Subgroups — #16726621

Order 3328.

2564 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719524	C₂	1	2	yes	maximal
2	#13123447	C₂	1	16	yes	maximal, complemented
3	C₂×#11718786	C₂	1	12	yes	maximal, complemented
4	C₂×C₁₃×#568	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
6	C₁₃×#568	C₄	1	16	yes	—
7	C₂×C₁₃×#131	C₂²	1	8	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	6	yes	—
9	#11719524	—	2	8	no	—
10	#11719524	C₂²	1	16	yes	—
11	C₂×#88202	—	2	8	no	—
12	C₂²×#6099	C₂²	1	12	yes	—
13	#11718786	C₂²	1	16	yes	—
14	#11718786	C₂²	1	48	yes	complemented
15	C₂×#88059	C₂²	1	48	yes	complemented
16	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
17	C₂×C₁₃×#41	D₄	1	8	yes	—
18	C₂×C₁₃×#41	Q₈	1	4	yes	—
19	C₂×C₁₃×#41	C₂³	1	4	yes	—
20	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
21	C₂²×D₄×C₁₃	Q₈	1	3	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
23	C₁₃×#131	C₂×C₄	1	64	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	48	yes	—
25	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
26	C₂×C₁₃×M₁₆	C₂³	1	12	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	12	yes	—
28	#88202	—	2	64	no	—
29	C₂×#6099	—	2	12	no	—
30	C₂×#6099	—	2	24	no	—
31	C₂×#6099	C₂³	1	8	yes	—
32	C₂×#6099	C₂³	1	48	yes	—
33	C₂²×#1055	—	2	12	no	—
34	#88059	C₂³	1	64	yes	—
35	#60113	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	C₂×Q₈	1	3	yes	—
37	C₁₃×#41	C₄⋊C₄	1	32	yes	—
38	C₁₃×#41	C₂²×C₄	1	16	yes	—
39	C₁₃×#41	C₂×D₄	1	16	yes	—
40	C₂×Q₈×C₁₃	C₄⋊C₄	1	8	yes	—
41	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
42	C₂×D₄×C₁₃	C₄⋊C₄	1	24	yes	—
43	C₂×D₄×C₁₃	C₂×Q₈	1	12	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
46	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
47	C₂²×C₄×C₁₃	C₂×Q₈	1	6	yes	—
48	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
49	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₁₃×M₁₆	C₂²×C₄	1	16	yes	—
51	C₁₃×M₁₆	C₂²×C₄	1	48	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	48	yes	—
53	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
54	#6099	—	2	8	no	—
55	#6099	—	2	48	no	—
56	#6099	C₂⁴	1	16	yes	—
57	C₂×#1055	—	2	8	no	—
58	C₂×#1055	—	2	48	no	—
59	C₂×#568	—	13	2	no	—
60	C₂×#568	D₁₃	1	1	yes	characteristic
61	#2916	—	13	16	no	—
62	C₂×#430	—	13	12	no	—
63	C₂³×C₁₃	—	8	3	no	—
64	C₂³×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
65	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
67	Q₈×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
68	D₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
69	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	12	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	24	yes	—
71	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
72	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
75	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×Q₈	1	6	yes	—
77	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
78	C₈×C₁₃	C₂³×C₄	1	16	yes	—
79	#1055	—	2	16	no	—
80	C₂²×#41	C₂×D₁₃	1	1	yes	characteristic
81	#568	—	13	16	no	—
82	#568	—	26	8	no	—
83	#568	Dic₁₃	1	16	yes	—
84	C₂×#131	—	26	8	no	—
85	C₂×#131	C₂×D₁₃	1	8	yes	—
86	C₂²×M₁₆	—	13	12	no	—
87	C₂²×M₁₆	C₂×D₁₃	1	6	yes	—
88	#430	—	13	16	no	—
89	#430	—	13	48	no	—
90	C₂×#108	—	13	48	no	—
