Subgroups — #29835519

Order 15616.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456355	C₂	1	2	yes	maximal
2	#21448103	C₂	1	4	yes	maximal, complemented
3	#21448096	C₂	1	2	yes	maximal, complemented
4	#21448093	C₂	1	2	yes	maximal, complemented
5	#21447605	C₂	1	2	yes	maximal
6	#21447603	C₂	1	1	yes	maximal, characteristic
7	#21447415	C₂	1	1	yes	maximal, characteristic
8	C₂×C₆₁×#366	C₂	1	1	yes	maximal, characteristic
9	#16840393	—	2	2	no	—
10	C₂×#14519717	C₂²	1	1	yes	characteristic
11	#16840170	—	2	4	no	—
12	#16840166	C₂²	1	4	yes	—
13	#16840163	C₂²	1	4	yes	—
14	#16839802	—	2	2	no	—
15	#16839800	C₂²	1	2	yes	—
16	#16839717	C₂²	1	1	yes	characteristic
17	#16839590	C₂²	1	2	yes	—
18	#16839589	C₂²	1	4	yes	—
19	#16839588	C₂²	1	1	yes	characteristic
20	#16839588	C₂²	1	1	yes	characteristic
21	#16839587	C₂²	1	4	yes	—
22	#16839537	C₂²	1	2	yes	—
23	C₁₆×Dic₆₁	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×#107×C₆₁	C₂²	1	2	yes	—
26	C₆₁×#366	C₂²	1	2	yes	—
27	C₆₁×#366	C₂²	1	2	yes	—
28	C₂×#11758194	D₄	1	1	yes	characteristic
29	#14519776	—	2	4	no	—
30	#14519776	—	4	2	no	—
31	#14519768	—	2	4	no	—
32	#14519768	—	4	2	no	—
33	#14519756	—	4	4	no	—
34	#14519736	—	2	2	no	—
35	#14519736	D₄	1	1	yes	characteristic
36	#14519717	Q₈	1	4	yes	complemented
37	C₂×#11758185	C₂³	1	2	yes	—
38	#14519711	C₂³	1	2	yes	—
39	#14519711	C₂³	1	2	yes	—
40	#14519710	—	2	4	no	—
41	#14519709	—	2	2	no	—
42	#14519708	—	2	4	no	—
43	#14519687	D₄	1	2	yes	—
44	C₈×Dic₆₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₁₆×C₆₁	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
50	#107×C₆₁	—	2	2	no	—
51	#107×C₆₁	C₂³	1	4	yes	—
52	C₂²×C₄×C₆₁	—	4	2	no	—
53	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₂₂	—	4	4	no	—
55	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	—	2	4	no	—
60	C₄⋊C₄×C₆₁	—	4	2	no	—
61	C₂²⋊Dic₆₁	—	4	2	no	—
62	C₂²⋊Dic₆₁	D₈	1	2	yes	—
63	C₄⋊Dic₆₁	—	2	2	no	—
64	C₄⋊Dic₆₁	—	2	2	no	—
65	C₄⋊Dic₆₁	—	4	4	no	—
66	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
67	C₄⋊Dic₆₁	D₈	1	1	yes	characteristic
68	#11758196	—	4	4	no	—
69	#11758196	—	4	4	no	—
70	C₄×Dic₆₁	—	4	2	no	—
71	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
72	#11758194	Q₁₆	1	4	yes	—
73	C₂×#92418	C₂×D₄	1	2	yes	—
74	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
76	#11758185	C₂×Q₈	1	4	yes	—
77	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	—	4	4	no	—
79	C₂×C₄×C₆₁	—	16	2	no	—
80	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₆₁	—	4	2	no	—
85	C₂×Dic₆₁	—	4	2	no	—
86	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
87	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
88	Dic₁₂₂	—	16	4	no	—
89	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
91	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
92	#92418	C₂×Q₁₆	1	4	yes	—
93	#33729	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	1	no	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	#500	1	2	yes	—
97	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₆₁	—	8	4	no	—
99	C₄×C₆₁	#500	1	2	yes	—
100	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₆₁	—	2	2	no	—
103	Dic₆₁	—	2	2	no	—
104	Dic₆₁	—	8	2	no	—
105	Dic₆₁	—	8	2	no	—
106	#3164	—	61	2	no	—
107	#2255	—	61	4	no	—
108	#2253	—	61	2	no	—
109	#2252	—	61	2	no	—
110	#2159	—	61	2	no	—
111	#2158	—	61	1	no	—
112	C₂×#366	D₆₁	1	1	yes	characteristic
113	#2108	—	61	1	no	—
