Subgroups — #29596402

Order 15616.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
2	#21445991	C₂	1	1	yes	maximal, characteristic
3	#21443912	C₂	1	4	yes	maximal
4	C₆₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₂²	1	2	yes	—
6	C₂×#14519713	—	2	1	no	—
7	C₂×#14519713	C₂²	1	2	yes	—
8	#16839786	—	2	1	no	—
9	C₄×#11758185	C₄	1	2	yes	—
10	C₄×#11758185	C₄	1	2	yes	—
11	C₂×#108×C₆₁	—	2	1	no	—
12	C₂×#108×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#11758194	—	2	2	no	—
16	C₂×#11758185	—	2	2	no	—
17	C₂×#11758185	—	2	2	no	—
18	C₂×#11758185	C₂×C₄	1	4	yes	—
19	#14519713	—	2	4	no	—
20	#14519713	—	4	2	no	—
21	C₄⋊#92418	—	4	2	no	—
22	C₂×M₁₆×C₆₁	—	2	2	no	—
23	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	#108×C₆₁	—	2	4	no	—
27	#108×C₆₁	—	4	2	no	—
28	C₄⋊C₈×C₆₁	—	4	2	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	M₁₆×C₆₁	—	4	2	no	—
36	M₁₆×C₆₁	—	4	2	no	—
37	C₂×C₈×C₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₈×C₆₁	SD₁₆	1	2	yes	—
42	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₆₁	#41	1	2	yes	—
45	C₂×C₈×C₆₁	#41	1	2	yes	—
46	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₁	D₈	1	2	yes	—
49	#11758194	—	4	2	no	—
50	#11758194	—	4	2	no	—
51	C₂×#92418	—	4	2	no	—
52	#11758185	—	2	4	no	—
53	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₆₁	—	2	4	no	—
55	C₂×C₄×C₆₁	#107	1	2	yes	—
56	C₂×C₄×C₆₁	#107	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₆₁	—	8	2	no	—
64	C₈×C₆₁	#106	1	2	yes	—
65	C₈×C₆₁	#106	1	2	yes	—
66	C₈×C₆₁	#117	1	2	yes	—
67	C₈×C₆₁	#125	1	2	yes	—
68	#92418	—	8	2	no	—
69	#12891	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
72	C₂²×C₆₁	#487	1	1	yes	characteristic
73	C₂²×C₆₁	#487	1	1	yes	characteristic
74	C₄×C₆₁	—	2	2	no	—
75	C₄×C₆₁	#368	1	4	yes	—
76	C₄×C₆₁	#382	1	1	yes	characteristic
77	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
78	C₄×C₆₁	#486	1	1	yes	characteristic
79	C₄×C₆₁	#486	1	1	yes	characteristic
80	C₂×C₄×C₁₆	—	61	1	no	—
81	#1848	—	61	1	no	—
82	#1848	D₆₁	1	1	yes	characteristic
83	#1381	—	61	4	no	—
84	C₂×C₆₁	—	2	1	no	—
85	C₂×C₆₁	—	2	1	no	—
86	C₂×C₆₁	#1834	1	1	yes	characteristic
87	C₂×C₆₁	C₂×#368	1	2	yes	—
88	C₂²×C₁₆	—	61	2	no	—
89	C₂×#108	—	2	1	no	—
90	C₂×#108	—	61	2	no	—
91	C₂×#108	—	122	1	no	—
92	C₂×#108	C₂×D₆₁	1	2	yes	—
93	#423	—	2	1	no	—
94	#423	—	122	1	no	—
95	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
96	C₄×C₁₆	—	61	2	no	—
97	C₄×C₁₆	—	61	2	no	—
98	C₆₁	#12891	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	122	2	no	—
101	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂×C₁₆	—	61	4	no	—
105	C₂×C₁₆	—	122	2	no	—
106	C₂×C₁₆	—	122	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	122	4	no	—
110	#108	—	244	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	244	2	no	—
113	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
114	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
115	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
116	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
117	C₂²×C₄	#11758202	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	244	2	no	—
122	M₁₆	—	244	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	244	2	no	—
125	C₂×C₈	#11758196	1	1	yes	characteristic
126	C₂×C₈	#11758196	1	1	yes	characteristic
127	C₂×C₈	#11758200	1	2	yes	—
128	C₂×C₈	#11758201	1	2	yes	—
129	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
131	C₂×C₈	#11758224	1	2	yes	—
132	C₂×C₈	#11758226	1	2	yes	—
133	C₄²	#11758196	1	1	yes	characteristic
134	C₄²	#11758196	1	1	yes	characteristic
135	C₄²	#11758199	1	2	yes	—
136	C₁₆	—	122	4	no	—
137	C₂³	C₂×#11758202	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#14519710	1	2	yes	—
140	C₂×C₄	#14519710	1	2	yes	—
141	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
142	C₂×C₄	#14519764	1	1	yes	characteristic
143	C₂×C₄	#14519766	1	1	yes	characteristic
144	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
145	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
146	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	488	2	no	—
149	C₈	#14519708	1	2	yes	—
150	C₈	#14519708	1	2	yes	—
151	C₈	#14519735	1	2	yes	—
152	C₈	#14519768	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#14519710	1	1	yes	characteristic
155	C₂²	#16840112	1	1	yes	characteristic
156	C₂²	#16840114	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#16839591	1	4	yes	—
159	C₄	#16839627	1	1	yes	characteristic
160	C₄	C₂×#14519708	1	1	yes	characteristic
161	C₄	#16840108	1	1	yes	characteristic
162	C₄	#16840110	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#21445946	1	1	yes	characteristic
166	C₂	C₂×#16839591	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1848	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1848	D₆₁
hypocenter	C₆₁	#12891
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
98. C₆₁	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11758185	2
#21445991	2
#21443912	2
C₆₁×#1848	2
#12891	61