Subgroups — #29591586

Order 15616.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447575	C₂	1	1	yes	maximal, characteristic
2	#21444102	C₂	1	1	yes	maximal, characteristic
3	#21443888	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	—	2	1	no	—
6	#16839815	C₄	1	1	yes	characteristic
7	#16839815	C₄	1	1	yes	characteristic
8	C₄⋊#11758185	—	2	1	no	—
9	C₄⋊#11758185	C₂²	1	2	yes	—
10	#16839483	C₄	1	2	yes	—
11	C₈×#92418	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₆₁	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×C₄⋊C₁₆	—	2	1	no	—
15	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	#14519717	—	4	2	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₄⋊#92418	—	2	1	no	—
19	C₄⋊#92418	C₈	1	2	yes	—
20	C₂²⋊#92418	—	2	1	no	—
21	C₂²⋊#92418	C₈	1	2	yes	—
22	C₄×#92418	C₂×C₄	1	2	yes	—
23	C₄×#92418	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	—	4	2	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	D₈	1	2	yes	—
40	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
41	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
46	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×#92418	—	2	2	no	—
48	C₂×#92418	C₂×C₈	1	4	yes	—
49	C₁₆×C₆₁	—	4	2	no	—
50	#11758185	—	4	2	no	—
51	C₂³×C₆₁	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	4	1	no	—
54	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#107	1	2	yes	—
58	C₂×C₄×C₆₁	#108	1	2	yes	—
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
62	C₈×C₆₁	—	2	1	no	—
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	#107	1	2	yes	—
65	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
66	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
67	#92418	—	2	4	no	—
68	#92418	—	4	2	no	—
69	#12005	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	1	no	—
71	C₂²×C₆₁	—	2	1	no	—
72	C₂²×C₆₁	#335	1	2	yes	—
73	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	#335	1	2	yes	—
78	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
79	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
80	C₄×C₆₁	#504	1	2	yes	—
81	#2150	—	61	1	no	—
82	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
83	#1450	—	61	1	no	—
84	#1370	—	61	4	no	—
85	C₂×C₆₁	—	2	2	no	—
86	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
87	C₂×C₆₁	#2151	1	1	yes	characteristic
88	C₂×C₆₁	#2153	1	1	yes	characteristic
89	C₂×M₃₂	—	122	1	no	—
90	C₂²×C₁₆	—	2	1	no	—
91	#431	—	61	1	no	—
92	#431	—	61	1	no	—
93	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	—	61	2	no	—
96	C₄⋊C₁₆	—	122	1	no	—
97	C₄⋊C₁₆	C₂×D₆₁	1	2	yes	—
98	#322	—	61	2	no	—
99	C₈²	—	61	2	no	—
100	C₆₁	#12005	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
104	M₃₂	—	244	2	no	—
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	4	2	no	—
107	C₂×C₁₆	—	122	2	no	—
108	C₄⋊C₈	—	61	2	no	—
109	C₄⋊C₈	—	122	1	no	—
110	C₂²⋊C₈	—	61	2	no	—
111	C₂²⋊C₈	—	122	1	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	—	61	2	no	—
114	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
115	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
118	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
119	C₂²×C₄	#11758189	1	1	yes	characteristic
120	C₂²×C₄	#11758196	1	1	yes	characteristic
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	61	4	no	—
123	C₂×C₈	—	122	2	no	—
124	C₂×C₈	#11758196	1	1	yes	characteristic
125	C₂×C₈	#11758196	1	1	yes	characteristic
126	C₂×C₈	#11758199	1	2	yes	—
127	C₂×C₈	#11758202	1	2	yes	—
128	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
130	C₄²	C₈×D₆₁	1	1	yes	characteristic
131	C₄²	#11758189	1	1	yes	characteristic
132	C₄²	#11758196	1	1	yes	characteristic
133	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₁₆	—	244	2	no	—
136	C₂³	#14519706	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	#14519706	1	1	yes	characteristic
140	C₂×C₄	#14519706	1	1	yes	characteristic
141	C₂×C₄	#14519706	1	1	yes	characteristic
142	C₂×C₄	#14519710	1	2	yes	—
143	C₂×C₄	#14519712	1	2	yes	—
144	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
145	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
146	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	122	4	no	—
150	C₈	—	244	2	no	—
151	C₈	#14519710	1	2	yes	—
152	C₈	C₂×#11758199	1	1	yes	characteristic
153	C₈	C₂×#11758202	1	1	yes	characteristic
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#16839514	1	2	yes	—
157	C₂²	C₂×#14519706	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#16839514	1	2	yes	—
162	C₄	C₂×#14519710	1	1	yes	characteristic
163	C₄	C₂×#14519712	1	1	yes	characteristic
164	C₄	#16840178	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	C₂×#16839514	1	1	yes	characteristic
167	C₂	#21447577	1	1	yes	characteristic
168	C₂	#21447584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂³×C₆₁	C₄⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#12005
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #21443888	C₂	semidirect
100. C₆₁	#12005	semidirect

Maximal subgroups

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

subgroup	index
#21447575	2
#21444102	2
#21443888	2
C₂×C₆₁×C₄⋊C₁₆	2
#12005	61