Subgroups — #30599499

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461752	C₂	1	2	yes	maximal
2	C₂×#16840624	C₂	1	2	yes	maximal
3	C₂×#16840623	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519844	—	2	4	no	—
6	C₂×C₄⋊#92420	C₂²	1	4	yes	—
7	#16840624	—	2	4	no	—
8	#16840623	—	2	4	no	—
9	C₂×C₄×#92418	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
16	C₂²×#92420	—	2	4	no	—
17	#14519844	—	4	8	no	—
18	C₄⋊#92420	—	2	4	no	—
19	C₄⋊#92420	—	2	4	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
24	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
25	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
26	C₄×#92418	—	2	2	no	—
27	C₈×Dic₆₁	—	2	2	no	—
28	C₈×Dic₆₁	C₈	1	4	yes	complemented
29	C₈×Dic₆₁	D₄	1	2	yes	—
30	C₈×Dic₆₁	D₄	1	2	yes	—
31	C₈×Dic₆₁	Q₈	1	2	yes	—
32	C₈×Dic₆₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₆₁	C₄⋊C₄	1	2	yes	—
40	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₂×#92420	—	4	4	no	—
43	C₂×#92420	—	4	8	no	—
44	C₂×C₈×C₆₁	—	2	1	no	—
45	C₂×C₈×C₆₁	—	2	1	no	—
46	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
50	C₄²×C₆₁	—	2	1	no	—
51	C₄²×C₆₁	—	2	1	no	—
52	C₄×Dic₆₁	—	2	4	no	—
53	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
54	C₄×Dic₆₁	M₁₆	1	2	yes	—
55	C₄×Dic₆₁	SD₁₆	1	8	yes	—
56	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
58	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
59	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×#92418	—	2	1	no	—
61	C₂×#92418	—	2	1	no	—
62	C₂×#92418	C₄⋊C₄	1	2	yes	—
63	C₂×#92418	C₄⋊C₄	1	2	yes	—
64	C₂×#92418	C₂×C₈	1	2	yes	—
65	C₂×#92418	M₁₆	1	2	yes	—
66	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	2	no	—
68	C₂×C₄×C₆₁	—	2	2	no	—
69	C₂×C₄×C₆₁	#106	1	4	yes	—
70	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₆₁	—	2	2	no	—
77	C₂×Dic₆₁	C₄⋊C₈	1	4	yes	—
78	C₂×Dic₆₁	#106	1	4	yes	—
79	C₂×Dic₆₁	#108	1	4	yes	—
80	C₂×Dic₆₁	#127	1	1	yes	characteristic
81	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×Dic₆₁	C₂×SD₁₆	1	4	yes	—
85	#92420	—	8	4	no	—
86	C₈×C₆₁	—	2	2	no	—
87	C₈×C₆₁	C₄⋊C₈	1	4	yes	complemented
88	#92418	—	2	2	no	—
89	#92418	C₄⋊C₈	1	4	yes	complemented
90	#8805	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	#384	1	1	yes	characteristic
92	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂²×C₆₁	#423	1	1	yes	characteristic
94	C₂²×C₆₁	C₂×#106	1	2	yes	—
95	C₂²×C₆₁	C₂×#108	1	2	yes	—
96	C₄×C₆₁	—	2	4	no	—
97	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₆₁	#423	1	1	yes	characteristic
100	C₄×C₆₁	#423	1	1	yes	characteristic
101	Dic₆₁	—	2	1	no	—
102	Dic₆₁	—	2	1	no	—
103	Dic₆₁	#334	1	8	yes	complemented
104	Dic₆₁	#492	1	2	yes	—
105	Dic₆₁	#498	1	2	yes	—
106	#1842	—	61	2	no	—
107	C₄²×C₈	—	61	1	no	—
108	C₂×#334	—	61	2	no	—
109	C₂×#334	—	61	2	no	—
110	C₂×C₆₁	C₂×#334	1	2	yes	—
111	C₂×C₆₁	#1566	1	2	yes	—
112	C₂×C₆₁	#1842	1	1	yes	characteristic
113	C₂×C₆₁	#1844	1	1	yes	characteristic
114	C₂×C₆₁	#1848	1	1	yes	characteristic
115	C₂×C₄⋊C₈	—	61	4	no	—
116	C₂×C₄⋊C₈	—	122	4	no	—
117	C₂×C₄×C₈	—	61	1	no	—
118	C₂×C₄×C₈	—	61	1	no	—
119	C₂×C₄×C₈	—	61	1	no	—
120	C₂×C₄×C₈	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₂×C₄×C₈	#6923	1	1	yes	characteristic
