Subgroups — C₂³×#137×C₆₁ #29566515

Order 15616.

11166 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#137×C₆₁	C₂	1	56	yes	maximal, direct product factor
4	C₂³×Q₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
5	C₂³×SD₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₆₁	—	2	14	no	—
8	C₂²×#41×C₆₁	C₂²	1	28	yes	—
9	C₂³×Q₈×C₆₁	—	2	14	no	—
10	C₂³×Q₈×C₆₁	—	2	1	no	—
11	C₂³×Q₈×C₆₁	C₂²	1	2	yes	—
12	C₂³×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₆₁	C₂²	1	28	yes	complemented
14	C₂³×D₄×C₆₁	—	2	1	no	—
15	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₁	—	2	1	no	—
17	C₂⁴×C₄×C₆₁	—	2	1	no	—
18	C₂⁴×C₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#137×C₆₁	C₂²	1	448	yes	direct product factor
20	C₂²×Q₁₆×C₆₁	C₂²	1	56	yes	complemented
21	C₂²×SD₁₆×C₆₁	C₂²	1	56	yes	complemented
22	C₂²×M₁₆×C₆₁	C₂²	1	28	yes	complemented
23	C₂³×C₈×C₆₁	C₂²	1	2	yes	—
24	C₂⁵×C₆₁	—	2	1	no	—
25	C₂⁵×C₆₁	D₄	1	1	yes	characteristic
26	C₂×#41×C₆₁	—	2	168	no	—
27	C₂×#41×C₆₁	C₂³	1	112	yes	—
28	C₂²×Q₈×C₆₁	—	2	14	no	—
29	C₂²×Q₈×C₆₁	—	2	28	no	—
30	C₂²×Q₈×C₆₁	—	2	168	no	—
31	C₂²×Q₈×C₆₁	—	2	7	no	—
32	C₂²×Q₈×C₆₁	C₂³	1	28	yes	—
33	C₂²×Q₈×C₆₁	C₂³	1	14	yes	—
34	C₂²×Q₈×C₆₁	C₂³	1	112	yes	complemented
35	C₂²×D₄×C₆₁	—	2	28	no	—
36	C₂²×D₄×C₆₁	—	2	7	no	—
37	C₂²×D₄×C₆₁	C₂³	1	14	yes	—
38	C₂³×C₄×C₆₁	—	2	28	no	—
39	C₂³×C₄×C₆₁	—	2	1	no	—
40	C₂³×C₄×C₆₁	—	2	2	no	—
41	C₂³×C₄×C₆₁	—	2	28	no	—
42	C₂³×C₄×C₆₁	D₄	1	14	yes	—
43	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₁	C₂³	1	14	yes	—
45	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
46	#137×C₆₁	C₂³	1	512	yes	direct product factor
47	C₂×Q₁₆×C₆₁	C₂³	1	224	yes	complemented
48	C₂×SD₁₆×C₆₁	C₂³	1	224	yes	complemented
49	C₂×M₁₆×C₆₁	C₂³	1	112	yes	complemented
50	C₂²×C₈×C₆₁	C₂³	1	28	yes	—
51	C₂⁴×C₆₁	—	2	8	no	—
52	C₂⁴×C₆₁	—	2	14	no	—
53	C₂⁴×C₆₁	—	4	8	no	—
54	C₂⁴×C₆₁	C₂×D₄	1	14	yes	—
55	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
56	#41×C₆₁	—	2	224	no	—
57	#41×C₆₁	C₂⁴	1	64	yes	—
58	C₂×Q₈×C₆₁	—	2	84	no	—
59	C₂×Q₈×C₆₁	—	2	112	no	—
60	C₂×Q₈×C₆₁	—	2	224	no	—
61	C₂×Q₈×C₆₁	—	2	42	no	—
62	C₂×Q₈×C₆₁	C₂⁴	1	56	yes	—
63	C₂×Q₈×C₆₁	C₂⁴	1	28	yes	—
64	C₂×Q₈×C₆₁	C₂⁴	1	64	yes	complemented
65	C₂×D₄×C₆₁	—	2	112	no	—
66	C₂×D₄×C₆₁	—	2	42	no	—
67	C₂×D₄×C₆₁	C₂⁴	1	28	yes	—
68	C₂²×C₄×C₆₁	—	2	14	no	—
69	C₂²×C₄×C₆₁	—	2	28	no	—
70	C₂²×C₄×C₆₁	—	2	112	no	—
71	C₂²×C₄×C₆₁	—	2	112	no	—
