₆₁	1	7	yes	—
179	Q₈	—	122	128	no	—
180	C₂×C₄	—	2	56	no	—
181	C₂×C₄	—	2	128	no	—
182	C₂×C₄	—	61	16	no	—
183	C₂×C₄	—	61	16	no	—
184	C₂×C₄	—	122	112	no	—
185	C₂×C₄	—	122	112	no	—
186	C₂×C₄	C₂²×C₂²⋊D₆₁	1	112	yes	—
187	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
188	C₂×C₄	C₂⁴×D₆₁	1	8	yes	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	8	no	—
191	C₂²	—	2	8	no	—
192	C₂²	C₂×#14519756	1	112	yes	direct product factor
193	C₂²	C₂³×C₂²⋊D₆₁	1	14	yes	—
194	C₂²	C₂³×Dic₁₂₂	1	14	yes	—
195	C₂²	C₂³×Dic₁₂₂	1	16	yes	complemented
196	C₂²	C₂²×#11758222	1	14	yes	—
197	C₂²	C₂⁵×D₆₁	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	122	16	no	—
200	C₄	—	122	16	no	—
201	C₄	C₂³×C₂²⋊D₆₁	1	16	yes	—
202	C₂	—	2	16	no	—
203	C₂	C₂²×#14519756	1	28	yes	direct product factor
204	C₂	C₂⁴×C₂²⋊D₆₁	1	1	yes	characteristic
205	C₂	C₂⁴×Dic₁₂₂	1	1	yes	characteristic
206	C₂	C₂³×#11758222	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₂₂	C₂	semidirect
2. C₂²×#14519756	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₆₁	C₂	semidirect
4. C₂³×#11758196	C₂	semidirect
8. C₂³×Dic₁₂₂	C₂²	semidirect
11. C₂×#14519756	C₂²	direct (N × H)
12. C₂²×C₄⋊Dic₆₁	C₂²	semidirect
13. C₂²×#11758196	C₂²	semidirect
21. C₂²×Dic₁₂₂	C₂³	semidirect
26. #14519756	C₂³	direct (N × H)
27. C₂×C₄⋊Dic₆₁	C₂³	semidirect
28. C₂×#11758196	C₂³	semidirect
32. C₂⁵×C₆₁	Q₈	semidirect
46. C₂⁴×C₆₁	C₂×Q₈	semidirect
61. C₂×Dic₁₂₂	C₂⁴	semidirect
63. C₄⋊Dic₆₁	C₂⁴	semidirect
64. #11758196	C₂⁴	semidirect
68. C₂³×C₆₁	C₂²⋊Q₈	semidirect
71. C₂³×C₆₁	C₂²×Q₈	semidirect
89. C₂²×C₆₁	C₂×C₂²⋊Q₈	semidirect
92. C₂²×C₆₁	C₂³×Q₈	semidirect
106. C₂×C₆₁	C₂²×C₂²⋊Q₈	semidirect
123. C₆₁	C₂³×C₂²⋊Q₈	semidirect
126. C₂⁵	Dic₁₂₂	semidirect
149. C₂⁴	C₂×Dic₁₂₂	semidirect
172. C₂³	#14519756	direct (N × H)
175. C₂³	C₂²×Dic₁₂₂	semidirect
192. C₂²	C₂×#14519756	direct (N × H)
195. C₂²	C₂³×Dic₁₂₂	semidirect
203. C₂	C₂²×#14519756	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×#14519756	2
C₂³×C₄⋊Dic₆₁	2
C₂³×#11758196	2
C₂³×C₂²⋊Dic₆₁	2
C₂⁵×C₄×C₆₁	2
C₂³×C₂²⋊Q₈	61

Subgroups — C23×#14519756 #30586656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519756 #30586656