Subgroups — C₂³×(C₂×#6923)⋊C₄ #30624535

Order 15616.

14334 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×#6923	C₂	1	2	yes	maximal
2	C₂²×(C₂×#6923)⋊C₄	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×#6923	—	2	16	no	—
5	C₂⁴×#6923	—	2	16	no	—
6	C₂⁴×#6923	C₄	1	32	yes	complemented
7	C₂⁴×#6923	C₂²	1	28	yes	—
8	C₂×(C₂×#6923)⋊C₄	C₂²	1	112	yes	direct product factor
9	C₂⁵×D₆₁	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₆₁	C₄	1	1	yes	characteristic, complemented
11	C₂³×C₄×D₆₁	—	2	16	no	—
12	C₂³×C₄×D₆₁	C₄	1	14	yes	complemented
13	C₂³×C₄×D₆₁	C₂²	1	14	yes	—
14	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₂³×#6923	—	2	256	no	—
16	C₂³×#6923	—	2	112	no	—
17	C₂³×#6923	—	2	112	no	—
18	C₂³×#6923	C₂×C₄	1	224	yes	complemented
19	C₂³×#6923	C₂³	1	56	yes	—
20	(C₂×#6923)⋊C₄	C₂³	1	64	yes	direct product factor
21	C₂⁴×D₆₁	C₂×C₄	1	7	yes	—
22	C₂⁴×D₆₁	C₂×C₄	1	16	yes	—
23	C₂⁴×D₆₁	D₄	1	8	yes	—
24	C₂⁴×D₆₁	D₄	1	16	yes	—
25	C₂⁴×D₆₁	Q₈	1	8	yes	—
26	C₂⁴×D₆₁	C₂³	1	7	yes	—
27	C₂³×Dic₆₁	—	2	8	no	—
28	C₂³×Dic₆₁	C₂×C₄	1	14	yes	complemented
29	C₂²×C₄×D₆₁	—	2	112	no	—
30	C₂²×C₄×D₆₁	—	2	112	no	—
31	C₂²×C₄×D₆₁	C₂×C₄	1	84	yes	complemented
32	C₂²×C₄×D₆₁	C₂³	1	28	yes	—
33	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₁	—	2	8	no	—
35	C₂³×C₄×C₆₁	C₂×C₄	1	14	yes	complemented
36	C₂⁴×C₆₁	C₄²	1	8	yes	—
37	C₂⁴×C₆₁	C₂²⋊C₄	1	8	yes	—
38	C₂⁴×C₆₁	C₄⋊C₄	1	8	yes	—
39	C₂⁴×C₆₁	C₂²×C₄	1	7	yes	—
40	C₂³×D₆₁	C₄²	1	56	yes	—
41	C₂³×D₆₁	C₂²⋊C₄	1	56	yes	—
42	C₂³×D₆₁	C₂²⋊C₄	1	128	yes	—
43	C₂³×D₆₁	C₄⋊C₄	1	56	yes	—
44	C₂³×D₆₁	C₄⋊C₄	1	128	yes	—
45	C₂³×D₆₁	C₂²×C₄	1	21	yes	—
46	C₂³×D₆₁	C₂²×C₄	1	56	yes	—
47	C₂³×D₆₁	C₂×D₄	1	28	yes	—
48	C₂³×D₆₁	C₂×D₄	1	56	yes	—
49	C₂³×D₆₁	C₂×Q₈	1	28	yes	—
50	C₂³×D₆₁	C₂⁴	1	7	yes	—
51	C₂²×#6923	—	2	896	no	—
52	C₂²×#6923	—	2	112	no	—
53	C₂²×#6923	—	2	112	no	—
54	C₂²×#6923	C₂²×C₄	1	224	yes	complemented
55	C₂²×#6923	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₆₁	—	2	56	no	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	28	yes	complemented
58	C₂²×Dic₆₁	—	2	56	no	—
59	C₂²×Dic₆₁	C₂²×C₄	1	28	yes	complemented
60	C₂×C₄×D₆₁	—	2	336	no	—
61	C₂×C₄×D₆₁	—	2	112	no	—
62	C₂×C₄×D₆₁	C₂²×C₄	1	56	yes	complemented