72	C₂²×C₄×C₆₁	C₂×D₄	1	14	yes	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	7	yes	—
74	C₂²×C₄×C₆₁	C₂×D₄	1	84	yes	—
75	C₂²×C₄×C₆₁	C₂⁴	1	7	yes	—
76	C₂²×C₄×C₆₁	C₂⁴	1	28	yes	—
77	Q₁₆×C₆₁	C₂⁴	1	128	yes	complemented
78	SD₁₆×C₆₁	C₂⁴	1	128	yes	complemented
79	M₁₆×C₆₁	C₂⁴	1	64	yes	complemented
80	C₂×C₈×C₆₁	C₂⁴	1	56	yes	—
81	C₂³×C₆₁	—	2	56	no	—
82	C₂³×C₆₁	—	2	28	no	—
83	C₂³×C₆₁	—	4	56	no	—
84	C₂³×C₆₁	#137	1	8	yes	direct product factor
85	C₂³×C₆₁	C₂²×D₄	1	28	yes	—
86	C₂³×C₆₁	C₂²×D₄	1	7	yes	—
87	Q₈×C₆₁	—	2	56	no	—
88	Q₈×C₆₁	—	2	64	no	—
89	Q₈×C₆₁	—	2	28	no	—
90	Q₈×C₆₁	C₂⁵	1	16	yes	—
91	Q₈×C₆₁	C₂⁵	1	8	yes	—
92	D₄×C₆₁	—	2	64	no	—
93	D₄×C₆₁	—	2	28	no	—
94	D₄×C₆₁	C₂⁵	1	8	yes	—
95	C₂×C₄×C₆₁	—	2	28	no	—
96	C₂×C₄×C₆₁	—	2	56	no	—
97	C₂×C₄×C₆₁	—	2	64	no	—
98	C₂×C₄×C₆₁	—	2	64	no	—
99	C₂×C₄×C₆₁	C₂²×D₄	1	28	yes	—
100	C₂×C₄×C₆₁	C₂²×D₄	1	21	yes	—
101	C₂×C₄×C₆₁	C₂²×D₄	1	56	yes	—
102	C₂×C₄×C₆₁	C₂⁵	1	7	yes	—
103	C₂×C₄×C₆₁	C₂⁵	1	8	yes	—
104	C₈×C₆₁	C₂⁵	1	16	yes	—
105	C₂³×#137	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₆₁	—	2	56	no	—
107	C₂²×C₆₁	—	2	8	no	—
108	C₂²×C₆₁	—	4	56	no	—
109	C₂²×C₆₁	C₂×#137	1	28	yes	direct product factor
110	C₂²×C₆₁	C₂³×D₄	1	8	yes	—
111	C₂²×C₆₁	C₂³×D₄	1	7	yes	—
112	C₄×C₆₁	—	2	8	no	—
113	C₄×C₆₁	—	2	16	no	—
114	C₄×C₆₁	C₂³×D₄	1	8	yes	—
115	C₄×C₆₁	C₂³×D₄	1	7	yes	—
116	C₄×C₆₁	C₂⁶	1	1	yes	characteristic
117	C₂³×#41	C₂×C₆₁	1	1	yes	characteristic
118	C₂⁴×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
119	C₂²×#137	C₂×C₆₁	1	56	yes	direct product factor
120	C₂³×Q₁₆	C₂×C₆₁	1	2	yes	complemented
121	C₂³×SD₁₆	C₂×C₆₁	1	2	yes	complemented
122	C₂³×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
123	C₂×C₆₁	—	2	8	no	—
124	C₂×C₆₁	—	4	8	no	—
125	C₂×C₆₁	C₂²×#137	1	14	yes	direct product factor
126	C₂×C₆₁	C₂⁴×D₄	1	1	yes	characteristic
127	C₂²×#41	—	2	14	no	—
128	C₂²×#41	C₂²×C₆₁	1	28	yes	—
129	C₂³×Q₈	—	2	1	no	—
130	C₂³×Q₈	—	2	14	no	—
131	C₂³×Q₈	C₂²×C₆₁	1	2	yes	—
132	C₂³×Q₈	C₂²×C₆₁	1	1	yes	characteristic
133	C₂³×Q₈	C₂²×C₆₁	1	28	yes	complemented
134	C₂³×D₄	—	2	1	no	—
135	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
136	C₂⁴×C₄	—	2	1	no	—
137	C₂⁴×C₄	—	2	1	no	—
138	C₂⁴×C₄	C₂²×C₆₁	1	1	yes	characteristic
139	C₂×#137	C₂²×C₆₁	1	448	yes	direct product factor
140	C₂²×Q₁₆	C₂²×C₆₁	1	56	yes	complemented
141	C₂²×SD₁₆	C₂²×C₆₁	1	56	yes	complemented
142	C₂²×M₁₆	C₂²×C₆₁	1	28	yes	complemented
143	C₂³×C₈	C₂²×C₆₁	1	2	yes	—
