Subgroups — #29579837

Order 15616.

494 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×#16840167	C₂	1	1	yes	maximal, characteristic
2	#21446780	C₂	1	2	yes	maximal
3	#21446621	C₂	1	2	yes	maximal
4	#21445947	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519777	C₂²	1	2	yes	—
7	#16840167	—	2	2	no	—
8	#16840167	C₄	1	4	yes	complemented
9	C₂×#14519711	—	2	2	no	—
10	C₂×#14519702	—	2	4	no	—
11	#16839632	C₂²	1	2	yes	—
12	#16839525	—	2	2	no	—
13	#16839525	—	2	4	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#322	—	2	2	no	—
18	C₂²×Dic₁₂₂	—	2	2	no	—
19	#14519777	—	2	2	no	—
20	#14519777	—	2	2	no	—
21	#14519777	—	2	2	no	—
22	#14519777	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₆₁	—	2	2	no	—
24	C₂×C₄⋊Dic₆₁	—	2	2	no	—
25	C₂×C₄⋊Dic₆₁	—	2	2	no	—
26	C₂×C₄⋊Dic₆₁	—	2	2	no	—
27	C₂×C₄⋊Dic₆₁	—	2	2	no	—
28	#14519711	—	2	8	no	—
29	#14519702	—	4	8	no	—
30	C₂²×C₈×C₆₁	D₄	1	2	yes	—
31	C₂²×C₈×C₆₁	D₄	1	2	yes	—
32	C₂²×C₈×C₆₁	D₄	1	2	yes	—
33	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₆₁	—	2	1	no	—
35	C₄×C₈×C₆₁	—	2	1	no	—
36	C₄×C₈×C₆₁	—	2	1	no	—
37	C₄×C₈×C₆₁	—	2	1	no	—
38	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₁	D₄	1	2	yes	—
40	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	—	4	2	no	—
44	C₂²×Dic₆₁	—	4	2	no	—
45	C₂×Dic₁₂₂	—	2	4	no	—
46	C₂×Dic₁₂₂	—	4	4	no	—
47	C₂×Dic₁₂₂	—	4	4	no	—
48	C₂×Dic₁₂₂	—	4	2	no	—
49	C₂×C₈×C₆₁	—	2	2	no	—
50	C₂×C₈×C₆₁	—	2	2	no	—
51	C₂×C₈×C₆₁	—	2	4	no	—
52	C₂×C₈×C₆₁	—	2	2	no	—
53	C₂×C₈×C₆₁	—	2	4	no	—
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
56	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₆₁	#41	1	1	yes	characteristic
59	C₄²×C₆₁	#41	1	1	yes	characteristic
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	2	4	no	—
62	C₄⋊Dic₆₁	—	4	2	no	—
63	C₄⋊Dic₆₁	—	4	2	no	—
64	C₄⋊Dic₆₁	—	4	4	no	—
65	C₄⋊Dic₆₁	—	4	4	no	—
66	C₄⋊Dic₆₁	—	4	4	no	—
67	C₂³×C₆₁	#127	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₆₁	#135	1	2	yes	—
76	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	4	yes	—
77	C₂×C₄×C₆₁	#137	1	4	yes	—
78	C₂×Dic₆₁	—	4	4	no	—
79	C₂×Dic₆₁	—	8	2	no	—
80	C₂×Dic₆₁	—	8	2	no	—
81	C₂×Dic₆₁	—	8	2	no	—
82	C₂×Dic₆₁	—	8	2	no	—
83	C₂×Dic₆₁	—	8	2	no	—
84	Dic₁₂₂	—	4	8	no	—
85	Dic₁₂₂	—	8	4	no	—
86	C₈×C₆₁	—	2	4	no	—
87	C₈×C₆₁	—	2	4	no	—
88	C₈×C₆₁	—	4	2	no	—
89	#9834	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	#390	1	1	yes	characteristic
91	C₂²×C₆₁	#494	1	1	yes	characteristic
92	C₂²×C₆₁	#495	1	1	yes	characteristic
93	C₂²×C₆₁	#496	1	2	yes	—
94	C₂²×C₆₁	#497	1	2	yes	—
95	C₄×C₆₁	#419	1	2	yes	—
96	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
97	C₄×C₆₁	#442	1	2	yes	—
98	C₄×C₆₁	#467	1	2	yes	—
99	C₄×C₆₁	#469	1	2	yes	—
100	C₄×C₆₁	#491	1	2	yes	—
101	Dic₆₁	—	8	4	no	—
102	Dic₆₁	—	16	2	no	—
103	C₂×#500	—	61	1	no	—
104	#1990	—	61	2	no	—
105	#1956	—	61	2	no	—
106	#1834	—	61	1	no	—
107	C₂×#322	D₆₁	1	1	yes	characteristic
108	C₂×C₆₁	#1719	1	2	yes	—
109	C₂×C₆₁	#1723	1	2	yes	—
110	C₂×C₆₁	#1960	1	1	yes	characteristic
111	C₂×C₆₁	#1969	1	1	yes	characteristic
112	C₂×C₆₁	#1972	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	61	2	no	—
