Subgroups — #29835543

Order 15616.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456333	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840164	C₂	1	1	yes	maximal, characteristic
3	#21448105	C₂	1	4	yes	maximal, complemented
4	#21448098	C₂	1	4	yes	maximal, complemented
5	#21447607	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
7	#16840319	—	2	1	no	—
8	C₂×#14519769	—	2	1	no	—
9	#16840167	C₂²	1	2	yes	—
10	#16840164	—	2	1	no	—
11	#16840164	—	2	1	no	—
12	#16840164	—	2	1	no	—
13	#16840164	C₂²	1	2	yes	—
14	#16840164	C₂²	1	2	yes	—
15	#16840164	C₂²	1	2	yes	—
16	#16840160	—	2	1	no	—
17	#16839800	—	2	1	no	—
18	#16839800	C₂²	1	2	yes	—
19	C₂×#14519711	C₂²	1	2	yes	—
20	C₂×#14519709	—	2	1	no	—
21	#16839590	C₂²	1	8	yes	—
22	#16839588	C₂²	1	8	yes	—
23	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#346	C₂²	1	2	yes	—
26	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
27	#14519777	—	2	2	no	—
28	#14519769	—	2	2	no	—
29	#14519769	—	2	2	no	—
30	#14519769	—	2	2	no	—
31	#14519769	—	4	2	no	—
32	#14519756	—	4	2	no	—
33	C₄×Dic₁₂₂	—	4	2	no	—
34	#14519736	—	2	2	no	—
35	C₂×C₄⋊Dic₆₁	—	2	2	no	—
36	C₂×#11758196	—	4	2	no	—
37	#14519711	—	2	2	no	—
38	#14519711	—	2	2	no	—
39	#14519711	C₂³	1	4	yes	—
40	#14519711	C₂³	1	4	yes	—
41	#14519709	—	2	2	no	—
42	#14519709	—	2	2	no	—
43	#14519709	—	2	2	no	—
44	#14519709	—	2	2	no	—
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
48	M₃₂×C₆₁	Q₈	1	8	yes	complemented
49	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	D₈	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₆₁	—	4	2	no	—
57	C₂×Dic₁₂₂	—	4	2	no	—
58	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
59	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₄²×C₆₁	D₈	1	2	yes	—
65	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²⋊Dic₆₁	—	4	2	no	—
68	C₄⋊Dic₆₁	—	2	4	no	—
69	C₄⋊Dic₆₁	—	2	4	no	—
70	C₄⋊Dic₆₁	—	4	2	no	—
71	C₄⋊Dic₆₁	—	4	2	no	—
72	#11758196	—	4	4	no	—
73	#11758196	—	4	4	no	—
74	#11758196	—	4	4	no	—
75	C₄×Dic₆₁	—	4	2	no	—
76	C₁₆×C₆₁	C₂×Q₈	1	8	yes	—
77	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₆₁	—	4	2	no	—
79	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
82	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
85	C₂×Dic₆₁	—	4	4	no	—
86	C₂×Dic₆₁	—	4	4	no	—
87	C₂×Dic₆₁	—	8	4	no	—
88	Dic₁₂₂	—	8	4	no	—
89	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
91	C₈×C₆₁	C₂×Q₁₆	1	2	yes	—
92	C₈×C₆₁	C₂²×Q₈	1	2	yes	—
93	#33732	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	2	2	no	—
95	C₂²×C₆₁	#500	1	2	yes	—
96	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₆₁	—	2	2	no	—
98	C₄×C₆₁	—	2	2	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	—	8	4	no	—
103	Dic₆₁	—	8	4	no	—
104	#3160	—	61	1	no	—
105	C₂×#499	—	61	1	no	—
106	#2255	—	61	4	no	—
107	#2253	—	61	4	no	—
108	#2159	—	61	4	no	—
109	C₄×M₃₂	D₆₁	1	1	yes	characteristic
110	C₂×C₆₁	—	2	2	no	—
111	C₂×C₆₁	C₂×#500	1	1	yes	characteristic
112	C₂×C₆₁	#3414	1	2	yes	—
113	#533	—	122	1	no	—
114	C₂×#125	—	122	1	no	—
115	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
116	#500	—	61	2	no	—
117	#499	—	61	2	no	—
118	#499	—	61	2	no	—
119	#499	—	61	2	no	—
120	#499	—	122	1	no	—
121	#499	—	122	1	no	—
122	#499	—	122	1	no	—
123	#498	—	122	1	no	—
124	#427	—	61	2	no	—
125	#427	—	122	1	no	—
126	C₂×#107	—	61	2	no	—
127	C₂×#106	—	122	1	no	—
