Subgroups — #29591657

Order 15616.

242 conjugacy classes of subgroups.

The subgroup lattice has 172 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	#21444578	C₂	1	1	yes	maximal, characteristic
3	#21443892	C₂	1	4	yes	maximal, complemented
4	C₆₁×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11758185	C₄	1	2	yes	—
6	#16839786	—	2	1	no	—
7	C₄×#11758185	C₄	1	1	yes	characteristic
8	C₄×#11758185	C₄	1	1	yes	characteristic
9	C₄×#11758185	C₂²	1	2	yes	—
10	#16839515	—	2	1	no	—
11	#16839515	C₂²	1	2	yes	—
12	C₆₁×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#335	—	2	1	no	—
15	C₆₁×#335	C₂²	1	2	yes	—
16	C₂×#11758194	—	4	1	no	—
17	C₂×#11758185	—	2	4	no	—
18	C₂×#11758185	C₂×C₄	1	2	yes	—
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₄⋊#92418	—	2	2	no	—
21	C₄⋊#92418	—	4	1	no	—
22	C₂×M₁₆×C₆₁	—	4	1	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	2	2	no	—
27	C₄⋊C₈×C₆₁	—	4	1	no	—
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
35	M₁₆×C₆₁	—	8	2	no	—
36	C₂×C₈×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	—	2	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	C₂²×C₄	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	D₈	1	1	yes	characteristic
45	C₄²×C₆₁	Q₁₆	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
47	#11758194	—	8	2	no	—
48	C₂×#92418	—	4	2	no	—
49	#11758185	—	2	2	no	—
50	#11758185	M₁₆	1	4	yes	complemented
51	C₂³×C₆₁	#107	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	1	no	—
55	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
58	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
59	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₆₁	—	2	1	no	—
64	C₈×C₆₁	—	2	1	no	—
65	C₈×C₆₁	—	8	2	no	—
66	C₈×C₆₁	#108	1	2	yes	—
67	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
68	#92418	—	8	2	no	—
69	#12023	—	61	1	no	maximal, Sylow
70	C₂²×C₆₁	—	2	1	no	—
71	C₂²×C₆₁	—	2	1	no	—
72	C₂²×C₆₁	#367	1	2	yes	—
73	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
74	C₄×C₆₁	—	2	1	no	—
75	C₄×C₆₁	—	2	1	no	—
76	C₄×C₆₁	#367	1	2	yes	—
77	C₄×C₆₁	#368	1	2	yes	—
78	C₄×C₆₁	#423	1	1	yes	characteristic
79	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
80	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
81	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
82	C₂×C₄×C₁₆	—	61	1	no	—
83	#1567	—	61	1	no	—
84	#1567	D₆₁	1	1	yes	characteristic
85	#1372	—	61	4	no	—
86	C₂×C₆₁	—	2	2	no	—
87	C₂×C₆₁	#1567	1	1	yes	characteristic
88	C₂×C₆₁	C₂×#367	1	1	yes	characteristic
89	C₂×C₆₁	C₂×#368	1	1	yes	characteristic
90	C₂²×C₁₆	—	61	2	no	—
91	#423	—	2	1	no	—
92	#423	—	122	1	no	—
93	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₄×C₁₆	—	61	1	no	—
95	C₄×C₁₆	—	61	1	no	—
96	C₄×C₁₆	—	61	2	no	—
97	#335	—	2	1	no	—
98	#335	—	61	2	no	—
99	#335	—	122	1	no	—
100	#335	C₂×D₆₁	1	2	yes	—
101	C₆₁	#12023	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	244	1	no	—
104	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
105	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
107	C₂×C₁₆	—	61	2	no	—
108	C₂×C₁₆	—	61	2	no	—
109	C₂×C₁₆	—	122	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	122	2	no	—
113	C₄⋊C₈	—	244	1	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	#11758196	1	1	yes	characteristic
119	C₂²×C₄	#11758199	1	1	yes	characteristic
120	C₂²×C₄	#11758202	1	1	yes	characteristic
121	M₁₆	—	8	2	no	—
122	M₁₆	—	488	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	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	C₂×Dic₁₂₂	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
131	C₄²	#11758196	1	1	yes	characteristic
132	C₄²	#11758199	1	1	yes	characteristic
133	C₄²	#11758202	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
135	C₁₆	—	61	4	no	—
136	C₁₆	—	122	2	no	—
137	C₂³	#14519710	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#14519710	1	1	yes	characteristic
142	C₂×C₄	#14519710	1	1	yes	characteristic
143	C₂×C₄	#14519710	1	1	yes	characteristic
144	C₂×C₄	#14519720	1	2	yes	—
145	C₂×C₄	#14519721	1	2	yes	—
146	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
147	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
148	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	488	2	no	—
153	C₈	#14519712	1	2	yes	—
154	C₈	M₁₆×D₆₁	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#16839589	1	2	yes	—
158	C₂²	C₂×#14519710	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#16839589	1	2	yes	—
162	C₄	#16839591	1	2	yes	—
163	C₄	#16839785	1	1	yes	characteristic
164	C₄	C₂×#14519712	1	1	yes	characteristic
165	C₄	C₂×#14519720	1	1	yes	characteristic
166	C₄	C₂×#14519721	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#21444577	1	1	yes	characteristic
169	C₂	C₂×#16839589	1	1	yes	characteristic
170	C₂	C₂×#16839591	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1567	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂³×C₆₁	#107
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1567	D₆₁
hypocenter	C₆₁	#12023
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #21443892	C₂	semidirect
50. #11758185	M₁₆	semidirect
101. C₆₁	#12023	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
#21444578	2
#21443892	2
C₆₁×#1567	2
#12023	61