63	C₂×C₄×D₆₁	C₂⁴	1	8	yes	—
64	C₂³×C₆₁	#95	1	64	yes	complemented
65	C₂³×C₆₁	C₂×C₄²	1	28	yes	—
66	C₂³×C₆₁	C₂×C₂²⋊C₄	1	28	yes	—
67	C₂³×C₆₁	C₂×C₄⋊C₄	1	28	yes	—
68	C₂³×C₆₁	C₂³×C₄	1	7	yes	—
69	C₂²×D₆₁	#95	1	448	yes	complemented
70	C₂²×D₆₁	C₂×C₄²	1	84	yes	—
71	C₂²×D₆₁	C₂×C₂²⋊C₄	1	224	yes	—
72	C₂²×D₆₁	C₂×C₂²⋊C₄	1	84	yes	—
73	C₂²×D₆₁	C₂×C₄⋊C₄	1	224	yes	—
74	C₂²×D₆₁	C₂×C₄⋊C₄	1	84	yes	—
75	C₂²×D₆₁	C₂³×C₄	1	28	yes	—
76	C₂²×D₆₁	C₂³×C₄	1	7	yes	—
77	C₂²×D₆₁	C₂²×D₄	1	14	yes	—
78	C₂²×D₆₁	C₂²×D₄	1	28	yes	—
79	C₂²×D₆₁	C₂²×Q₈	1	14	yes	—
80	C₂²×D₆₁	C₂⁵	1	1	yes	characteristic
81	C₂×#6923	—	2	448	no	—
82	C₂×#6923	—	2	16	no	—
83	C₂×#6923	—	2	16	no	—
84	C₂×#6923	C₂³×C₄	1	32	yes	complemented
85	C₂×C₄×C₆₁	—	2	56	no	—
86	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	complemented
87	C₂×Dic₆₁	—	2	56	no	—
88	C₂×Dic₆₁	C₂³×C₄	1	8	yes	complemented
89	C₄×D₆₁	—	2	112	no	—
90	C₄×D₆₁	—	2	16	no	—
91	C₂³×#95	—	61	1	no	maximal, Sylow
92	C₂²×C₆₁	C₂×#95	1	112	yes	complemented
93	C₂²×C₆₁	C₂²×C₄²	1	14	yes	—
94	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	14	yes	—
95	C₂²×C₆₁	C₂²×C₄⋊C₄	1	14	yes	—
96	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
97	C₂×D₆₁	C₂×#95	1	336	yes	complemented
98	C₂×D₆₁	C₂²×C₄²	1	14	yes	—
99	C₂×D₆₁	C₂²×C₂²⋊C₄	1	56	yes	—
100	C₂×D₆₁	C₂²×C₂²⋊C₄	1	14	yes	—
101	C₂×D₆₁	C₂²×C₄⋊C₄	1	56	yes	—
102	C₂×D₆₁	C₂²×C₄⋊C₄	1	14	yes	—
103	C₂×D₆₁	C₂⁴×C₄	1	2	yes	—
104	C₂×D₆₁	C₂³×D₄	1	1	yes	characteristic
105	C₂×D₆₁	C₂³×D₄	1	2	yes	—
106	C₂×D₆₁	C₂³×Q₈	1	1	yes	characteristic
107	#6923	—	2	32	no	—
108	C₄×C₆₁	—	2	8	no	—
109	Dic₆₁	—	2	8	no	—
110	C₂⁵×C₄	—	61	2	no	—
111	C₂⁵×C₄	—	61	1	no	—
112	C₂²×#95	—	61	28	no	—
113	C₂×C₆₁	C₂²×#95	1	28	yes	complemented
114	C₂×C₆₁	C₂³×C₄²	1	1	yes	characteristic
115	C₂×C₆₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₆₁	C₂³×C₄⋊C₄	1	1	yes	characteristic
117	D₆₁	C₂²×#95	1	28	yes	complemented
118	D₆₁	C₂³×C₂²⋊C₄	1	2	yes	—
119	D₆₁	C₂³×C₄⋊C₄	1	2	yes	—
120	C₂⁶	—	61	1	no	—
121	C₂⁴×C₄	—	61	32	no	—
122	C₂⁴×C₄	—	61	28	no	—
123	C₂⁴×C₄	—	61	1	no	—
124	C₂⁴×C₄	—	61	14	no	—
125	C₂⁴×C₄	—	61	14	no	—
126	C₂⁴×C₄	—	122	16	no	—
127	C₂⁴×C₄	—	122	16	no	—
128	C₂⁴×C₄	—	122	16	no	—
