Subgroups — C₂³×#14519692 #30543474

Order 15616.

7902 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊Dic₆₁	C₂	1	2	yes	maximal, complemented
2	C₂²×#14519692	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₆₁	—	2	2	no	—
5	C₂²×C₂²⋊Dic₆₁	—	2	14	no	—
6	C₂²×C₂²⋊Dic₆₁	C₂²	1	28	yes	complemented
7	C₂×#14519692	C₂²	1	112	yes	direct product factor
8	C₂⁶×C₆₁	C₄	1	1	yes	characteristic, complemented
9	C₂⁶×C₆₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₁	—	2	2	no	—
11	C₂³×D₄×C₆₁	—	2	14	no	—
12	C₂³×D₄×C₆₁	C₄	1	14	yes	complemented
13	C₂³×D₄×C₆₁	C₂²	1	14	yes	—
14	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic, complemented
15	C₂³×Dic₆₁	—	2	28	no	—
16	C₂³×Dic₆₁	—	4	16	no	—
17	C₂×C₂²⋊Dic₆₁	—	2	84	no	—
18	C₂×C₂²⋊Dic₆₁	C₂³	1	56	yes	complemented
19	#14519692	C₂³	1	64	yes	direct product factor
20	C₂⁵×C₆₁	—	2	7	no	—
21	C₂⁵×C₆₁	—	2	1	no	—
22	C₂⁵×C₆₁	—	4	8	no	—
23	C₂⁵×C₆₁	—	4	8	no	—
24	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂⁵×C₆₁	C₂×C₄	1	7	yes	—
26	C₂⁵×C₆₁	C₂×C₄	1	14	yes	complemented
27	C₂⁵×C₆₁	D₄	1	2	yes	—
28	C₂⁵×C₆₁	D₄	1	14	yes	—
29	C₂⁵×C₆₁	C₂³	1	7	yes	—
30	C₂²×D₄×C₆₁	—	2	28	no	—
31	C₂²×D₄×C₆₁	—	2	84	no	—
32	C₂²×D₄×C₆₁	—	2	28	no	—
33	C₂²×D₄×C₆₁	—	2	84	no	—
34	C₂²×D₄×C₆₁	C₂×C₄	1	84	yes	complemented
35	C₂²×D₄×C₆₁	C₂³	1	28	yes	—
36	C₂³×C₄×C₆₁	—	2	7	no	—
37	C₂³×C₄×C₆₁	—	2	1	no	—
38	C₂³×C₄×C₆₁	C₂×C₄	1	14	yes	complemented
39	C₂⁴×C₆₁	—	2	16	no	—
40	C₂⁴×C₆₁	—	2	8	no	—
41	C₂⁴×C₆₁	—	2	42	no	—
42	C₂⁴×C₆₁	—	2	14	no	—
43	C₂⁴×C₆₁	—	4	28	no	—
44	C₂⁴×C₆₁	—	4	56	no	—
45	C₂⁴×C₆₁	—	4	28	no	—
46	C₂⁴×C₆₁	—	4	56	no	—
47	C₂⁴×C₆₁	—	4	56	no	—
48	C₂⁴×C₆₁	—	4	8	no	—
49	C₂⁴×C₆₁	C₂²⋊C₄	1	7	yes	—
50	C₂⁴×C₆₁	C₂²⋊C₄	1	14	yes	—
51	C₂⁴×C₆₁	C₂²⋊C₄	1	42	yes	—
52	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂⁴×C₆₁	C₂²×C₄	1	7	yes	—
54	C₂⁴×C₆₁	C₂²×C₄	1	21	yes	—
55	C₂⁴×C₆₁	C₂²×C₄	1	28	yes	complemented
56	C₂⁴×C₆₁	C₂×D₄	1	14	yes	—
57	C₂⁴×C₆₁	C₂×D₄	1	42	yes	—
58	C₂⁴×C₆₁	C₂⁴	1	7	yes	—
59	C₂×D₄×C₆₁	—	2	168	no	—
60	C₂×D₄×C₆₁	—	2	168	no	—
61	C₂×D₄×C₆₁	—	2	56	no	—
62	C₂×D₄×C₆₁	—	2	56	no	—
63	C₂×D₄×C₆₁	C₂²×C₄	1	56	yes	complemented
64	C₂×D₄×C₆₁	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₆₁	—	2	42	no	—
66	C₂²×C₄×C₆₁	—	2	14	no	—
67	C₂²×C₄×C₆₁	C₂²×C₄	1	28	yes	complemented
68	C₂²×Dic₆₁	—	2	56	no	—
69	C₂²×Dic₆₁	—	4	112	no	—
70	C₂²⋊Dic₆₁	—	2	56	no	—
71	C₂²⋊Dic₆₁	C₂⁴	1	16	yes	complemented
72	C₂³×C₆₁	—	2	56	no	—
