Subgroups — #29573770

Order 15616.

362 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445935	C₂	1	2	yes	maximal
2	#21445564	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839514	C₂	1	2	yes	maximal
4	C₂×#16839513	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
9	#16839516	—	2	4	no	—
10	#16839516	C₄	1	2	yes	—
11	#16839516	C₄	1	2	yes	—
12	#16839514	—	2	4	no	—
13	#16839513	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#334	—	2	2	no	—
17	C₂²×#92418	—	2	2	no	—
18	C₂²×#92418	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₁	D₄	1	2	yes	—
21	C₂×C₄×Dic₆₁	Q₈	1	2	yes	—
22	C₄⋊#92418	—	2	2	no	—
23	C₄⋊#92418	—	2	2	no	—
24	C₄×#92418	D₄	1	2	yes	—
25	C₄×#92418	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₆₁	—	2	2	no	—
31	C₄⋊C₈×C₆₁	—	2	2	no	—
32	C₄×C₈×C₆₁	D₄	1	2	yes	—
33	C₄×C₈×C₆₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
37	C₂²×Dic₆₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₆₁	—	2	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	4	2	no	—
41	C₂×C₈×C₆₁	—	4	4	no	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
44	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₄×Dic₆₁	—	2	2	no	—
49	C₄×Dic₆₁	—	2	2	no	—
50	C₄×Dic₆₁	—	2	4	no	—
51	C₄×Dic₆₁	—	4	2	no	—
52	C₂×#92418	—	2	2	no	—
53	C₂×#92418	—	2	2	no	—
54	C₂×#92418	—	4	2	no	—
55	C₂×#92418	—	4	4	no	—
56	C₂×#92418	C₄⋊C₄	1	4	yes	—
57	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	—	2	2	no	—
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#131	1	2	yes	—
66	C₂×C₄×C₆₁	#131	1	2	yes	—
67	C₂×C₄×C₆₁	#135	1	4	yes	—
68	C₂×Dic₆₁	—	2	4	no	—
69	C₂×Dic₆₁	—	2	4	no	—
70	C₂×Dic₆₁	—	2	4	no	—
71	C₈×C₆₁	—	2	2	no	—
72	C₈×C₆₁	—	2	2	no	—
73	C₈×C₆₁	—	8	2	no	—
74	#92418	—	2	4	no	—
75	#92418	—	8	2	no	—
76	#8811	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	#384	1	1	yes	characteristic
78	C₂²×C₆₁	#424	1	2	yes	—
79	C₂²×C₆₁	#427	1	2	yes	—
80	C₂²×C₆₁	#430	1	2	yes	—
81	C₄×C₆₁	—	2	1	no	—
82	C₄×C₆₁	—	2	1	no	—
83	C₄×C₆₁	#424	1	2	yes	—
84	C₄×C₆₁	#424	1	1	yes	characteristic
85	C₄×C₆₁	#424	1	1	yes	characteristic
86	C₄×C₆₁	#495	1	2	yes	—
87	C₄×C₆₁	#499	1	2	yes	—
88	Dic₆₁	—	4	4	no	—
89	#1831	—	61	2	no	—
90	#1744	—	61	1	no	—
91	C₂×#335	—	61	2	no	—
92	C₂×#334	—	61	1	no	—
93	C₂×#334	D₆₁	1	1	yes	characteristic
94	C₂×C₆₁	#1572	1	4	yes	—
95	C₂×C₆₁	#1843	1	1	yes	characteristic
96	C₂×C₆₁	#1846	1	1	yes	characteristic
97	C₂×C₆₁	#1849	1	1	yes	characteristic
98	C₂×C₄⋊C₈	—	61	2	no	—
99	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
100	C₂×C₄×C₈	—	61	1	no	—
101	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
102	C₄³	—	61	1	no	—
103	#336	—	61	2	no	—
104	#336	—	61	2	no	—
105	#336	—	122	4	no	—
106	#335	—	122	4	no	—
107	#334	—	2	2	no	—
108	#334	—	122	2	no	—
109	C₆₁	#8811	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	61	2	no	—
112	C₂²×C₈	—	122	2	no	—
113	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
115	C₂×C₄²	—	61	2	no	—
116	C₂×C₄²	—	61	2	no	—
117	C₂×C₄²	—	61	2	no	—
118	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	122	2	no	—
122	C₄⋊C₈	—	122	2	no	—
123	C₄×C₈	—	61	2	no	—
124	C₄×C₈	—	61	2	no	—
125	C₄×C₈	Dic₁₂₂	1	2	yes	—
126	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
127	C₂²×C₄	—	61	4	no	—
128	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
129	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
130	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	61	4	no	—
136	C₂×C₈	—	122	2	no	—
137	C₂×C₈	—	122	2	no	—
138	C₂×C₈	—	244	2	no	—
139	C₂×C₈	—	244	4	no	—
140	C₂×C₈	#11758196	1	2	yes	—
141	C₂×C₈	#11758196	1	2	yes	—
142	C₄²	—	122	2	no	—
143	C₄²	—	122	2	no	—
144	C₄²	—	122	4	no	—
145	C₄²	—	244	2	no	—
146	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
147	C₄²	D₄×D₆₁	1	1	yes	characteristic
148	C₄²	Q₈×D₆₁	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂³	C₄⋊C₄×D₆₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	122	4	no	—
153	C₂×C₄	—	122	4	no	—
154	C₂×C₄	—	122	4	no	—
155	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
156	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
157	C₂×C₄	#14519773	1	1	yes	characteristic
158	C₂×C₄	#14519775	1	1	yes	characteristic
159	C₂×C₄	#14519776	1	1	yes	characteristic
160	C₂×C₄	#14519776	1	1	yes	characteristic
161	C₂×C₄	#14519783	1	2	yes	—
162	C₂×C₄	#14519785	1	2	yes	—
163	C₂×C₄	#14519798	1	4	yes	—
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	8	2	no	—
167	C₈	—	122	4	no	—
168	C₈	—	488	2	no	—
169	C₂²	#16839631	1	1	yes	characteristic
170	C₂²	#16839787	1	2	yes	—
171	C₂²	#16839797	1	2	yes	—
172	C₂²	#16839809	1	2	yes	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	244	4	no	—
176	C₄	#16839787	1	2	yes	—
177	C₄	#16839788	1	1	yes	characteristic
178	C₄	#16839790	1	1	yes	characteristic
179	C₄	#16840143	1	2	yes	—
180	C₄	#16840162	1	2	yes	—
181	C₂	#21444596	1	4	yes	—
182	C₂	#21445972	1	1	yes	characteristic
183	C₂	#21445980	1	1	yes	characteristic
184	C₂	#21445992	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#334	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#334	D₆₁
hypocenter	C₆₁	#8811
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
109. C₆₁	#8811	semidirect

Maximal subgroups

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

subgroup	index
#21445935	2
#21445564	2
C₂×#16839514	2
C₂×#16839513	2
C₂×C₆₁×#334	2
#8811	61