144	C₆₁	C₂³×#137	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁵	—	2	1	no	—
146	C₂⁵	D₄×C₆₁	1	1	yes	characteristic
147	C₂×#41	—	2	168	no	—
148	C₂×#41	C₂³×C₆₁	1	112	yes	—
149	C₂²×Q₈	—	2	14	no	—
150	C₂²×Q₈	—	2	28	no	—
151	C₂²×Q₈	—	2	168	no	—
152	C₂²×Q₈	—	2	7	no	—
153	C₂²×Q₈	C₂³×C₆₁	1	28	yes	—
154	C₂²×Q₈	C₂³×C₆₁	1	14	yes	—
155	C₂²×Q₈	C₂³×C₆₁	1	112	yes	complemented
156	C₂²×D₄	—	2	28	no	—
157	C₂²×D₄	—	2	7	no	—
158	C₂²×D₄	C₂³×C₆₁	1	14	yes	—
159	C₂³×C₄	—	2	1	no	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	—	2	28	no	—
162	C₂³×C₄	—	2	28	no	—
163	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₆₁	1	14	yes	—
165	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
166	C₂³×C₄	C₂³×C₆₁	1	14	yes	—
167	#137	C₂³×C₆₁	1	512	yes	direct product factor
168	C₂×Q₁₆	C₂³×C₆₁	1	224	yes	complemented
169	C₂×SD₁₆	C₂³×C₆₁	1	224	yes	complemented
170	C₂×M₁₆	C₂³×C₆₁	1	112	yes	complemented
171	C₂²×C₈	C₂³×C₆₁	1	28	yes	—
172	C₂⁴	—	2	8	no	—
173	C₂⁴	—	2	14	no	—
174	C₂⁴	—	4	8	no	—
175	C₂⁴	C₂×D₄×C₆₁	1	14	yes	—
176	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
177	#41	—	2	224	no	—
178	#41	C₂⁴×C₆₁	1	64	yes	—
179	C₂×Q₈	—	2	84	no	—
180	C₂×Q₈	—	2	112	no	—
181	C₂×Q₈	—	2	224	no	—
182	C₂×Q₈	—	2	42	no	—
183	C₂×Q₈	C₂⁴×C₆₁	1	56	yes	—
184	C₂×Q₈	C₂⁴×C₆₁	1	28	yes	—
185	C₂×Q₈	C₂⁴×C₆₁	1	64	yes	complemented
186	C₂×D₄	—	2	112	no	—
187	C₂×D₄	—	2	42	no	—
188	C₂×D₄	C₂⁴×C₆₁	1	28	yes	—
189	C₂²×C₄	—	2	14	no	—
190	C₂²×C₄	—	2	28	no	—
191	C₂²×C₄	—	2	112	no	—
192	C₂²×C₄	—	2	112	no	—
193	C₂²×C₄	C₂×D₄×C₆₁	1	14	yes	—
194	C₂²×C₄	C₂×D₄×C₆₁	1	7	yes	—
195	C₂²×C₄	C₂×D₄×C₆₁	1	84	yes	—
196	C₂²×C₄	C₂⁴×C₆₁	1	7	yes	—
197	C₂²×C₄	C₂⁴×C₆₁	1	28	yes	—
198	Q₁₆	C₂⁴×C₆₁	1	128	yes	complemented
199	SD₁₆	C₂⁴×C₆₁	1	128	yes	complemented
200	M₁₆	C₂⁴×C₆₁	1	64	yes	complemented
201	C₂×C₈	C₂⁴×C₆₁	1	56	yes	—
202	C₂³	—	2	56	no	—
203	C₂³	—	2	28	no	—
204	C₂³	—	4	56	no	—
205	C₂³	#137×C₆₁	1	8	yes	direct product factor
206	C₂³	C₂²×D₄×C₆₁	1	28	yes	—
207	C₂³	C₂²×D₄×C₆₁	1	7	yes	—
208	Q₈	—	2	56	no	—
209	Q₈	—	2	64	no	—
210	Q₈	—	2	28	no	—
211	Q₈	C₂⁵×C₆₁	1	16	yes	—
212	Q₈	C₂⁵×C₆₁	1	8	yes	—
213	D₄	—	2	64	no	—
214	D₄	—	2	28	no	—
215	D₄	C₂⁵×C₆₁	1	8	yes	—
216	C₂×C₄	—	2	28	no	—
217	C₂×C₄	—	2	56	no	—
218	C₂×C₄	—	2	64	no	—
219	C₂×C₄	—	2	64	no	—
220	C₂×C₄	C₂²×D₄×C₆₁	1	28	yes	—
221	C₂×C₄	C₂²×D₄×C₆₁	1	21	yes	—
222	C₂×C₄	C₂²×D₄×C₆₁	1	56	yes	—
223	C₂×C₄	C₂⁵×C₆₁	1	7	yes	—
224	C₂×C₄	C₂⁵×C₆₁	1	8	yes	—