73	C₂³×C₆₁	—	2	28	no	—
74	C₂³×C₆₁	—	2	28	no	—
75	C₂³×C₆₁	—	2	56	no	—
76	C₂³×C₆₁	—	2	28	no	—
77	C₂³×C₆₁	—	2	28	no	—
78	C₂³×C₆₁	—	4	84	no	—
79	C₂³×C₆₁	—	4	42	no	—
80	C₂³×C₆₁	—	4	84	no	—
81	C₂³×C₆₁	—	4	14	no	—
82	C₂³×C₆₁	—	4	168	no	—
83	C₂³×C₆₁	—	4	56	no	—
84	C₂³×C₆₁	—	4	56	no	—
85	C₂³×C₆₁	#99	1	8	yes	complemented
86	C₂³×C₆₁	C₂×C₂²⋊C₄	1	21	yes	—
87	C₂³×C₆₁	C₂×C₂²⋊C₄	1	42	yes	—
88	C₂³×C₆₁	C₂×C₂²⋊C₄	1	42	yes	—
89	C₂³×C₆₁	C₂×C₂²⋊C₄	1	7	yes	—
90	C₂³×C₆₁	C₂³×C₄	1	7	yes	—
91	C₂³×C₆₁	C₂³×C₄	1	7	yes	—
92	C₂³×C₆₁	C₂³×C₄	1	8	yes	complemented
93	C₂³×C₆₁	C₂²×D₄	1	14	yes	—
94	C₂³×C₆₁	C₂²×D₄	1	14	yes	—
95	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
96	D₄×C₆₁	—	2	112	no	—
97	D₄×C₆₁	—	2	16	no	—
98	C₂×C₄×C₆₁	—	2	28	no	—
99	C₂×C₄×C₆₁	—	2	28	no	—
100	C₂×C₄×C₆₁	C₂³×C₄	1	8	yes	complemented
101	C₂×Dic₆₁	—	2	16	no	—
102	C₂×Dic₆₁	—	4	112	no	—
103	C₂³×#99	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	—	2	28	no	—
105	C₂²×C₆₁	—	2	42	no	—
106	C₂²×C₆₁	—	2	14	no	—
107	C₂²×C₆₁	—	2	84	no	—
108	C₂²×C₆₁	—	2	8	no	—
109	C₂²×C₆₁	—	4	42	no	—
110	C₂²×C₆₁	—	4	7	no	—
111	C₂²×C₆₁	—	4	14	no	—
112	C₂²×C₆₁	—	4	1	no	—
113	C₂²×C₆₁	—	4	8	no	—
114	C₂²×C₆₁	—	4	56	no	—
115	C₂²×C₆₁	—	4	56	no	—
116	C₂²×C₆₁	C₂×#99	1	28	yes	complemented
117	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	7	yes	—
118	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	14	yes	—
119	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	7	yes	—
120	C₂²×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
121	C₂²×C₆₁	C₂³×D₄	1	2	yes	—
122	C₄×C₆₁	—	2	8	no	—
123	Dic₆₁	—	4	16	no	—
124	C₂⁴×D₄	D₆₁	1	1	yes	characteristic
125	C₂³×C₂²⋊C₄	—	61	2	no	—
126	C₂²×#99	—	61	28	no	—
127	C₂×C₆₁	—	2	2	no	—
128	C₂×C₆₁	—	2	7	no	—
129	C₂×C₆₁	—	2	1	no	—
130	C₂×C₆₁	—	2	14	no	—
131	C₂×C₆₁	—	4	8	no	—
132	C₂×C₆₁	C₂²×#99	1	14	yes	complemented
133	C₂×C₆₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
134	C₂⁶	Dic₆₁	1	1	yes	characteristic, complemented
135	C₂⁶	C₂×D₆₁	1	1	yes	characteristic
136	C₂³×D₄	—	2	14	no	—
137	C₂³×D₄	—	2	2	no	—
138	C₂³×D₄	Dic₆₁	1	14	yes	complemented
139	C₂³×D₄	C₂×D₆₁	1	14	yes	—
140	C₂⁴×C₄	—	122	2	no	—
141	C₂⁴×C₄	Dic₆₁	1	1	yes	characteristic, complemented
142	C₂²×C₂²⋊C₄	—	61	28	no	—
143	C₂²×C₂²⋊C₄	—	122	14	no	—
144	C₂×#99	—	61	112	no	—
145	C₆₁	C₂³×#99	1	1	yes	characteristic, complemented, Sylow
146	C₂⁵	—	2	7	no	—
147	C₂⁵	—	2	1	no	—
148	C₂⁵	—	4	8	no	—
