Subgroups — #29594903

Order 15616.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519713	C₂	1	2	yes	maximal
2	#21443913	C₂	1	4	yes	maximal, complemented
3	C₂²×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758194	—	2	2	no	—
5	C₂×#14519713	—	2	2	no	—
6	C₂×#14519713	—	2	4	no	—
7	C₂×#14519713	C₄	1	8	yes	—
8	C₂×#14519713	C₂²	1	4	yes	—
9	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₆₁	—	2	4	no	—
11	C₂×M₃₂×C₆₁	C₄	1	4	yes	—
12	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
13	C₂×#11758194	—	2	8	no	—
14	C₂×#11758194	—	2	4	no	—
15	C₂×#11758194	—	4	2	no	—
16	C₂×#11758194	—	4	4	no	—
17	C₂²×#92418	—	4	2	no	—
18	#14519713	—	2	8	no	—
19	#14519713	—	2	4	no	—
20	#14519713	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₆₁	D₄	1	4	yes	—
26	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
28	M₃₂×C₆₁	—	2	8	no	—
29	C₂×C₁₆×C₆₁	—	2	4	no	—
30	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
31	C₂⁴×C₆₁	Q₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	D₈	1	2	yes	—
35	C₂²×C₄×C₆₁	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₄²	1	4	yes	—
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
40	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	#11758194	—	2	8	no	—
48	#11758194	—	4	8	no	—
49	#11758194	—	4	4	no	—
50	C₂×#92418	—	4	8	no	—
51	C₂×#92418	—	4	4	no	—
52	C₁₆×C₆₁	—	2	4	no	—
53	C₂³×C₆₁	—	4	2	no	—
54	C₂³×C₆₁	#103	1	4	yes	—
55	C₂³×C₆₁	#107	1	2	yes	—
56	C₂³×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	#95	1	4	yes	—
58	C₂×C₄×C₆₁	#102	1	4	yes	—
59	C₂×C₄×C₆₁	#102	1	4	yes	—
60	C₂×C₄×C₆₁	#103	1	4	yes	—
61	C₂×C₄×C₆₁	#106	1	4	yes	—
62	C₂×C₄×C₆₁	#107	1	2	yes	—
63	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
67	C₈×C₆₁	#95	1	4	yes	—
68	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
71	#92418	—	4	8	no	—
72	#12559	—	61	1	no	maximal, Sylow
73	C₂²×C₆₁	—	2	4	no	—
74	C₂²×C₆₁	—	2	2	no	—
75	C₂²×C₆₁	—	4	4	no	—
76	C₂²×C₆₁	#340	1	4	yes	—
77	C₂²×C₆₁	C₂×#103	1	2	yes	—
78	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
79	C₄×C₆₁	#340	1	4	yes	—
80	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
81	C₄×C₆₁	C₂×#102	1	2	yes	—
82	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
83	C₂²×M₃₂	D₆₁	1	1	yes	characteristic
84	C₂²×#108	—	61	2	no	—
85	#1381	—	61	4	no	—
86	C₂×C₆₁	—	2	4	no	—
87	C₂×C₆₁	—	2	2	no	—
88	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
89	C₂×C₆₁	#2161	1	2	yes	—
90	C₂²×M₁₆	—	122	2	no	—
91	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
92	C₂×M₃₂	—	2	4	no	—
93	C₂×M₃₂	Dic₆₁	1	4	yes	—
94	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
95	C₂×#108	—	61	8	no	—
96	C₂×#108	—	61	4	no	—
97	C₂×#108	—	122	2	no	—
98	C₂×#108	—	122	4	no	—
99	C₆₁	#12559	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₄	D₂₄₄	1	1	yes	characteristic
101	C₂×M₁₆	—	122	8	no	—
102	C₂×M₁₆	—	122	4	no	—
103	C₂×M₁₆	—	244	4	no	—
104	C₂×M₁₆	—	244	2	no	—
105	C₂²×C₈	—	244	2	no	—
106	C₂²×C₈	Dic₁₂₂	1	2	yes	—
107	C₂²×C₈	C₄×D₆₁	1	4	yes	—
108	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₆₁	1	2	yes	—
110	C₂²×C₈	C₂²⋊D₆₁	1	4	yes	—
111	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×Dic₆₁	1	4	yes	—
115	#108	—	61	8	no	—
116	#108	—	122	8	no	—
117	#108	—	122	4	no	—
118	C₂⁴	Dic₂₄₄	1	1	yes	characteristic
119	C₂²×C₄	#11758190	1	4	yes	—
120	C₂²×C₄	D₄₈₈	1	2	yes	—
121	C₂²×C₄	Dic₂₄₄	1	1	yes	characteristic
122	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
123	C₂²×C₄	#11758198	1	4	yes	—
124	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
125	M₁₆	—	122	8	no	—
126	M₁₆	—	244	8	no	—
127	M₁₆	—	244	4	no	—
128	C₂×C₈	—	244	8	no	—
129	C₂×C₈	—	244	4	no	—
130	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
131	C₂×C₈	#11758196	1	8	yes	—
132	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
133	C₂×C₈	#11758198	1	4	yes	—
134	C₂×C₈	C₂²⋊Dic₆₁	1	4	yes	—
135	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
137	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#14519702	1	4	yes	—
142	C₂³	#14519711	1	2	yes	—
143	C₂³	C₂×Dic₂₄₄	1	1	yes	characteristic
144	C₂×C₄	#14519684	1	4	yes	—
145	C₂×C₄	#14519699	1	4	yes	—
146	C₂×C₄	#14519699	1	4	yes	—
147	C₂×C₄	#14519702	1	4	yes	—
148	C₂×C₄	#14519709	1	4	yes	—
149	C₂×C₄	#14519711	1	2	yes	—
150	C₂×C₄	C₂×#11758198	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
153	C₂×C₄	C₂×#11758190	1	2	yes	—
154	C₈	—	244	8	no	—
155	C₈	#14519684	1	4	yes	—
156	C₈	C₂×C₄×Dic₆₁	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
158	C₈	C₂×#11758196	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#16839525	1	4	yes	—
163	C₂²	C₂×#14519702	1	2	yes	—
164	C₂²	C₂×#14519711	1	1	yes	characteristic
165	C₄	#16839525	1	4	yes	—
166	C₄	C₂×#14519684	1	1	yes	characteristic
167	C₄	C₂×#14519699	1	2	yes	—
168	C₄	C₂×#14519709	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×#16839525	1	1	yes	characteristic
172	C₂	#21447614	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₈₈
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₂²×M₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂⁴×C₆₁	Q₁₆
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₂²×M₃₂	D₆₁
hypocenter	C₆₁	#12559
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #21443913	C₂	semidirect
99. C₆₁	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519713	2
#21443913	2
C₂²×M₃₂×C₆₁	2
#12559	61