Subgroups — #29578570

Order 15616.

378 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	#21446537	C₂	1	2	yes	maximal
2	C₄×#14519709	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
4	#21443659	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519709	C₄	1	2	yes	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×#14519706	—	2	2	no	—
9	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	#16839525	C₄	1	1	yes	characteristic
11	#16839525	C₄	1	1	yes	characteristic
12	#16839516	—	2	2	no	—
13	#16839513	—	2	2	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×#92418	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₆₁	—	2	2	no	—
22	#14519709	—	2	4	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄⋊#92418	—	2	2	no	—
25	#14519706	—	4	4	no	—
26	#14519684	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	D₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₆₁	—	2	1	no	—
33	C₄×C₈×C₆₁	—	2	1	no	—
34	C₄×C₈×C₆₁	—	2	1	no	—
35	C₄×C₈×C₆₁	D₄	1	2	yes	—
36	C₄×C₈×C₆₁	Q₈	1	2	yes	—
37	C₄×C₈×C₆₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
41	C₂²×Dic₆₁	—	2	2	no	—
42	C₂²×Dic₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	2	2	no	—
45	C₂×C₈×C₆₁	—	2	2	no	—
46	C₂×C₈×C₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	4	1	no	—
49	C₂×C₈×C₆₁	—	4	1	no	—
50	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₆₁	#41	1	1	yes	characteristic
57	C₄⋊Dic₆₁	—	2	2	no	—
58	C₄⋊Dic₆₁	—	2	2	no	—
59	C₄⋊Dic₆₁	—	2	2	no	—
60	C₄⋊Dic₆₁	M₁₆	1	4	yes	—
61	C₄×Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	—	4	2	no	—
63	C₂×#92418	—	4	2	no	—
64	C₂×#92418	—	4	2	no	—
65	C₂×#92418	—	4	2	no	—
66	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	—	2	1	no	—
68	C₂×C₄×C₆₁	—	2	1	no	—
69	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	#125	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₆₁	#131	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₆₁	#135	1	2	yes	—
79	C₂×Dic₆₁	—	2	4	no	—
80	C₂×Dic₆₁	—	4	2	no	—
81	C₂×Dic₆₁	—	4	2	no	—
82	C₂×Dic₆₁	—	4	2	no	—
83	C₂×Dic₆₁	—	4	2	no	—
84	C₂×Dic₆₁	—	4	2	no	—
85	C₈×C₆₁	—	2	2	no	—
86	C₈×C₆₁	—	2	2	no	—
87	C₈×C₆₁	—	2	4	no	—
88	C₈×C₆₁	—	4	2	no	—
89	#92418	—	8	2	no	—
90	#9597	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	#389	1	1	yes	characteristic
92	C₂²×C₆₁	#435	1	2	yes	—
93	C₂²×C₆₁	C₄×SD₁₆	1	2	yes	—
94	C₂²×C₆₁	#444	1	1	yes	characteristic
95	C₂²×C₆₁	#444	1	1	yes	characteristic
96	C₄×C₆₁	—	2	1	no	—
97	C₄×C₆₁	—	2	1	no	—
98	C₄×C₆₁	#423	1	1	yes	characteristic
99	C₄×C₆₁	#424	1	1	yes	characteristic
100	C₄×C₆₁	#446	1	2	yes	—
101	C₄×C₆₁	#465	1	2	yes	—
102	C₄×C₆₁	#498	1	1	yes	characteristic
103	C₄×C₆₁	#499	1	1	yes	characteristic
104	Dic₆₁	—	4	4	no	—
105	Dic₆₁	—	8	2	no	—
106	#1939	—	61	2	no	—
107	C₄×#106	—	61	1	no	—
108	C₂×#334	—	61	1	no	—
109	C₂×#322	D₆₁	1	1	yes	characteristic
110	#1279	—	61	2	no	—
111	C₂×C₆₁	#1582	1	2	yes	—
112	C₂×C₆₁	#1583	1	2	yes	—
113	C₂×C₆₁	#1940	1	1	yes	characteristic
114	C₂×C₆₁	#1941	1	1	yes	characteristic
115	C₂×C₆₁	#1946	1	1	yes	characteristic
116	C₂×#106	—	61	2	no	—