225	C₈	C₂⁵×C₆₁	1	16	yes	—
226	C₂²	—	2	56	no	—
227	C₂²	—	2	8	no	—
228	C₂²	—	4	56	no	—
229	C₂²	C₂×#137×C₆₁	1	28	yes	direct product factor
230	C₂²	C₂³×D₄×C₆₁	1	8	yes	—
231	C₂²	C₂³×D₄×C₆₁	1	7	yes	—
232	C₄	—	2	8	no	—
233	C₄	—	2	16	no	—
234	C₄	C₂³×D₄×C₆₁	1	8	yes	—
235	C₄	C₂³×D₄×C₆₁	1	7	yes	—
236	C₄	C₂⁶×C₆₁	1	1	yes	characteristic
237	C₂	—	2	8	no	—
238	C₂	—	4	8	no	—
239	C₂	C₂²×#137×C₆₁	1	14	yes	direct product factor
240	C₂	C₂⁴×D₄×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₁	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₆₁
prime radical	C₂³×#41×C₆₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₁ ◁ C₂⁴×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Q₈×C₆₁	C₂	semidirect
3. C₂²×#137×C₆₁	C₂	direct (N × H)
4. C₂³×Q₁₆×C₆₁	C₂	semidirect
5. C₂³×SD₁₆×C₆₁	C₂	semidirect
6. C₂³×M₁₆×C₆₁	C₂	semidirect
13. C₂³×Q₈×C₆₁	C₂²	semidirect
19. C₂×#137×C₆₁	C₂²	direct (N × H)
20. C₂²×Q₁₆×C₆₁	C₂²	semidirect
21. C₂²×SD₁₆×C₆₁	C₂²	semidirect
22. C₂²×M₁₆×C₆₁	C₂²	semidirect
34. C₂²×Q₈×C₆₁	C₂³	semidirect
46. #137×C₆₁	C₂³	direct (N × H)
47. C₂×Q₁₆×C₆₁	C₂³	semidirect
48. C₂×SD₁₆×C₆₁	C₂³	semidirect
49. C₂×M₁₆×C₆₁	C₂³	semidirect
64. C₂×Q₈×C₆₁	C₂⁴	semidirect
77. Q₁₆×C₆₁	C₂⁴	semidirect
78. SD₁₆×C₆₁	C₂⁴	semidirect
79. M₁₆×C₆₁	C₂⁴	semidirect
84. C₂³×C₆₁	#137	direct (N × H)
105. C₂³×#137	C₆₁	direct (N × H)
109. C₂²×C₆₁	C₂×#137	direct (N × H)
118. C₂⁴×Q₈	C₂×C₆₁	semidirect
119. C₂²×#137	C₂×C₆₁	direct (N × H)
120. C₂³×Q₁₆	C₂×C₆₁	semidirect
121. C₂³×SD₁₆	C₂×C₆₁	semidirect
122. C₂³×M₁₆	C₂×C₆₁	semidirect
125. C₂×C₆₁	C₂²×#137	direct (N × H)
133. C₂³×Q₈	C₂²×C₆₁	semidirect
139. C₂×#137	C₂²×C₆₁	direct (N × H)
140. C₂²×Q₁₆	C₂²×C₆₁	semidirect
141. C₂²×SD₁₆	C₂²×C₆₁	semidirect
142. C₂²×M₁₆	C₂²×C₆₁	semidirect
144. C₆₁	C₂³×#137	direct (N × H)
155. C₂²×Q₈	C₂³×C₆₁	semidirect
167. #137	C₂³×C₆₁	direct (N × H)
168. C₂×Q₁₆	C₂³×C₆₁	semidirect
169. C₂×SD₁₆	C₂³×C₆₁	semidirect
170. C₂×M₁₆	C₂³×C₆₁	semidirect
185. C₂×Q₈	C₂⁴×C₆₁	semidirect
198. Q₁₆	C₂⁴×C₆₁	semidirect
199. SD₁₆	C₂⁴×C₆₁	semidirect
200. M₁₆	C₂⁴×C₆₁	semidirect
205. C₂³	#137×C₆₁	direct (N × H)
229. C₂²	C₂×#137×C₆₁	direct (N × H)
239. C₂	C₂²×#137×C₆₁	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×#41×C₆₁	2
C₂⁴×Q₈×C₆₁	2
C₂²×#137×C₆₁	2
C₂³×Q₁₆×C₆₁	2
C₂³×SD₁₆×C₆₁	2
C₂³×M₁₆×C₆₁	2
C₂³×#137	61

Subgroups — C23×#137×C61 #29566515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#137×C₆₁ #29566515