Subgroups — #30598038

Order 15616.

446 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519846	C₂	1	1	yes	maximal, characteristic
2	#21461612	C₂	1	2	yes	maximal
3	C₈×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#21460653	C₂	1	2	yes	maximal
5	#335×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519846	C₄	1	2	yes	—
7	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×#6923	—	2	2	no	—
10	C₄×C₄⋊#6923	C₂²	1	2	yes	—
11	#16840636	C₄	1	2	yes	—
12	(C₂×#6923)⋊C₈	—	2	2	no	—
13	C₈×#92420	C₄	1	2	yes	complemented
14	C₈×#92420	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
16	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
17	#16839515	C₄	1	1	yes	characteristic, complemented
18	#16839514	C₄	1	2	yes	complemented
19	C₆₁×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×D₆₁⋊C₈	—	2	1	no	—
21	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
22	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄⋊#6923	C₂×C₄	1	4	yes	—
24	C₂×C₄×#6923	—	2	2	no	—
25	#14519846	C₈	1	8	yes	complemented
26	C₄×#92420	—	2	1	no	—
27	C₄×#92420	C₂×C₄	1	2	yes	—
28	C₄×#92420	C₂×C₄	1	2	yes	—
29	C₄×#92420	Q₈	1	2	yes	—
30	C₈×#6923	—	4	4	no	—
31	(C₂×#6923)⋊C₄	—	2	2	no	—
32	C₂×C₈×D₆₁	—	2	2	no	—
33	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₆₁	C₂³	1	1	yes	characteristic
36	C₄⋊#92418	C₂×C₄	1	2	yes	—
37	#14519706	—	2	2	no	—
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6923	—	2	4	no	—
44	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
47	C₂×#92420	—	2	2	no	—
48	C₂×#92420	C₄⋊C₄	1	2	yes	—
49	C₂×#92420	C₄⋊C₄	1	2	yes	—
50	C₄⋊#6923	—	2	4	no	—
51	C₄⋊#6923	C₂×C₈	1	8	yes	—
52	C₄×#6923	—	4	2	no	—
53	C₄×#6923	—	4	2	no	—
54	D₆₁⋊C₈	—	2	4	no	—
55	D₆₁⋊C₈	—	2	1	no	—
56	D₆₁⋊C₈	—	2	1	no	—
57	D₆₁⋊C₈	D₈	1	2	yes	—
58	D₆₁⋊C₈	Q₁₆	1	2	yes	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
62	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₂×#92418	—	2	2	no	—
67	C₂×#92418	C₄²	1	1	yes	characteristic
68	C₂×#92418	C₄²	1	1	yes	characteristic
69	C₈×D₆₁	—	2	2	no	—
70	C₈×D₆₁	—	4	2	no	—
71	C₈×D₆₁	—	4	2	no	—
72	C₈×D₆₁	C₂×C₈	1	4	yes	—
73	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
74	C₂×#6923	—	2	8	no	—
75	C₂×#6923	—	4	4	no	—
76	C₂×#6923	—	4	4	no	—
77	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
80	C₂×Dic₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×Dic₆₁	#117	1	1	yes	characteristic
82	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
83	C₄×D₆₁	—	2	2	no	—
84	C₄×D₆₁	C₂²×C₈	1	2	yes	—
85	C₄×D₆₁	#131	1	2	yes	—
86	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
87	C₄×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
88	C₄×D₆₁	#135	1	1	yes	characteristic
89	C₄×D₆₁	#135	1	1	yes	characteristic
90	#92420	—	2	4	no	—
91	#92420	—	2	2	no	—
92	#92420	#107	1	4	yes	complemented
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	—	4	2	no	—
95	C₈×C₆₁	C₄×C₈	1	2	yes	complemented
96	#92418	—	2	1	no	—
97	#92418	—	4	2	no	—
98	#92418	C₄×C₈	1	2	yes	complemented
99	C₈×#107	—	61	1	no	maximal, Sylow
100	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₆₁	C₈×D₄	1	2	yes	—
102	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
103	C₂×D₆₁	C₄×Q₁₆	1	1	yes	characteristic
104	C₂×D₆₁	#443	1	2	yes	—
105	#6923	—	4	8	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₆₁	#405	1	1	yes	characteristic
109	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
110	C₄×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
111	Dic₆₁	—	2	1	no	—
112	Dic₆₁	C₂×#107	1	1	yes	characteristic
113	Dic₆₁	#427	1	1	yes	characteristic
114	Dic₆₁	C₈×Q₈	1	2	yes	—
115	C₄×#107	—	61	1	no	—
116	C₈×C₄⋊C₄	—	61	2	no	—
117	C₂×#335	—	61	1	no	—
118	C₂×C₈²	—	61	1	no	—
119	#1279	—	61	2	no	—