123	C₄³	—	61	1	no	—
124	#334	—	122	4	no	—
125	#334	—	122	4	no	—
126	C₆₁	#8805	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	61	1	no	—
128	C₂²×C₈	—	61	1	no	—
129	C₂²×C₈	—	122	4	no	—
130	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
131	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
132	C₂×C₄²	—	61	2	no	—
133	C₂×C₄²	—	61	1	no	—
134	C₂×C₄²	—	61	1	no	—
135	C₂×C₄²	—	61	2	no	—
136	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
137	C₄⋊C₈	—	122	4	no	—
138	C₄⋊C₈	—	122	4	no	—
139	C₄⋊C₈	—	244	8	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	61	4	no	—
142	C₄×C₈	—	61	2	no	—
143	C₄×C₈	—	61	2	no	—
144	C₄×C₈	—	61	2	no	—
145	C₄×C₈	—	61	2	no	—
146	C₄×C₈	—	122	2	no	—
147	C₄×C₈	—	122	2	no	—
148	C₂²×C₄	—	61	1	no	—
149	C₂²×C₄	—	61	2	no	—
150	C₂²×C₄	—	61	1	no	—
151	C₂²×C₄	C₄⋊#6923	1	2	yes	—
152	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	61	2	no	—
156	C₂×C₈	—	61	2	no	—
157	C₂×C₈	—	61	2	no	—
158	C₂×C₈	—	61	2	no	—
159	C₂×C₈	—	122	1	no	—
160	C₂×C₈	—	122	1	no	—
161	C₂×C₈	—	244	4	no	—
162	C₂×C₈	—	244	8	no	—
163	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
164	C₂×C₈	C₄⋊#6923	1	2	yes	—
165	C₂×C₈	C₄⋊#6923	1	2	yes	—
166	C₂×C₈	#11758217	1	2	yes	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	61	8	no	—
170	C₄²	—	61	1	no	—
171	C₄²	—	61	1	no	—
172	C₄²	—	61	1	no	—
173	C₄²	—	61	1	no	—
174	C₄²	—	61	2	no	—
175	C₄²	—	61	2	no	—
176	C₄²	—	122	4	no	—
177	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	61	1	no	—
181	C₂×C₄	—	61	1	no	—
182	C₂×C₄	—	61	4	no	—
183	C₂×C₄	—	61	4	no	—
184	C₂×C₄	—	61	4	no	—
185	C₂×C₄	—	61	4	no	—
186	C₂×C₄	—	61	1	no	—
187	C₂×C₄	—	61	1	no	—
188	C₂×C₄	—	122	2	no	—
189	C₂×C₄	#14519845	1	4	yes	—
190	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
192	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
193	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
194	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
195	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	61	4	no	—
198	C₈	—	122	2	no	—
199	C₈	—	488	4	no	—
200	C₈	#14519844	1	4	yes	complemented
201	C₂²	#16840708	1	1	yes	characteristic
202	C₂²	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
203	C₂²	#16840765	1	1	yes	characteristic
204	C₂²	C₂×#14519845	1	2	yes	—
205	C₂²	D₆₁⋊#108	1	2	yes	—
206	C₄	—	2	4	no	—
207	C₄	—	61	8	no	—
208	C₄	—	61	2	no	—
209	C₄	—	61	2	no	—
210	C₄	—	122	1	no	—
211	C₄	—	122	1	no	—
212	C₄	D₆₁⋊(C₄⋊C₈)	1	1	yes	characteristic
213	C₄	C₂×#14519844	1	1	yes	characteristic
214	C₄	#16840765	1	1	yes	characteristic
215	C₄	#16840767	1	1	yes	characteristic
216	C₂	D₆₁⋊#334	1	2	yes	—
217	C₂	#21461373	1	2	yes	—
218	C₂	#21461752	1	1	yes	characteristic
219	C₂	#21461754	1	1	yes	characteristic
220	C₂	#21461759	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8805
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
28. C₈×Dic₆₁	C₈	semidirect
87. C₈×C₆₁	C₄⋊C₈	semidirect
89. #92418	C₄⋊C₈	semidirect
103. Dic₆₁	#334	semidirect
126. C₆₁	#8805	semidirect
200. C₈	#14519844	semidirect

Maximal subgroups

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

subgroup	index
#21461752	2
C₂×#16840624	2
C₂×#16840623	2
C₄×C₈×Dic₆₁	2
#8805	61