129	C₂⁴×C₄	#6923	1	1	yes	characteristic, complemented
130	C₂×#95	—	61	112	no	—
131	C₆₁	C₂³×#95	1	1	yes	characteristic, complemented, Sylow
132	C₂⁵	—	61	16	no	—
133	C₂⁵	—	61	8	no	—
134	C₂⁵	—	61	8	no	—
135	C₂⁵	—	61	16	no	—
136	C₂⁵	—	61	7	no	—
137	C₂⁵	—	61	7	no	—
138	C₂⁵	C₂×#6923	1	1	yes	characteristic
139	C₂³×C₄	—	2	8	no	—
140	C₂³×C₄	—	61	224	no	—
141	C₂³×C₄	—	61	56	no	—
142	C₂³×C₄	—	61	14	no	—
143	C₂³×C₄	—	61	84	no	—
144	C₂³×C₄	—	61	28	no	—
145	C₂³×C₄	—	122	112	no	—
146	C₂³×C₄	—	122	112	no	—
147	C₂³×C₄	—	122	256	no	—
148	C₂³×C₄	—	122	112	no	—
149	C₂³×C₄	—	122	112	no	—
150	C₂³×C₄	—	122	8	no	—
151	C₂³×C₄	C₂×#6923	1	14	yes	complemented
152	#95	—	61	64	no	—
153	C₂⁴	—	61	128	no	—
154	C₂⁴	—	61	128	no	—
155	C₂⁴	—	61	56	no	—
156	C₂⁴	—	61	56	no	—
157	C₂⁴	—	61	56	no	—
158	C₂⁴	—	61	56	no	—
159	C₂⁴	—	61	28	no	—
160	C₂⁴	—	61	56	no	—
161	C₂⁴	—	61	28	no	—
162	C₂⁴	—	61	7	no	—
163	C₂⁴	—	61	21	no	—
164	C₂⁴	C₄×#6923	1	8	yes	—
165	C₂⁴	C₄⋊#6923	1	8	yes	—
166	C₂⁴	C₂²⋊#6923	1	8	yes	—
167	C₂⁴	C₂²×#6923	1	7	yes	—
168	C₂²×C₄	—	2	56	no	—
169	C₂²×C₄	—	61	16	no	—
170	C₂²×C₄	—	61	224	no	—
171	C₂²×C₄	—	61	28	no	—
172	C₂²×C₄	—	61	8	no	—
173	C₂²×C₄	—	61	56	no	—
174	C₂²×C₄	—	122	336	no	—
175	C₂²×C₄	—	122	112	no	—
176	C₂²×C₄	—	122	112	no	—
177	C₂²×C₄	—	122	112	no	—
178	C₂²×C₄	—	122	896	no	—
179	C₂²×C₄	—	122	56	no	—
180	C₂²×C₄	C₂²×#6923	1	28	yes	complemented
181	C₂³	—	61	84	no	—
182	C₂³	—	61	28	no	—
183	C₂³	—	61	84	no	—
184	C₂³	—	61	28	no	—
185	C₂³	—	61	84	no	—
186	C₂³	—	61	14	no	—
187	C₂³	—	61	14	no	—
188	C₂³	—	61	7	no	—
189	C₂³	—	61	1	no	—
190	C₂³	—	61	448	no	—
191	C₂³	—	61	224	no	—
192	C₂³	—	61	224	no	—
193	C₂³	(C₂×#6923)⋊C₄	1	64	yes	direct product factor
194	C₂³	C₂×C₄×#6923	1	28	yes	—
195	C₂³	C₂×C₂²⋊#6923	1	28	yes	—
196	C₂³	C₂×C₄⋊#6923	1	28	yes	—
197	C₂³	C₂³×#6923	1	7	yes	—
198	C₂×C₄	—	2	56	no	—
199	C₂×C₄	—	61	32	no	—
200	C₂×C₄	—	61	8	no	—
201	C₂×C₄	—	122	16	no	—
202	C₂×C₄	—	122	112	no	—
203	C₂×C₄	—	122	16	no	—
204	C₂×C₄	—	122	16	no	—
205	C₂×C₄	—	122	448	no	—
206	C₂×C₄	—	122	56	no	—
207	C₂×C₄	C₂³×#6923	1	8	yes	complemented
208	C₂²	—	61	2	no	—
209	C₂²	—	61	14	no	—
210	C₂²	—	61	2	no	—
211	C₂²	—	61	14	no	—
212	C₂²	—	61	1	no	—
