Subgroups — #29596461

Order 15616.

314 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	C₂×#16839591	C₂	1	4	yes	maximal
2	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic
3	#21445991	C₂	1	1	yes	maximal, characteristic
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	#16839591	—	2	8	no	—
10	C₄×#11758185	C₄	1	2	yes	—
11	C₄×#11758185	C₄	1	2	yes	—
12	C₂×#108×C₆₁	—	2	1	no	—
13	C₂×#108×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#11758194	—	2	2	no	—
17	C₂×#11758185	C₂×C₄	1	4	yes	—
18	C₂×#11758185	D₄	1	4	yes	—
19	C₂×#11758185	Q₈	1	4	yes	—
20	#14519713	—	2	4	no	—
21	#14519713	—	4	2	no	—
22	C₄⋊#92418	—	4	2	no	—
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
27	#108×C₆₁	—	2	4	no	—
28	#108×C₆₁	—	4	2	no	—
29	C₄⋊C₈×C₆₁	—	4	2	no	—
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	Q₁₆	1	2	yes	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	M₁₆×C₆₁	—	4	2	no	—
37	M₁₆×C₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₁	D₈	1	2	yes	—
42	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
46	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₆₁	D₈	1	2	yes	—
50	#11758194	—	4	2	no	—
51	#11758194	—	4	2	no	—
52	C₂×#92418	—	4	2	no	—
53	#11758185	C₄⋊C₄	1	8	yes	—
54	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	4	no	—
56	C₂×C₄×C₆₁	#107	1	2	yes	—
57	C₂×C₄×C₆₁	#107	1	2	yes	—
58	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₆₁	—	8	2	no	—
65	C₈×C₆₁	#107	1	2	yes	—
66	C₈×C₆₁	#107	1	2	yes	—
67	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₈×C₆₁	C₄⋊Q₈	1	2	yes	—
69	#92418	—	8	2	no	—
70	#12909	—	61	1	no	maximal, Sylow
71	C₂²×C₆₁	—	2	2	no	—
72	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
73	C₂²×C₆₁	#494	1	1	yes	characteristic
74	C₂²×C₆₁	#500	1	1	yes	characteristic
75	C₄×C₆₁	—	2	2	no	—
76	C₄×C₆₁	#368	1	4	yes	complemented
77	C₄×C₆₁	#384	1	1	yes	characteristic
78	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
79	C₄×C₆₁	#493	1	1	yes	characteristic
80	C₄×C₆₁	#500	1	1	yes	characteristic
81	C₂×#368	—	61	4	no	—
82	C₂×C₄×C₁₆	—	61	1	no	—
83	#1848	—	61	1	no	—
84	#1848	D₆₁	1	1	yes	characteristic
85	C₂×C₆₁	—	2	1	no	—
86	C₂×C₆₁	—	2	1	no	—
87	C₂×C₆₁	#1845	1	1	yes	characteristic
88	C₂×C₆₁	C₂×#368	1	2	yes	—
89	C₂²×C₁₆	—	61	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	61	2	no	—
92	C₂×#108	—	122	1	no	—
93	C₂×#108	C₂×D₆₁	1	2	yes	—
94	#423	—	2	1	no	—
95	#423	—	122	1	no	—
96	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
97	#368	—	122	8	no	—
98	C₄×C₁₆	—	61	2	no	—
99	C₄×C₁₆	—	61	2	no	—
100	C₆₁	#12909	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	122	2	no	—
103	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
106	C₂×C₁₆	—	61	4	no	—
107	C₂×C₁₆	—	61	4	no	—
108	C₂×C₁₆	—	61	4	no	—
109	#108	—	2	4	no	—
110	#108	—	4	2	no	—
111	#108	—	122	4	no	—
112	#108	—	244	2	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	244	2	no	—
115	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
116	C₄×C₈	Dic₁₂₂	1	1	yes	characteristic
117	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
118	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
119	C₂²×C₄	#11758202	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	244	2	no	—
124	M₁₆	—	244	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	244	2	no	—
127	C₂×C₈	#11758196	1	1	yes	characteristic
128	C₂×C₈	#11758196	1	1	yes	characteristic
129	C₂×C₈	#11758199	1	2	yes	—
130	C₂×C₈	#11758202	1	2	yes	—
131	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₈	D₄×D₆₁	1	2	yes	—
134	C₂×C₈	Q₈×D₆₁	1	2	yes	—
135	C₄²	#11758196	1	1	yes	characteristic
136	C₄²	#11758196	1	1	yes	characteristic
137	C₄²	#11758199	1	2	yes	—
138	C₁₆	—	61	8	no	—
139	C₂³	C₂×#11758202	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#14519710	1	2	yes	—
142	C₂×C₄	#14519710	1	2	yes	—
143	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
144	C₂×C₄	#14519773	1	1	yes	characteristic
145	C₂×C₄	#14519775	1	1	yes	characteristic
146	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
147	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
148	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	488	2	no	—
151	C₈	#14519710	1	2	yes	—
152	C₈	#14519710	1	2	yes	—
153	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
154	C₈	#14519776	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14519710	1	1	yes	characteristic
157	C₂²	#16840141	1	1	yes	characteristic
158	C₂²	#16840165	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#16839591	1	4	yes	complemented
161	C₄	#16839631	1	1	yes	characteristic
162	C₄	C₂×#14519710	1	1	yes	characteristic
163	C₄	#16840138	1	1	yes	characteristic
164	C₄	#16840165	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#21445978	1	1	yes	characteristic
168	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	#12909
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
76. C₄×C₆₁	#368	semidirect
100. C₆₁	#12909	semidirect
160. C₄	#16839591	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839591	2
C₂×C₄×#11758185	2
#21445991	2
C₆₁×#1848	2
#12909	61