Subgroups — #29598615

Order 15616.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
2	#21445979	C₂	1	1	yes	maximal, characteristic
3	#21443910	C₂	1	4	yes	maximal
4	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519711	—	2	1	no	—
7	C₂×#14519711	C₂²	1	2	yes	—
8	#16839632	—	2	1	no	—
9	C₄×#11758185	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
11	C₆₁×#500	—	2	2	no	—
12	C₆₁×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₆₁	—	2	1	no	—
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₆₁	—	2	2	no	—
17	C₂×C₄⋊Dic₆₁	—	4	1	no	—
18	C₂×C₄⋊Dic₆₁	—	4	1	no	—
19	C₂×#11758185	—	2	2	no	—
20	C₂×#11758185	—	2	2	no	—
21	C₂×#11758185	Q₈	1	4	yes	complemented
22	#14519711	—	2	4	no	—
23	#14519711	—	4	2	no	—
24	C₂²×Q₈×C₆₁	—	4	1	no	—
25	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₆₁	—	2	4	no	—
28	C₄⋊Q₈×C₆₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
31	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
32	#107×C₆₁	—	2	4	no	—
33	#107×C₆₁	—	2	2	no	—
34	#107×C₆₁	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₆₁	D₄	1	2	yes	—
37	C₂×Q₈×C₆₁	—	4	4	no	—
38	C₂×Q₈×C₆₁	—	8	2	no	—
39	C₂×Q₈×C₆₁	—	8	2	no	—
40	C₂²×C₄×C₆₁	—	4	2	no	—
41	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
48	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₆₁	#41	1	2	yes	—
52	C₄⋊C₄×C₆₁	—	2	4	no	—
53	C₄⋊C₄×C₆₁	—	4	4	no	—
54	C₄⋊C₄×C₆₁	—	4	2	no	—
55	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₆₁	SD₁₆	1	2	yes	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	C₄⋊Dic₆₁	—	4	2	no	—
59	C₄⋊Dic₆₁	—	8	2	no	—
60	C₄⋊Dic₆₁	—	8	2	no	—
61	#11758185	—	2	4	no	—
62	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₆₁	—	8	8	no	—
64	C₂×C₄×C₆₁	—	4	4	no	—
65	C₂×C₄×C₆₁	—	8	2	no	—
66	C₂×C₄×C₆₁	—	8	2	no	—
67	C₂×C₄×C₆₁	#102	1	2	yes	—
68	C₂×C₄×C₆₁	#102	1	2	yes	—
69	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
70	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
71	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
72	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
73	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₆₁	—	8	2	no	—
80	C₂×Dic₆₁	—	8	2	no	—
81	C₂×Dic₆₁	—	8	2	no	—
82	C₈×C₆₁	#103	1	4	yes	—
83	C₈×C₆₁	C₄×Q₈	1	2	yes	—
84	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
85	#13282	—	61	1	no	maximal, Sylow
86	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
87	C₂²×C₆₁	#486	1	1	yes	characteristic
88	C₂²×C₆₁	#500	1	1	yes	characteristic
89	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
90	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
91	C₂²×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
92	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
93	C₄×C₆₁	—	8	4	no	—
94	C₄×C₆₁	#357	1	4	yes	—
95	C₄×C₆₁	#358	1	4	yes	—
96	C₄×C₆₁	#391	1	1	yes	characteristic
97	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
98	C₄×C₆₁	#487	1	1	yes	characteristic
99	C₄×C₆₁	#498	1	1	yes	characteristic
100	Dic₆₁	—	16	2	no	—
101	C₂×#500	D₆₁	1	1	yes	characteristic
102	C₂×C₄×C₁₆	—	61	1	no	—
103	#1845	—	61	1	no	—
104	#1380	—	61	4	no	—
105	C₂×C₆₁	#1990	1	1	yes	characteristic
106	C₂×C₆₁	C₂×#357	1	1	yes	characteristic
107	C₂×C₆₁	C₂×#358	1	1	yes	characteristic
108	C₂×C₆₁	#2240	1	1	yes	characteristic
109	C₂×C₆₁	#2241	1	1	yes	characteristic
110	C₂×C₆₁	#2252	1	1	yes	characteristic
111	C₂×C₆₁	#2254	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂²×C₁₆	—	61	2	no	—