149	C₂⁵	—	4	8	no	—
150	C₂⁵	C₂×Dic₆₁	1	1	yes	characteristic
151	C₂⁵	C₂×Dic₆₁	1	7	yes	—
152	C₂⁵	C₂×Dic₆₁	1	14	yes	complemented
153	C₂⁵	C₂²⋊D₆₁	1	2	yes	—
154	C₂⁵	C₂²⋊D₆₁	1	14	yes	—
155	C₂⁵	C₂²×D₆₁	1	7	yes	—
156	C₂²×D₄	—	2	28	no	—
157	C₂²×D₄	—	2	84	no	—
158	C₂²×D₄	—	2	28	no	—
159	C₂²×D₄	—	2	84	no	—
160	C₂²×D₄	C₂×Dic₆₁	1	84	yes	complemented
161	C₂²×D₄	C₂²×D₆₁	1	28	yes	—
162	C₂³×C₄	—	2	7	no	—
163	C₂³×C₄	—	2	1	no	—
164	C₂³×C₄	—	122	28	no	—
165	C₂³×C₄	—	244	16	no	—
166	C₂³×C₄	C₂×Dic₆₁	1	14	yes	complemented
167	C₂×C₂²⋊C₄	—	61	56	no	—
168	C₂×C₂²⋊C₄	—	122	84	no	—
169	#99	—	61	64	no	—
170	C₂⁴	—	2	16	no	—
171	C₂⁴	—	2	8	no	—
172	C₂⁴	—	2	42	no	—
173	C₂⁴	—	2	14	no	—
174	C₂⁴	—	4	28	no	—
175	C₂⁴	—	4	56	no	—
176	C₂⁴	—	4	28	no	—
177	C₂⁴	—	4	56	no	—
178	C₂⁴	—	4	56	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	C₂²⋊Dic₆₁	1	7	yes	—
181	C₂⁴	C₂²⋊Dic₆₁	1	14	yes	—
182	C₂⁴	C₂²⋊Dic₆₁	1	42	yes	—
183	C₂⁴	C₂²⋊Dic₆₁	1	1	yes	characteristic
184	C₂⁴	C₂²×Dic₆₁	1	7	yes	—
185	C₂⁴	C₂²×Dic₆₁	1	21	yes	—
186	C₂⁴	C₂²×Dic₆₁	1	28	yes	complemented
187	C₂⁴	C₂×C₂²⋊D₆₁	1	14	yes	—
188	C₂⁴	C₂×C₂²⋊D₆₁	1	42	yes	—
189	C₂⁴	C₂³×D₆₁	1	7	yes	—
190	C₂×D₄	—	2	56	no	—
191	C₂×D₄	—	2	56	no	—
192	C₂×D₄	—	2	168	no	—
193	C₂×D₄	—	2	168	no	—
194	C₂×D₄	C₂²×Dic₆₁	1	56	yes	complemented
195	C₂×D₄	C₂³×D₆₁	1	8	yes	—
196	C₂²×C₄	—	2	42	no	—
197	C₂²×C₄	—	2	14	no	—
198	C₂²×C₄	—	122	56	no	—
199	C₂²×C₄	—	244	112	no	—
200	C₂²×C₄	C₂²×Dic₆₁	1	28	yes	complemented
201	C₂²⋊C₄	—	61	16	no	—
202	C₂²⋊C₄	—	122	56	no	—
203	C₂³	—	2	56	no	—
204	C₂³	—	2	28	no	—
205	C₂³	—	2	28	no	—
206	C₂³	—	2	56	no	—
207	C₂³	—	2	28	no	—
208	C₂³	—	2	28	no	—
209	C₂³	—	4	84	no	—
210	C₂³	—	4	42	no	—
211	C₂³	—	4	84	no	—
212	C₂³	—	4	14	no	—
213	C₂³	—	4	56	no	—
214	C₂³	—	4	168	no	—
215	C₂³	—	4	56	no	—
216	C₂³	#14519692	1	8	yes	direct product factor
217	C₂³	C₂×C₂²⋊Dic₆₁	1	21	yes	—
218	C₂³	C₂×C₂²⋊Dic₆₁	1	42	yes	—
219	C₂³	C₂×C₂²⋊Dic₆₁	1	42	yes	—
220	C₂³	C₂×C₂²⋊Dic₆₁	1	7	yes	—
221	C₂³	C₂³×Dic₆₁	1	7	yes	—
222	C₂³	C₂³×Dic₆₁	1	7	yes	—
223	C₂³	C₂³×Dic₆₁	1	8	yes	complemented
224	C₂³	C₂²×C₂²⋊D₆₁	1	14	yes	—
225	C₂³	C₂²×C₂²⋊D₆₁	1	14	yes	—
226	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
227	D₄	—	2	112	no	—
228	D₄	—	2	16	no	—
229	C₂×C₄	—	2	28	no	—
230	C₂×C₄	—	2	28	no	—
231	C₂×C₄	—	122	16	no	—
232	C₂×C₄	—	244	112	no	—
233	C₂×C₄	C₂³×Dic₆₁	1	8	yes	complemented
234	C₂²	—	2	28	no	—
235	C₂²	—	2	14	no	—
236	C₂²	—	2	42	no	—