213	C₂²	—	61	14	no	—
214	C₂²	—	61	1	no	—
215	C₂²	—	61	56	no	—
216	C₂²	—	61	56	no	—
217	C₂²	—	61	336	no	—
218	C₂²	C₂×(C₂×#6923)⋊C₄	1	112	yes	direct product factor
219	C₂²	C₂²×C₄×#6923	1	14	yes	—
220	C₂²	C₂²×C₂²⋊#6923	1	14	yes	—
221	C₂²	C₂²×C₄⋊#6923	1	14	yes	—
222	C₂²	C₂⁴×#6923	1	1	yes	characteristic
223	C₄	—	2	8	no	—
224	C₄	—	122	32	no	—
225	C₄	—	122	8	no	—
226	C₂	—	61	2	no	—
227	C₂	—	61	2	no	—
228	C₂	—	61	28	no	—
229	C₂	C₂²×(C₂×#6923)⋊C₄	1	28	yes	direct product factor
230	C₂	C₂³×C₄×#6923	1	1	yes	characteristic
231	C₂	C₂³×C₂²⋊#6923	1	1	yes	characteristic
232	C₂	C₂³×C₄⋊#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₄²
Frattini	C₂²	C₂⁴×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle	C₂⁵×C₆₁	C₂×C₄
prime radical	C₂⁵×D₆₁	C₂²
cosocle	C₂²×D₆₁	C₂⁵
hypercenter	C₂⁴×C₄	#6923
hypocenter	C₆₁	C₂³×#95
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×(C₂×#6923)⋊C₄	C₂	direct (N × H)
6. C₂⁴×#6923	C₄	semidirect
8. C₂×(C₂×#6923)⋊C₄	C₂²	direct (N × H)
10. C₂⁴×Dic₆₁	C₄	semidirect
12. C₂³×C₄×D₆₁	C₄	semidirect
14. C₂⁴×C₄×C₆₁	C₄	semidirect
18. C₂³×#6923	C₂×C₄	semidirect
20. (C₂×#6923)⋊C₄	C₂³	direct (N × H)
28. C₂³×Dic₆₁	C₂×C₄	semidirect
31. C₂²×C₄×D₆₁	C₂×C₄	semidirect
35. C₂³×C₄×C₆₁	C₂×C₄	semidirect
54. C₂²×#6923	C₂²×C₄	semidirect
57. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
59. C₂²×Dic₆₁	C₂²×C₄	semidirect
62. C₂×C₄×D₆₁	C₂²×C₄	semidirect
64. C₂³×C₆₁	#95	semidirect
69. C₂²×D₆₁	#95	semidirect
84. C₂×#6923	C₂³×C₄	semidirect
86. C₂×C₄×C₆₁	C₂³×C₄	semidirect
88. C₂×Dic₆₁	C₂³×C₄	semidirect
92. C₂²×C₆₁	C₂×#95	semidirect
97. C₂×D₆₁	C₂×#95	semidirect
113. C₂×C₆₁	C₂²×#95	semidirect
117. D₆₁	C₂²×#95	semidirect
129. C₂⁴×C₄	#6923	semidirect
131. C₆₁	C₂³×#95	semidirect
151. C₂³×C₄	C₂×#6923	semidirect
180. C₂²×C₄	C₂²×#6923	semidirect
193. C₂³	(C₂×#6923)⋊C₄	direct (N × H)
207. C₂×C₄	C₂³×#6923	semidirect
218. C₂²	C₂×(C₂×#6923)⋊C₄	direct (N × H)
229. C₂	C₂²×(C₂×#6923)⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×#6923	2
C₂²×(C₂×#6923)⋊C₄	2
C₂⁴×C₄×D₆₁	2
C₂³×#95	61

Subgroups — C23×(C2×#6923)⋊C4 #30624535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×(C₂×#6923)⋊C₄ #30624535