Subgroups — #29571320

Order 15616.

334 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519713	C₂	1	1	yes	maximal, characteristic
2	#21444567	C₂	1	1	yes	maximal, characteristic
3	#21444108	C₂	1	2	yes	maximal
4	#21443656	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×#14519713	C₄	1	2	yes	—
8	C₄×#11758194	C₂²	1	2	yes	—
9	#16839527	C₄	1	2	yes	—
10	#16839515	C₄	1	2	yes	—
11	#16839513	C₄	1	2	yes	—
12	#16839484	—	2	2	no	—
13	#16839483	—	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	C₄	1	2	yes	—
18	C₆₁×#322	C₄	1	2	yes	—
19	C₂×#11758194	C₂×C₄	1	4	yes	—
20	#14519713	—	2	4	no	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₂²⋊#92418	—	2	2	no	—
23	#14519688	—	2	2	no	—
24	C₄×#92418	—	2	2	no	—
25	C₄×#92418	—	2	2	no	—
26	C₂²×C₈×C₆₁	—	2	1	no	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₁	—	2	1	no	—
32	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₆₁	D₄	1	2	yes	—
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	—	2	2	no	—
44	C₂×C₈×C₆₁	—	4	2	no	—
45	C₂×C₈×C₆₁	—	4	2	no	—
46	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
52	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₆₁	#41	1	1	yes	characteristic
56	#11758194	—	2	4	no	—
57	#11758194	—	2	4	no	—
58	C₂×#92418	—	2	4	no	—
59	C₂×#92418	—	2	4	no	—
60	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	—	2	2	no	—
62	C₂×C₄×C₆₁	—	2	2	no	—
63	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₆₁	#131	1	2	yes	—
71	C₈×C₆₁	—	2	1	no	—
72	C₈×C₆₁	—	2	1	no	—
73	C₈×C₆₁	—	2	4	no	—
74	C₈×C₆₁	—	4	2	no	—
75	C₈×C₆₁	#97	1	2	yes	—
76	C₈×C₆₁	#97	1	2	yes	—
77	#92418	—	4	4	no	—
78	#92418	—	4	4	no	—
79	#8295	—	61	1	no	maximal, Sylow
80	C₂²×C₆₁	—	2	1	no	—
81	C₂²×C₆₁	—	2	1	no	—
82	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₆₁	#446	1	2	yes	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
87	C₄×C₆₁	#405	1	1	yes	characteristic
88	C₄×C₆₁	#430	1	2	yes	—
89	C₄×C₆₁	#436	1	2	yes	—
90	C₄×C₆₁	#444	1	2	yes	—
91	C₄×#108	—	61	1	no	—
92	#1564	—	61	1	no	—
93	#1451	—	61	2	no	—
94	C₂×#322	D₆₁	1	1	yes	characteristic
95	#1278	—	61	2	no	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	#1770	1	1	yes	characteristic
98	C₂×C₆₁	#1773	1	1	yes	characteristic
99	C₂×C₆₁	#1778	1	1	yes	characteristic
100	#431	—	61	2	no	—
101	C₂×#108	—	61	2	no	—
102	C₄×M₁₆	—	61	2	no	—
103	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#341	—	61	2	no	—
107	#335	—	61	2	no	—
108	#334	—	61	2	no	—
109	#322	—	122	2	no	—
110	#322	—	122	2	no	—
111	#322	Dic₆₁	1	2	yes	—
112	#322	Dic₆₁	1	2	yes	—
113	C₆₁	#8295	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	—	61	4	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	Dic₁₂₂	1	2	yes	—
117	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	#108	—	122	4	no	—
121	C₄⋊C₈	—	61	4	no	—
122	C₂²⋊C₈	—	122	2	no	—
123	#97	—	122	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	122	2	no	—
126	C₄×C₈	—	122	2	no	—
127	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
128	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
129	C₄×C₈	D₂₄₄	1	2	yes	—
130	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
131	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
134	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂²×C₄	#11758222	1	1	yes	characteristic
137	M₁₆	—	122	4	no	—
138	M₁₆	—	122	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	122	4	no	—
144	C₂×C₈	—	122	4	no	—
145	C₂×C₈	#11758189	1	4	yes	—
146	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
149	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
151	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
152	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
153	C₄²	#11758222	1	1	yes	characteristic
154	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
155	C₂³	C₄×Dic₁₂₂	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
160	C₂×C₄	#14519736	1	1	yes	characteristic
161	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
163	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
164	C₂×C₄	C₂×#11758189	1	2	yes	—
165	C₂×C₄	#14519783	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	4	no	—
169	C₈	—	4	2	no	—
170	C₈	—	244	4	no	—
171	C₈	—	244	4	no	—
172	C₈	#14519687	1	2	yes	—
173	C₈	#14519687	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
177	C₂²	#16839892	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×#14519687	1	1	yes	characteristic
181	C₄	#16839717	1	1	yes	characteristic
182	C₄	#16839811	1	2	yes	—
183	C₄	#16839841	1	2	yes	—
184	C₄	#16839885	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#21445686	1	1	yes	characteristic
187	C₂	#21445703	1	1	yes	characteristic
188	C₂	#21445725	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#8295
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
113. C₆₁	#8295	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519713	2
#21444567	2
#21444108	2
#21443656	2
C₂×C₆₁×#322	2
#8295	61