237	C₂²	—	2	84	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	7	no	—
240	C₂²	—	4	14	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	42	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	56	no	—
245	C₂²	—	4	56	no	—
246	C₂²	C₂×#14519692	1	28	yes	direct product factor
247	C₂²	C₂²×C₂²⋊Dic₆₁	1	7	yes	—
248	C₂²	C₂²×C₂²⋊Dic₆₁	1	14	yes	—
249	C₂²	C₂²×C₂²⋊Dic₆₁	1	7	yes	—
250	C₂²	C₂⁴×Dic₆₁	1	1	yes	characteristic
251	C₂²	C₂³×C₂²⋊D₆₁	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	244	16	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	7	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	14	no	—
258	C₂	—	4	8	no	—
259	C₂	C₂²×#14519692	1	14	yes	direct product factor
260	C₂	C₂³×C₂²⋊Dic₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₆₁
commutator	C₂²×C₆₁	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₁
Fitting, prime radical	C₂⁴×D₄×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂³×C₆₁	C₂⁵
hypercenter	C₂⁴×D₄	D₆₁
hypocenter	C₆₁	C₂³×#99
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂²⋊Dic₆₁	C₂	semidirect
2. C₂²×#14519692	C₂	direct (N × H)
6. C₂²×C₂²⋊Dic₆₁	C₂²	semidirect
7. C₂×#14519692	C₂²	direct (N × H)
8. C₂⁶×C₆₁	C₄	semidirect
12. C₂³×D₄×C₆₁	C₄	semidirect
14. C₂⁴×C₄×C₆₁	C₄	semidirect
18. C₂×C₂²⋊Dic₆₁	C₂³	semidirect
19. #14519692	C₂³	direct (N × H)
26. C₂⁵×C₆₁	C₂×C₄	semidirect
34. C₂²×D₄×C₆₁	C₂×C₄	semidirect
38. C₂³×C₄×C₆₁	C₂×C₄	semidirect
55. C₂⁴×C₆₁	C₂²×C₄	semidirect
63. C₂×D₄×C₆₁	C₂²×C₄	semidirect
67. C₂²×C₄×C₆₁	C₂²×C₄	semidirect
71. C₂²⋊Dic₆₁	C₂⁴	semidirect
85. C₂³×C₆₁	#99	semidirect
92. C₂³×C₆₁	C₂³×C₄	semidirect
100. C₂×C₄×C₆₁	C₂³×C₄	semidirect
116. C₂²×C₆₁	C₂×#99	semidirect
132. C₂×C₆₁	C₂²×#99	semidirect
134. C₂⁶	Dic₆₁	semidirect
138. C₂³×D₄	Dic₆₁	semidirect
141. C₂⁴×C₄	Dic₆₁	semidirect
145. C₆₁	C₂³×#99	semidirect
152. C₂⁵	C₂×Dic₆₁	semidirect
160. C₂²×D₄	C₂×Dic₆₁	semidirect
166. C₂³×C₄	C₂×Dic₆₁	semidirect
186. C₂⁴	C₂²×Dic₆₁	semidirect
194. C₂×D₄	C₂²×Dic₆₁	semidirect
200. C₂²×C₄	C₂²×Dic₆₁	semidirect
216. C₂³	#14519692	direct (N × H)
223. C₂³	C₂³×Dic₆₁	semidirect
233. C₂×C₄	C₂³×Dic₆₁	semidirect
246. C₂²	C₂×#14519692	direct (N × H)
259. C₂	C₂²×#14519692	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊Dic₆₁	2
C₂²×#14519692	2
C₂⁴×D₄×C₆₁	2
C₂³×#99	61

Subgroups — C23×#14519692 #30543474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519692 #30543474