Subgroups — #29578915

Order 15616.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#21446755	C₂	1	2	yes	maximal
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839494	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₄₄	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₂₂	C₂²	1	2	yes	—
8	C₄×Dic₂₄₄	—	2	2	no	—
9	C₄×Dic₂₄₄	—	2	2	no	—
10	C₂×#14519711	C₄	1	1	yes	characteristic
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×#14519702	C₄	1	2	yes	—
13	#16839516	—	2	4	no	—
14	#16839513	—	2	2	no	—
15	#16839494	—	2	4	no	—
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#322	—	2	2	no	—
20	C₂²×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₄₄	—	2	4	no	—
22	C₂×Dic₂₄₄	—	2	2	no	—
23	C₂²×#92418	—	2	2	no	—
24	C₄×Dic₁₂₂	—	2	2	no	—
25	C₄×Dic₁₂₂	—	2	2	no	—
26	C₄×Dic₁₂₂	—	2	2	no	—
27	C₄×Dic₁₂₂	—	2	2	no	—
28	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×#11758196	—	2	2	no	—
30	C₂×C₄×Dic₆₁	—	2	2	no	—
31	#14519711	—	2	2	no	—
32	C₄⋊#92418	—	2	2	no	—
33	C₄⋊#92418	—	2	2	no	—
34	#14519702	—	2	4	no	—
35	C₂²×C₈×C₆₁	—	2	1	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₁	D₄	1	2	yes	—
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₁	—	2	1	no	—
41	C₄×C₈×C₆₁	—	2	1	no	—
42	C₄×C₈×C₆₁	—	2	1	no	—
43	C₄×C₈×C₆₁	D₄	1	2	yes	—
44	C₄×C₈×C₆₁	D₄	1	2	yes	—
45	C₄×C₈×C₆₁	D₄	1	2	yes	—
46	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	2	2	no	—
50	C₂²×Dic₆₁	—	2	2	no	—
51	C₂×Dic₁₂₂	—	2	2	no	—
52	C₂×Dic₁₂₂	—	2	2	no	—
53	C₂×Dic₁₂₂	—	2	2	no	—
54	C₂×Dic₁₂₂	—	2	2	no	—
55	C₂×Dic₁₂₂	—	2	2	no	—
56	C₂×Dic₁₂₂	M₁₆	1	4	yes	—
57	C₂×C₈×C₆₁	—	2	4	no	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	—	2	2	no	—
62	C₂×C₈×C₆₁	—	2	4	no	—
63	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₁	#41	1	1	yes	characteristic
69	C₄⋊Dic₆₁	—	2	2	no	—
70	C₄⋊Dic₆₁	—	2	2	no	—
71	#11758196	—	4	2	no	—
72	#11758196	—	4	2	no	—
73	C₄×Dic₆₁	—	4	2	no	—
74	C₄×Dic₆₁	—	4	2	no	—
75	C₂×#92418	—	4	2	no	—
76	C₂×#92418	—	4	4	no	—
77	Dic₂₄₄	—	2	8	no	—
78	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
84	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
85	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₆₁	#131	1	2	yes	—
88	C₂×C₄×C₆₁	#137	1	4	yes	—
89	C₂×Dic₆₁	—	2	4	no	—
90	C₂×Dic₆₁	—	4	2	no	—
91	C₂×Dic₆₁	—	4	2	no	—
92	C₂×Dic₆₁	—	4	2	no	—
93	C₂×Dic₆₁	—	4	4	no	—
94	Dic₁₂₂	—	2	4	no	—
95	Dic₁₂₂	—	2	4	no	—
96	Dic₁₂₂	—	4	2	no	—
97	Dic₁₂₂	—	4	2	no	—
98	C₈×C₆₁	—	2	4	no	—
99	C₈×C₆₁	—	2	2	no	—
100	C₈×C₆₁	—	2	2	no	—
101	C₈×C₆₁	—	4	2	no	—
102	#92418	—	8	2	no	—
103	#9664	—	61	1	no	maximal, Sylow
104	C₂²×C₆₁	#390	1	1	yes	characteristic
105	C₂²×C₆₁	#435	1	2	yes	—
106	C₂²×C₆₁	#441	1	2	yes	—
107	C₂²×C₆₁	#443	1	2	yes	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	—	2	1	no	—
110	C₄×C₆₁	#407	1	1	yes	characteristic
111	C₄×C₆₁	#408	1	1	yes	characteristic
112	C₄×C₆₁	#435	1	2	yes	—
113	C₄×C₆₁	#470	1	2	yes	—
114	C₄×C₆₁	#497	1	2	yes	—
115	Dic₆₁	—	4	4	no	—
116	Dic₆₁	—	8	2	no	—
117	C₂×C₄×Q₁₆	—	61	1	no	—
118	#1984	—	61	2	no	—
119	C₂×#334	—	61	1	no	—
120	C₂×Q₈⋊C₈	—	61	2	no	—
121	C₂×#322	D₆₁	1	1	yes	characteristic
122	C₂×C₆₁	#1587	1	4	yes	—
123	C₂×C₆₁	#1955	1	1	yes	characteristic
124	C₂×C₆₁	#1971	1	1	yes	characteristic
125	C₂×C₆₁	#1974	1	1	yes	characteristic
126	C₂²×Q₁₆	—	61	1	no	—
127	C₂×C₄×Q₈	—	61	2	no	—
128	C₄×Q₁₆	—	122	2	no	—
129	C₄×Q₁₆	—	122	2	no	—
