Subgroups — #29571317

Order 15616.

358 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	#21444567	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#92418	C₂	1	2	yes	maximal
3	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
4	#21443657	C₂	1	2	yes	maximal
5	C₄×#108×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#16839815	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	#16839515	C₄	1	2	yes	—
10	#16839513	C₄	1	2	yes	—
11	#16839483	—	2	2	no	—
12	#16839483	C₄	1	2	yes	—
13	#16839483	C₄	1	2	yes	—
14	C₈×#92418	—	2	2	no	—
15	C₂×#108×C₆₁	C₄	1	2	yes	—
16	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#341	C₄	1	2	yes	—
19	C₂²×#92418	—	2	1	no	—
20	C₂²×#92418	Q₈	1	2	yes	—
21	C₄⋊#92418	C₂×C₄	1	4	yes	—
22	C₂²⋊#92418	—	2	2	no	—
23	C₄×#92418	—	2	2	no	—
24	C₄×#92418	—	2	1	no	—
25	C₄×#92418	C₂×C₄	1	2	yes	—
26	C₄×#92418	C₂×C₄	1	2	yes	—
27	C₄×#92418	D₄	1	2	yes	—
28	C₂×M₁₆×C₆₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#108×C₆₁	C₈	1	8	yes	complemented
33	#97×C₆₁	—	2	2	no	—
34	C₄×C₈×C₆₁	—	2	2	no	—
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	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
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	M₁₆×C₆₁	—	2	4	no	—
43	M₁₆×C₆₁	C₂×C₈	1	8	yes	—
44	C₂×C₈×C₆₁	—	2	4	no	—
45	C₂×C₈×C₆₁	—	2	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	C₂×C₈	1	4	yes	—
51	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₆₁	#41	1	1	yes	characteristic
55	C₂×#92418	—	2	2	no	—
56	C₂×#92418	—	2	4	no	—
57	C₂×#92418	—	4	2	no	—
58	C₂×#92418	—	4	2	no	—
59	C₂×#92418	C₄⋊C₄	1	4	yes	—
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	C₂²×C₈	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	8	no	—
74	C₈×C₆₁	C₄×C₈	1	2	yes	—
75	C₈×C₆₁	C₄×C₈	1	2	yes	—
76	#92418	—	2	4	no	—
77	#92418	—	4	2	no	—
78	#92418	—	4	4	no	—
79	#8294	—	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	C₈×Q₈	1	2	yes	—
84	C₄×C₆₁	—	2	1	no	—
85	C₄×C₆₁	—	2	1	no	—
86	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
87	C₄×C₆₁	#405	1	1	yes	characteristic
88	C₄×C₆₁	#430	1	2	yes	—
89	C₄×C₆₁	C₈×D₄	1	2	yes	—
90	C₄×C₆₁	#444	1	2	yes	—
91	C₄×#108	D₆₁	1	1	yes	characteristic
92	#1564	—	61	1	no	—
93	C₈×M₁₆	—	61	2	no	—
94	C₂×#322	—	61	1	no	—
95	#1278	—	61	2	no	—
96	C₂×C₆₁	—	2	2	no	—
97	C₂×C₆₁	C₈×C₄⋊C₄	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	Dic₆₁	1	2	yes	—
102	C₄×M₁₆	C₂×D₆₁	1	2	yes	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	—	61	1	no	—
105	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
106	#341	Dic₆₁	1	2	yes	—
107	#335	—	61	2	no	—
108	#334	—	61	2	no	—
109	#322	—	61	2	no	—
110	#322	—	61	2	no	—
111	#322	—	122	2	no	—
112	C₈²	—	122	2	no	—
113	C₆₁	#8294	1	1	yes	characteristic, complemented, Sylow
114	C₂×M₁₆	C₂×Dic₆₁	1	4	yes	—
115	C₂²×C₈	—	61	2	no	—
116	C₂²×C₈	—	122	1	no	—
117	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	#108	#92418	1	8	yes	complemented
121	C₄⋊C₈	—	61	4	no	—
122	C₂²⋊C₈	—	122	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	61	2	no	—
126	C₄×C₈	—	61	2	no	—
127	C₄×C₈	—	61	2	no	—
128	C₄×C₈	—	122	2	no	—
129	C₄×C₈	—	122	1	no	—
130	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
131	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
132	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
133	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
134	C₂²×C₄	Q₈×D₆₁	1	1	yes	characteristic
135	C₂²×C₄	#11758226	1	1	yes	characteristic
136	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×#92418	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	—	122	2	no	—
143	C₂×C₈	—	122	4	no	—
144	C₂×C₈	—	244	2	no	—
145	C₂×C₈	—	244	2	no	—
146	C₂×C₈	C₂×#92418	1	4	yes	—
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	1	yes	characteristic
151	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
153	C₄²	D₄×D₆₁	1	1	yes	characteristic
154	C₄²	#11758224	1	1	yes	characteristic
155	C₂³	Q₈×Dic₆₁	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	C₄⋊C₄×D₆₁	1	1	yes	characteristic
160	C₂×C₄	#14519735	1	1	yes	characteristic
161	C₂×C₄	#14519739	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
163	C₂×C₄	#14519742	1	1	yes	characteristic
164	C₂×C₄	C₂²×#92418	1	2	yes	—
165	C₂×C₄	#14519784	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	8	no	—
169	C₈	—	122	4	no	—
170	C₈	—	244	2	no	—
171	C₈	—	244	4	no	—
172	C₈	C₄×#92418	1	2	yes	—
173	C₈	C₄×#92418	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₆₁	1	1	yes	characteristic
177	C₂²	Q₈×#92418	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×#92418	1	1	yes	characteristic
181	C₄	#16839718	1	1	yes	characteristic
182	C₄	#16839809	1	2	yes	—
183	C₄	D₄×#92418	1	2	yes	—
184	C₄	#16839883	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#92418	1	1	yes	characteristic
187	C₂	#21445706	1	1	yes	characteristic
188	C₂	#21445727	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#108×C₆₁	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#108	D₆₁
hypocenter	C₆₁	#8294
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
32. #108×C₆₁	C₈	semidirect
113. C₆₁	#8294	semidirect
120. #108	#92418	semidirect

Maximal subgroups

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

subgroup	index
#21444567	2
M₁₆×#92418	2
C₂×#16839483	2
#21443657	2
C₄×#108×C₆₁	2
#8294	61