117	C₂×C₄⋊C₈	—	61	2	no	—
118	C₂×C₄⋊C₈	—	122	2	no	—
119	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	61	2	no	—
123	#340	—	61	1	no	—
124	#340	—	61	1	no	—
125	#336	—	122	2	no	—
126	#334	—	122	2	no	—
127	#322	—	2	2	no	—
128	C₆₁	#9597	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	122	2	no	—
131	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
133	C₂²×C₈	D₂₄₄	1	2	yes	—
134	C₂×C₄⋊C₄	—	61	2	no	—
135	C₂×C₄⋊C₄	—	61	2	no	—
136	C₂×C₄²	—	122	2	no	—
137	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
138	#106	—	122	4	no	—
139	C₄⋊C₈	—	122	2	no	—
140	C₄⋊C₈	—	122	2	no	—
141	C₄⋊C₈	—	244	4	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	Dic₁₂₂	1	2	yes	—
146	C₄×C₈	Dic₁₂₂	1	2	yes	—
147	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
148	#95	—	122	2	no	—
149	C₂²×C₄	—	122	2	no	—
150	C₂²×C₄	—	122	2	no	—
151	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
153	C₂²×C₄	#11758222	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	244	2	no	—
162	C₂×C₈	—	244	2	no	—
163	C₂×C₈	—	244	2	no	—
164	C₂×C₈	#11758190	1	4	yes	—
165	C₂×C₈	#11758196	1	2	yes	—
166	C₂×C₈	#11758196	1	2	yes	—
167	C₄⋊C₄	—	61	4	no	—
168	C₄⋊C₄	—	122	2	no	—
169	C₄⋊C₄	—	122	2	no	—
170	C₄⋊C₄	—	122	2	no	—
171	C₄²	—	244	2	no	—
172	C₄²	—	244	2	no	—
173	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
175	C₄²	#11758222	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂³	C₄×D₂₄₄	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	122	4	no	—
181	C₂×C₄	—	244	2	no	—
182	C₂×C₄	—	244	2	no	—
183	C₂×C₄	—	244	2	no	—
184	C₂×C₄	—	244	2	no	—
185	C₂×C₄	—	244	2	no	—
186	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
187	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
188	C₂×C₄	#14519751	1	1	yes	characteristic
189	C₂×C₄	#14519756	1	1	yes	characteristic
190	C₂×C₄	#14519769	1	1	yes	characteristic
191	C₂×C₄	#14519777	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
193	C₂×C₄	#14519785	1	2	yes	—
194	C₂×C₄	C₂×#11758190	1	2	yes	—
195	C₂×C₄	#14519799	1	2	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₈	—	4	2	no	—
200	C₈	—	488	2	no	—
201	C₂²	#16839660	1	1	yes	characteristic
202	C₂²	#16839835	1	2	yes	—
203	C₂²	C₄×#11758190	1	2	yes	—
204	C₂²	#16839885	1	1	yes	characteristic
205	C₂²	#16839885	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	488	2	no	—
209	C₄	—	244	4	no	—
210	C₄	#16839785	1	1	yes	characteristic
211	C₄	#16839790	1	1	yes	characteristic
212	C₄	#16839890	1	2	yes	—
213	C₄	#16839991	1	2	yes	—
214	C₄	#16840160	1	1	yes	characteristic
215	C₄	#16840164	1	1	yes	characteristic
216	C₂	#21444657	1	2	yes	—
217	C₂	#21444664	1	2	yes	—
218	C₂	#21446542	1	1	yes	characteristic
219	C₂	#21446547	1	1	yes	characteristic
220	C₂	#21446572	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	#9597
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
128. C₆₁	#9597	semidirect

Maximal subgroups

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

subgroup	index
#21446537	2
C₄×#14519709	2
C₂×#16839513	2
#21443659	2
C₂×C₆₁×#322	2
#9597	61