114	C₂×C₆₁	—	2	2	no	—
115	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
116	C₂×C₆₁	#3415	1	1	yes	characteristic
117	C₂×C₆₁	#3417	1	1	yes	characteristic
118	#544	—	122	2	no	—
119	C₂×M₃₂	—	61	1	no	—
120	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
121	#501	—	122	4	no	—
122	#500	—	61	4	no	—
123	#499	—	61	4	no	—
124	#428	—	122	2	no	—
125	#427	—	61	2	no	—
126	C₂×#107	C₂×D₆₁	1	2	yes	—
127	#405	—	61	1	no	—
128	#367	—	61	2	no	—
129	#367	—	61	4	no	—
130	#366	—	61	1	no	—
131	#366	—	61	1	no	—
132	#366	—	61	4	no	—
133	#366	C₂×D₆₁	1	2	yes	—
134	#366	C₂×D₆₁	1	2	yes	—
135	#346	—	61	2	no	—
136	C₄×C₁₆	—	61	2	no	—
137	C₆₁	#33729	1	1	yes	characteristic, complemented, Sylow
138	C₂×M₁₆	—	61	1	no	—
139	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
140	C₄⋊Q₈	—	122	4	no	—
141	C₄⋊Q₈	—	244	2	no	—
142	#125	—	122	4	no	—
143	#125	—	244	2	no	—
144	C₂²⋊Q₈	—	244	4	no	—
145	#117	—	61	1	no	—
146	#117	—	122	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	M₃₂	—	61	4	no	—
149	C₂×C₁₆	—	61	2	no	—
150	C₂×C₁₆	Dic₁₂₂	1	4	yes	complemented
151	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
153	#107	—	2	2	no	—
154	#107	—	61	2	no	—
155	#107	—	61	2	no	—
156	#107	—	122	4	no	—
157	#107	C₂²×D₆₁	1	4	yes	—
158	#106	—	122	4	no	—
159	#106	—	122	2	no	—
160	#97	—	61	2	no	—
161	C₄×C₈	—	61	2	no	—
162	C₂×Q₈	—	244	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
165	M₁₆	—	61	4	no	—
166	C₂×C₈	—	61	2	no	—
167	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
168	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₆₁	1	2	yes	—
170	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	61	1	no	—
174	C₄⋊C₄	—	61	1	no	—
175	C₄⋊C₄	—	122	2	no	—
176	C₄⋊C₄	—	122	2	no	—
177	C₄⋊C₄	—	244	4	no	—
178	C₄⋊C₄	—	244	4	no	—
179	C₄⋊C₄	—	244	4	no	—
180	C₂²⋊C₄	—	61	2	no	—
181	C₂²⋊C₄	—	244	2	no	—
182	C₄²	—	61	2	no	—
183	C₄²	—	244	2	no	—
184	C₁₆	—	61	4	no	—
185	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
186	C₁₆	C₂×Dic₁₂₂	1	2	yes	—
187	C₂³	D₈×D₆₁	1	1	yes	characteristic
188	Q₈	—	976	4	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	—	16	2	no	—
191	C₂×C₄	—	61	2	no	—
192	C₂×C₄	—	61	2	no	—
193	C₂×C₄	—	244	2	no	—
194	C₂×C₄	—	244	2	no	—
195	C₂×C₄	#14519776	1	2	yes	—
196	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
199	C₈	—	61	4	no	—
200	C₈	#14519776	1	2	yes	—
201	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
202	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#16840166	1	2	yes	—
206	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
207	C₄	—	8	4	no	—
208	C₄	—	122	2	no	—
209	C₄	—	122	2	no	—
210	C₄	—	488	2	no	—
211	C₄	—	488	2	no	—
212	C₄	#16840166	1	2	yes	—
213	C₄	C₂×#14519776	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#16840166	1	1	yes	characteristic
217	C₂	#21458956	1	1	yes	characteristic
218	C₂	#21458968	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#366	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₂×#366	D₆₁
hypocenter	C₆₁	#33729
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #21448103	C₂	semidirect
3. #21448096	C₂	semidirect
4. #21448093	C₂	semidirect
36. #14519717	Q₈	semidirect
47. C₂×C₁₆×C₆₁	Q₈	semidirect
137. C₆₁	#33729	semidirect
150. C₂×C₁₆	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456355	2
#21448103	2
#21448096	2
#21448093	2
#21447605	2
#21447603	2
#21447415	2
C₂×C₆₁×#366	2
#33729	61