114	#500	—	2	2	no	—
115	#500	Dic₆₁	1	4	yes	complemented
116	C₂×#107	—	2	1	no	—
117	C₂×#107	—	61	2	no	—
118	C₂×#107	—	122	1	no	—
119	C₂×#107	C₂×D₆₁	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#384	—	122	1	no	—
122	C₄×C₁₆	—	122	2	no	—
123	C₆₁	#13282	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	4	1	no	—
125	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	122	2	no	—
132	C₂×C₄⋊C₄	—	244	1	no	—
133	C₂×C₄⋊C₄	—	244	1	no	—
134	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
135	C₂×C₁₆	—	61	4	no	—
136	C₂×C₁₆	—	122	2	no	—
137	C₂×C₁₆	—	122	2	no	—
138	#107	—	2	4	no	—
139	#107	—	2	2	no	—
140	#107	—	122	4	no	—
141	#107	—	244	2	no	—
142	#107	C₂×Dic₆₁	1	4	yes	—
143	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
144	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	244	2	no	—
150	C₂²×C₄	#11758199	1	1	yes	characteristic
151	C₂²×C₄	#11758199	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
153	C₂×C₈	#11758201	1	2	yes	—
154	C₂×C₈	#11758202	1	2	yes	—
155	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
156	C₂×C₈	Q₈×D₆₁	1	2	yes	—
157	C₂×C₈	#11758226	1	2	yes	—
158	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	488	2	no	—
164	C₄⋊C₄	—	488	2	no	—
165	C₄⋊C₄	—	244	2	no	—
166	C₄⋊C₄	—	244	2	no	—
167	C₄²	#11758200	1	2	yes	—
168	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
169	C₁₆	—	122	4	no	—
170	C₂³	C₂×#11758199	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	488	2	no	—
176	C₂×C₄	—	488	2	no	—
177	C₂×C₄	—	488	2	no	—
178	C₂×C₄	#14519698	1	2	yes	—
179	C₂×C₄	#14519698	1	2	yes	—
180	C₂×C₄	#14519718	1	2	yes	—
181	C₂×C₄	#14519720	1	2	yes	—
182	C₂×C₄	#14519721	1	2	yes	—
183	C₂×C₄	#14519723	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
185	C₂×C₄	#14519766	1	1	yes	characteristic
186	C₂×C₄	#14519775	1	1	yes	characteristic
187	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
188	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
189	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
190	C₈	#14519701	1	4	yes	—
191	C₈	Q₈×Dic₆₁	1	2	yes	—
192	C₈	#14519753	1	2	yes	—
193	C₂²	C₂×#14519698	1	1	yes	characteristic
194	C₂²	#16840110	1	1	yes	characteristic
195	C₂²	#16840165	1	1	yes	characteristic
196	C₂²	C₂×#14519718	1	1	yes	characteristic
197	C₂²	C₂×#14519720	1	1	yes	characteristic
198	C₂²	C₂×#14519721	1	1	yes	characteristic
199	C₂²	C₂×#14519723	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	976	2	no	—
202	C₄	#16839563	1	4	yes	—
203	C₄	#16839566	1	4	yes	—
204	C₄	#16839669	1	1	yes	characteristic
205	C₄	C₂×#14519701	1	1	yes	characteristic
206	C₄	#16840114	1	1	yes	characteristic
207	C₄	#16840158	1	1	yes	characteristic
208	C₂	#21446781	1	1	yes	characteristic
209	C₂	C₂×#16839563	1	1	yes	characteristic
210	C₂	C₂×#16839566	1	1	yes	characteristic
211	C₂	#21448044	1	1	yes	characteristic
212	C₂	#21448048	1	1	yes	characteristic
213	C₂	#21448092	1	1	yes	characteristic
214	C₂	#21448099	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle, prime radical	C₂³×C₆₁	C₂×D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	#13282
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₆₁×#500	C₄	semidirect
21. C₂×#11758185	Q₈	semidirect
115. #500	Dic₆₁	semidirect
123. C₆₁	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11758185	2
#21445979	2
#21443910	2
C₂×C₆₁×#500	2
#13282	61