Subgroups — #29598619

Order 15616.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839538	C₂	1	1	yes	maximal, characteristic
2	#21445983	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	C₂²	1	2	yes	—
7	C₂×#14519709	—	2	1	no	—
8	#16839628	—	2	1	no	—
9	#16839538	—	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×#11758185	—	2	2	no	—
18	C₂×#11758185	—	2	2	no	—
19	C₂×#11758185	Q₈	1	4	yes	complemented
20	#14519711	—	2	4	no	—
21	#14519709	—	4	2	no	—
22	#14519684	—	4	1	no	—
23	#14519684	—	4	1	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	4	no	—
39	C₂²×C₄×C₆₁	—	4	2	no	—
40	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	4	2	no	—
44	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₆₁	#41	1	2	yes	—
51	C₄⋊C₄×C₆₁	—	2	4	no	—
52	C₄⋊C₄×C₆₁	—	4	4	no	—
53	C₄⋊C₄×C₆₁	—	4	2	no	—
54	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₆₁	SD₁₆	1	2	yes	—
56	C₄⋊Dic₆₁	—	4	2	no	—
57	C₄⋊Dic₆₁	—	4	2	no	—
58	#11758185	—	2	4	no	—
59	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₆₁	—	8	8	no	—
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	—	4	4	no	—
64	C₂×C₄×C₆₁	—	8	4	no	—
65	C₂×C₄×C₆₁	#102	1	2	yes	—
66	C₂×C₄×C₆₁	#102	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₆₁	—	8	2	no	—
74	C₂×Dic₆₁	—	8	2	no	—
75	C₂×Dic₆₁	—	8	2	no	—
76	C₈×C₆₁	#103	1	4	yes	—
77	C₈×C₆₁	C₄×Q₈	1	2	yes	—
78	C₈×C₆₁	C₂²⋊Q₈	1	2	yes	—
79	#13283	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
81	C₂²×C₆₁	#486	1	1	yes	characteristic
82	C₂²×C₆₁	#500	1	1	yes	characteristic
83	C₂²×C₆₁	#509	1	2	yes	—
84	C₂²×C₆₁	#510	1	2	yes	—
85	C₄×C₆₁	—	2	2	no	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	—	8	4	no	—
88	C₄×C₆₁	#391	1	1	yes	characteristic
89	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
90	C₄×C₆₁	#487	1	1	yes	characteristic
91	C₄×C₆₁	#498	1	1	yes	characteristic
92	Dic₆₁	—	16	2	no	—
93	C₂×#500	D₆₁	1	1	yes	characteristic
94	C₂×#346	—	61	1	no	—
95	#1846	—	61	1	no	—
96	#1380	—	61	4	no	—
97	C₂×C₆₁	#1990	1	1	yes	characteristic
98	C₂×C₆₁	#2140	1	1	yes	characteristic
99	C₂×C₆₁	#2141	1	1	yes	characteristic
100	C₂×C₆₁	#2243	1	2	yes	—
101	C₂×C₆₁	#2255	1	2	yes	—
102	C₂×C₄⋊Q₈	—	2	1	no	—
103	C₂²×C₁₆	—	61	2	no	—
104	#500	—	2	2	no	—
105	#500	Dic₆₁	1	4	yes	complemented
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	61	2	no	—
108	C₂×#107	C₂×D₆₁	1	2	yes	—
109	C₂×#106	—	122	1	no	—
110	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
111	#382	—	122	1	no	—
112	#346	—	122	2	no	—
113	C₆₁	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	122	2	no	—
122	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
123	C₂×C₁₆	—	61	4	no	—
124	C₂×C₁₆	—	122	2	no	—
125	C₂×C₁₆	—	122	2	no	—
126	#107	—	2	4	no	—
127	#107	—	2	2	no	—
128	#107	—	122	4	no	—
129	#107	C₂×Dic₆₁	1	4	yes	—
130	#106	—	244	2	no	—
131	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
132	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
133	#95	—	244	1	no	—
134	#95	—	244	1	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	244	2	no	—
139	C₂²×C₄	#11758199	1	1	yes	characteristic
140	C₂²×C₄	#11758199	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
142	C₂×C₈	#11758201	1	2	yes	—
143	C₂×C₈	#11758202	1	2	yes	—
144	C₂×C₈	C₂²⋊Dic₆₁	1	2	yes	—
145	C₂×C₈	Q₈×D₆₁	1	2	yes	—
146	C₂×C₈	#11758226	1	2	yes	—
147	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	244	2	no	—
153	C₄⋊C₄	—	244	2	no	—
154	C₄²	#11758200	1	2	yes	—
155	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
156	C₁₆	—	122	4	no	—
157	C₂³	C₂×#11758199	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	488	2	no	—
164	C₂×C₄	—	488	2	no	—
165	C₂×C₄	—	488	2	no	—
166	C₂×C₄	#14519698	1	2	yes	—
167	C₂×C₄	#14519698	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
169	C₂×C₄	#14519766	1	1	yes	characteristic
170	C₂×C₄	#14519775	1	1	yes	characteristic
171	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
172	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
173	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
174	C₈	#14519701	1	4	yes	—
175	C₈	Q₈×Dic₆₁	1	2	yes	—
176	C₈	#14519753	1	2	yes	—
177	C₂²	C₂×#14519698	1	1	yes	characteristic
178	C₂²	#16840110	1	1	yes	characteristic
179	C₂²	#16840165	1	1	yes	characteristic
180	C₂²	#16840198	1	2	yes	—
181	C₂²	#16840203	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	976	2	no	—
186	C₄	#16839669	1	1	yes	characteristic
187	C₄	C₂×#14519701	1	1	yes	characteristic
188	C₄	#16840114	1	1	yes	characteristic
189	C₄	#16840158	1	1	yes	characteristic
190	C₂	#21446781	1	1	yes	characteristic
191	C₂	#21447535	1	1	yes	characteristic
192	C₂	#21447539	1	1	yes	characteristic
193	C₂	#21448058	1	2	yes	—
194	C₂	#21448102	1	2	yes	—

	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	#13283
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
19. C₂×#11758185	Q₈	semidirect
105. #500	Dic₆₁	semidirect
113. C₆₁	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839538	2
#21445983	2
#21443910	2
C₂×C₆₁×#500	2
#13283	61