114	#500	—	61	4	no	—
115	#500	—	122	2	no	—
116	C₂×#107	—	122	2	no	—
117	C₂×#103	—	122	4	no	—
118	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	#384	—	61	2	no	—
122	#340	—	122	2	no	—
123	#340	—	122	4	no	—
124	#322	—	2	2	no	—
125	C₆₁	#9834	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	122	2	no	—
127	C₂²×C₈	D₂₄₄	1	2	yes	—
128	C₂²×C₈	D₂₄₄	1	2	yes	—
129	C₂²×C₈	D₂₄₄	1	2	yes	—
130	C₄⋊Q₈	—	61	4	no	—
131	C₄⋊Q₈	—	122	2	no	—
132	C₄⋊Q₈	—	122	2	no	—
133	C₄⋊Q₈	—	122	2	no	—
134	C₂×C₄⋊C₄	—	122	2	no	—
135	C₂×C₄⋊C₄	—	122	2	no	—
136	C₂×C₄⋊C₄	—	122	2	no	—
137	C₂×C₄⋊C₄	—	122	2	no	—
138	C₂×C₄⋊C₄	—	122	2	no	—
139	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
140	#107	—	122	8	no	—
141	#103	—	244	8	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₄×D₆₁	1	2	yes	—
147	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
148	C₂×Q₈	—	122	4	no	—
149	C₂×Q₈	—	244	4	no	—
150	C₂×Q₈	—	244	4	no	—
151	C₂×Q₈	—	244	2	no	—
152	C₂²×C₄	—	244	2	no	—
153	C₂²×C₄	—	244	2	no	—
154	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
156	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	Dic₂₄₄	1	4	yes	—
163	C₂×C₈	#11758198	1	4	yes	—
164	C₄⋊C₄	—	122	4	no	—
165	C₄⋊C₄	—	122	4	no	—
166	C₄⋊C₄	—	244	4	no	—
167	C₄⋊C₄	—	244	4	no	—
168	C₄⋊C₄	—	244	4	no	—
169	C₄⋊C₄	—	244	2	no	—
170	C₄⋊C₄	—	244	2	no	—
171	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
172	C₄²	#11758222	1	1	yes	characteristic
173	C₄²	#11758222	1	1	yes	characteristic
174	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
175	C₂³	#14519774	1	1	yes	characteristic
176	Q₈	—	488	4	no	—
177	Q₈	—	244	8	no	—
178	C₂×C₄	—	488	2	no	—
179	C₂×C₄	—	488	2	no	—
180	C₂×C₄	—	488	2	no	—
181	C₂×C₄	—	488	2	no	—
182	C₂×C₄	—	488	2	no	—
183	C₂×C₄	—	244	4	no	—
184	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
185	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
186	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
187	C₂×C₄	#14519751	1	1	yes	characteristic
188	C₂×C₄	#14519751	1	1	yes	characteristic
189	C₂×C₄	#14519765	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
191	C₂×C₄	#14519799	1	2	yes	—
192	C₂×C₄	#14519805	1	4	yes	—
193	C₂×C₄	#14519810	1	4	yes	—
194	C₈	—	2	4	no	—
195	C₈	—	2	4	no	—
196	C₈	—	4	2	no	—
197	C₂²	#16839664	1	1	yes	characteristic
198	C₂²	#16840142	1	1	yes	characteristic
199	C₂²	#16840146	1	1	yes	characteristic
200	C₂²	#16840151	1	2	yes	—
201	C₂²	#16840156	1	2	yes	—
202	C₄	—	976	2	no	—
203	C₄	—	488	4	no	—
204	C₄	#16839772	1	2	yes	—
205	C₄	C₄×Dic₂₄₄	1	2	yes	—
206	C₄	#16839875	1	2	yes	—
207	C₄	#16840003	1	2	yes	—
208	C₄	#16840015	1	2	yes	—
209	C₄	#16840131	1	2	yes	—
210	C₂	#21445472	1	2	yes	—
211	C₂	#21445496	1	2	yes	—
212	C₂	#21446639	1	1	yes	characteristic
213	C₂	#21446680	1	1	yes	characteristic
214	C₂	#21446695	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#9834
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
8. #16840167	C₄	semidirect
125. C₆₁	#9834	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840167	2
#21446780	2
#21446621	2
#21445947	2
C₂×C₆₁×#322	2
#9834	61