128	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
129	#367	—	61	8	no	—
130	#366	—	61	8	no	—
131	#346	C₂×D₆₁	1	2	yes	—
132	C₄×C₁₆	C₂×D₆₁	1	2	yes	—
133	C₆₁	#33732	1	1	yes	characteristic, complemented, Sylow
134	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	122	2	no	—
137	#125	—	122	2	no	—
138	#125	—	122	2	no	—
139	#125	—	122	2	no	—
140	#125	—	244	2	no	—
141	C₂²⋊Q₈	—	244	2	no	—
142	C₄×Q₈	—	244	2	no	—
143	#117	—	122	2	no	—
144	C₂×C₄⋊C₄	—	122	2	no	—
145	C₂×C₄⋊C₄	—	244	2	no	—
146	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
147	M₃₂	Dic₁₂₂	1	8	yes	complemented
148	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
149	#107	—	61	4	no	—
150	#107	—	61	4	no	—
151	#107	—	122	2	no	—
152	#107	—	122	2	no	—
153	#106	—	122	2	no	—
154	#106	—	122	2	no	—
155	#106	—	122	2	no	—
156	#106	—	122	2	no	—
157	C₄×C₈	D₂₄₄	1	1	yes	characteristic
158	C₄×C₈	D₂₄₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
161	C₂×Q₈	—	244	2	no	—
162	C₂²×C₄	—	244	2	no	—
163	C₂²×C₄	D₄₈₈	1	2	yes	—
164	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
165	C₂×C₈	Dic₂₄₄	1	4	yes	—
166	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
167	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
168	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
169	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
170	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
171	C₄⋊C₄	—	122	4	no	—
172	C₄⋊C₄	—	122	4	no	—
173	C₄⋊C₄	—	244	2	no	—
174	C₄⋊C₄	—	244	4	no	—
175	C₄⋊C₄	—	244	4	no	—
176	C₄⋊C₄	—	244	4	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₂²⋊C₄	—	244	2	no	—
179	C₄²	—	244	2	no	—
180	C₄²	D₄₈₈	1	2	yes	—
181	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
182	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
183	C₁₆	C₂×Dic₁₂₂	1	8	yes	—
184	C₂³	C₂×D₄₈₈	1	1	yes	characteristic
185	Q₈	—	488	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	488	4	no	—
188	C₂×C₄	—	244	4	no	—
189	C₂×C₄	—	244	4	no	—
190	C₂×C₄	#14519777	1	2	yes	—
191	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
192	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
193	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
194	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
195	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
196	C₈	#14519777	1	2	yes	—
197	C₈	C₂×Dic₂₄₄	1	2	yes	—
198	C₈	C₂×Dic₂₄₄	1	2	yes	—
199	C₈	C₂²×Dic₁₂₂	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#16840167	1	2	yes	—
202	C₂²	C₂²×D₄₈₈	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	488	4	no	—
206	C₄	—	488	4	no	—
207	C₄	#16840167	1	2	yes	—
208	C₄	C₂×#14519777	1	1	yes	characteristic
209	C₄	C₂²×Dic₂₄₄	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#16840167	1	1	yes	characteristic
212	C₂	#21458955	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₈₈
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂³×C₆₁	C₂×D₈
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	#33732
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21448105	C₂	semidirect
4. #21448098	C₂	semidirect
48. M₃₂×C₆₁	Q₈	semidirect
133. C₆₁	#33732	semidirect
147. M₃₂	Dic₁₂₂	semidirect

Maximal subgroups

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

subgroup	index
#21456333	2
C₂×#16840164	2
#21448105	2
#21448098	2
#21447607	2
C₄×M₃₂×C₆₁	2
#33732	61