120	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₆₁	#1772	1	1	yes	characteristic
122	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
123	D₆₁	C₈×D₈	1	2	yes	—
124	D₆₁	C₈×Q₁₆	1	2	yes	—
125	C₂×#107	—	61	2	no	—
126	C₂×C₄⋊C₈	—	61	2	no	—
127	C₂×C₄×C₈	—	61	1	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	—	122	2	no	—
131	C₄×C₄⋊C₄	—	61	2	no	—
132	#340	—	61	2	no	—
133	#336	—	122	2	no	—
134	#335	—	61	2	no	—
135	#335	—	61	1	no	—
136	#335	#6923	1	1	yes	characteristic, complemented
137	C₈²	—	61	2	no	—
138	C₈²	—	61	2	no	—
139	C₆₁	C₈×#107	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	1	no	—
142	C₂²×C₈	—	61	1	no	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	122	2	no	—
145	C₂²×C₈	—	122	1	no	—
146	C₂×C₄⋊C₄	—	61	4	no	—
147	C₂×C₄²	—	61	1	no	—
148	C₂×C₄²	—	122	2	no	—
149	#107	—	61	8	no	—
150	C₄⋊C₈	—	61	2	no	—
151	C₄⋊C₈	—	122	2	no	—
152	C₄⋊C₈	C₂×#6923	1	2	yes	—
153	C₄×C₈	—	61	2	no	—
154	C₄×C₈	—	61	2	no	—
155	C₄×C₈	—	61	1	no	—
156	C₄×C₈	—	61	1	no	—
157	C₄×C₈	—	61	2	no	—
158	C₄×C₈	—	61	1	no	—
159	C₄×C₈	—	122	1	no	—
160	C₄×C₈	—	244	4	no	—
161	C₄×C₈	C₂×#6923	1	1	yes	characteristic
162	#95	—	122	2	no	—
163	C₂²×C₄	—	61	1	no	—
164	C₂²×C₄	—	61	1	no	—
165	C₂²×C₄	—	61	1	no	—
166	C₂²×C₄	—	122	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	61	2	no	—
169	C₂×C₈	—	61	2	no	—
170	C₂×C₈	—	61	2	no	—
171	C₂×C₈	—	61	1	no	—
172	C₂×C₈	—	61	4	no	—
173	C₂×C₈	—	61	2	no	—
174	C₂×C₈	—	61	1	no	—
175	C₂×C₈	—	122	2	no	—
176	C₂×C₈	—	122	2	no	—
177	C₂×C₈	—	122	1	no	—
178	C₂×C₈	—	122	1	no	—
179	C₂×C₈	—	122	4	no	—
180	C₂×C₈	—	122	2	no	—
181	C₂×C₈	—	244	2	no	—
182	C₂×C₈	—	244	2	no	—
183	C₂×C₈	C₄×#6923	1	1	yes	characteristic
184	C₂×C₈	C₄×#6923	1	1	yes	characteristic
185	C₄⋊C₄	—	61	8	no	—
186	C₄⋊C₄	—	122	4	no	—
187	C₄²	—	61	1	no	—
188	C₄²	—	61	1	no	—
189	C₄²	—	61	1	no	—
190	C₄²	—	244	2	no	—
191	C₄²	—	244	2	no	—
192	C₄²	C₂²×#6923	1	1	yes	characteristic
193	C₂³	—	61	1	no	—
194	C₂×C₄	—	61	1	no	—
195	C₂×C₄	—	61	1	no	—
196	C₂×C₄	—	61	2	no	—
197	C₂×C₄	—	61	2	no	—
198	C₂×C₄	—	61	1	no	—
199	C₂×C₄	—	61	1	no	—
200	C₂×C₄	—	61	1	no	—
201	C₂×C₄	—	61	1	no	—
202	C₂×C₄	—	61	1	no	—
203	C₂×C₄	—	122	2	no	—
204	C₂×C₄	—	122	8	no	—
205	C₂×C₄	—	244	4	no	—
206	C₂×C₄	—	244	4	no	—
207	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
208	C₂×C₄	D₄×#6923	1	1	yes	characteristic
209	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	61	4	no	—
213	C₈	—	61	2	no	—
214	C₈	—	122	2	no	—
215	C₈	—	122	4	no	—
216	C₈	—	122	1	no	—
217	C₈	—	244	2	no	—
218	C₈	C₈×#6923	1	2	yes	complemented
219	C₂²	—	61	2	no	—
220	C₂²	—	61	2	no	—
221	C₂²	—	61	1	no	—
222	C₂²	—	61	1	no	—
223	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	61	2	no	—
226	C₄	—	61	1	no	—
227	C₄	—	61	1	no	—
228	C₄	—	122	1	no	—
229	C₄	—	244	8	no	—
230	C₄	C₂×C₈×#6923	1	1	yes	characteristic
231	C₄	#16840728	1	1	yes	characteristic
232	C₄	D₈×#6923	1	1	yes	characteristic
233	C₄	Q₁₆×#6923	1	1	yes	characteristic
234	C₂	—	61	2	no	—
235	C₂	—	61	2	no	—
236	C₂	#21461612	1	1	yes	characteristic
237	C₂	#21461624	1	1	yes	characteristic
238	C₂	#107×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#335	C₄
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	#335	#6923
hypocenter	C₆₁	C₈×#107
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
13. C₈×#92420	C₄	semidirect
14. C₈×#92420	C₄	semidirect
17. #16839515	C₄	semidirect
18. #16839514	C₄	semidirect
19. C₆₁×#335	C₄	semidirect
25. #14519846	C₈	semidirect
92. #92420	#107	semidirect
95. C₈×C₆₁	C₄×C₈	semidirect
98. #92418	C₄×C₈	semidirect
136. #335	#6923	semidirect
139. C₆₁	C₈×#107	semidirect
218. C₈	C₈×#6923	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519846	2
#21461612	2
C₈×D₆₁⋊C₈	2
#21460653	2
#335×D₆₁	2
C₈×#107	61