130	C₂×#107	—	61	1	no	—
131	C₂×C₄⋊C₈	—	61	2	no	—
132	C₂×#103	—	61	2	no	—
133	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
136	#336	—	122	4	no	—
137	#334	—	122	2	no	—
138	Q₈⋊C₈	—	122	4	no	—
139	#322	—	2	2	no	—
140	C₆₁	#9664	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	61	2	no	—
142	C₂×Q₁₆	—	122	4	no	—
143	C₂×Q₁₆	—	122	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	122	2	no	—
146	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
148	C₂²×C₈	D₂₄₄	1	2	yes	—
149	C₄×Q₈	—	122	2	no	—
150	C₄×Q₈	—	122	2	no	—
151	C₄×Q₈	—	122	2	no	—
152	C₄×Q₈	—	122	2	no	—
153	C₂×C₄⋊C₄	—	61	2	no	—
154	C₂×C₄⋊C₄	—	122	2	no	—
155	C₂×C₄²	—	122	2	no	—
156	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
157	#107	—	122	2	no	—
158	C₄⋊C₈	—	122	2	no	—
159	C₄⋊C₈	—	122	2	no	—
160	#103	—	122	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₂₄₄	1	2	yes	—
165	C₄×C₈	D₂₄₄	1	2	yes	—
166	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
167	C₂×Q₈	—	61	4	no	—
168	C₂×Q₈	—	122	2	no	—
169	C₂×Q₈	—	122	2	no	—
170	C₂×Q₈	—	122	2	no	—
171	C₂×Q₈	—	122	2	no	—
172	C₂×Q₈	—	122	2	no	—
173	C₂²×C₄	—	122	2	no	—
174	C₂²×C₄	—	122	2	no	—
175	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
177	C₂²×C₄	#11758222	1	1	yes	characteristic
178	Q₁₆	—	122	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	244	4	no	—
186	C₂×C₈	—	244	2	no	—
187	C₂×C₈	#11758198	1	2	yes	—
188	C₂×C₈	#11758198	1	2	yes	—
189	C₄⋊C₄	—	122	2	no	—
190	C₄⋊C₄	—	122	2	no	—
191	C₄⋊C₄	—	244	2	no	—
192	C₄⋊C₄	—	244	2	no	—
193	C₄²	—	244	2	no	—
194	C₄²	—	244	2	no	—
195	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
196	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
197	C₄²	#11758222	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
199	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
200	Q₈	—	122	4	no	—
201	Q₈	—	122	4	no	—
202	Q₈	—	244	2	no	—
203	Q₈	—	244	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	122	4	no	—
206	C₂×C₄	—	244	4	no	—
207	C₂×C₄	—	244	2	no	—
208	C₂×C₄	—	244	2	no	—
209	C₂×C₄	—	244	2	no	—
210	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
211	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
212	C₂×C₄	#14519751	1	1	yes	characteristic
213	C₂×C₄	#14519751	1	1	yes	characteristic
214	C₂×C₄	#14519765	1	1	yes	characteristic
215	C₂×C₄	#14519774	1	1	yes	characteristic
216	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
217	C₂×C₄	#14519785	1	2	yes	—
218	C₂×C₄	#14519810	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	488	2	no	—
224	C₂²	#16839664	1	1	yes	characteristic
225	C₂²	#16839835	1	2	yes	—
226	C₂²	#16839870	1	2	yes	—
227	C₂²	#16839880	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	488	2	no	—
231	C₄	—	244	4	no	—
232	C₄	#16839725	1	1	yes	characteristic
233	C₄	#16839730	1	1	yes	characteristic
234	C₄	#16839835	1	2	yes	—
235	C₄	#16840020	1	2	yes	—
236	C₄	#16840156	1	2	yes	—
237	C₂	#21444692	1	4	yes	—
238	C₂	#21446617	1	1	yes	characteristic
239	C₂	#21446690	1	1	yes	characteristic
240	C₂	#21446707	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₄₄
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#9664
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
140. C₆₁	#9664	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₄₄	2
#21446755	2
C₂×#16839513	2
C₂×#16839494	2
C₂×C₆₁×#322	2
#9664	61