91	C₂²×C₁₃	—	4	1	no	—
92	C₂²×C₁₃	—	4	12	no	—
93	C₂²×C₁₃	—	4	6	no	—
94	C₂²×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
95	C₂²×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
96	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
97	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
98	C₄×C₁₃	C₂²×C₄⋊C₄	1	12	yes	—
99	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
101	C₂×#41	Dic₂₆	1	4	yes	—
102	C₂×#41	D₅₂	1	8	yes	—
103	C₂×#41	C₂×Dic₁₃	1	8	yes	—
104	C₂×#41	C₂²×D₁₃	1	4	yes	—
105	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
106	C₂²×D₄	Dic₂₆	1	3	yes	—
107	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
108	#131	—	26	64	no	—
109	#131	C₂×Dic₁₃	1	64	yes	—
110	C₂×M₁₆	—	13	8	no	—
111	C₂×M₁₆	—	13	48	no	—
112	C₂×M₁₆	—	26	12	no	—
113	C₂×M₁₆	—	26	24	no	—
114	C₂×M₁₆	C₂×Dic₁₃	1	48	yes	—
115	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
116	C₂×M₁₆	C₂²×D₁₃	1	12	yes	—
117	C₂²×C₈	—	26	12	no	—
118	C₂²×C₈	C₂²×D₁₃	1	12	yes	—
119	#108	—	13	64	no	—
120	C₂×C₁₃	—	2	2	no	—
121	C₂×C₁₃	—	2	1	no	—
122	C₂×C₁₃	—	2	12	no	—
123	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
124	C₂⁴	C₂×Dic₂₆	1	3	yes	—
125	#41	C₄⋊Dic₁₃	1	32	yes	—
126	#41	C₂×D₅₂	1	16	yes	—
127	#41	C₂²×Dic₁₃	1	16	yes	—
128	C₂×Q₈	C₄⋊Dic₁₃	1	8	yes	—
129	C₂×Q₈	C₂×Dic₂₆	1	4	yes	—
130	C₂×D₄	C₄⋊Dic₁₃	1	24	yes	—
131	C₂×D₄	C₂×Dic₂₆	1	12	yes	—
132	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
133	C₂²×C₄	C₂×Dic₂₆	1	6	yes	—
134	C₂²×C₄	C₂×D₅₂	1	12	yes	—
135	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
136	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
137	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
138	M₁₆	—	13	16	no	—
139	M₁₆	—	26	8	no	—
140	M₁₆	—	26	48	no	—
141	M₁₆	C₂²×Dic₁₃	1	16	yes	—
142	M₁₆	C₂²×Dic₁₃	1	48	yes	—
143	C₂×C₈	—	26	8	no	—
144	C₂×C₈	—	26	48	no	—
145	C₂×C₈	C₂²×Dic₁₃	1	48	yes	—
146	C₂×C₈	C₂³×D₁₃	1	8	yes	—
147	C₁₃	#60113	1	1	yes	characteristic, complemented, Sylow
148	C₂³	—	8	3	no	—
149	C₂³	C₂×C₄⋊Dic₁₃	1	12	yes	—
150	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
151	C₂³	C₂²×Dic₂₆	1	6	yes	—
152	Q₈	C₂×C₄⋊Dic₁₃	1	16	yes	—
153	D₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₁₃	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊Dic₁₃	1	24	yes	—
156	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₂₆	1	6	yes	—
158	C₂×C₄	C₂²×D₅₂	1	2	yes	—
159	C₂×C₄	C₂²×D₅₂	1	12	yes	—
160	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
161	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
162	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
163	C₈	—	26	16	no	—
164	C₈	C₂³×Dic₁₃	1	16	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	12	no	—
167	C₂²	—	4	6	no	—
168	C₂²	C₂²×C₄⋊Dic₁₃	1	2	yes	—
169	C₂²	C₂²×C₄⋊Dic₁₃	1	12	yes	—
170	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
171	C₄	C₂²×C₄⋊Dic₁₃	1	2	yes	—
172	C₄	C₂²×C₄⋊Dic₁₃	1	12	yes	—
173	C₄	C₂³×D₅₂	1	1	yes	characteristic
174	C₄	C₂⁴×Dic₁₃	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	12	no	—
178	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×D₅₂
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#568	C₂
socle	C₂×C₁₃	C₂³×C₄⋊C₄
prime radical	C₂²×C₁₃×#41	C₂²
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂×#568	D₁₃
hypocenter	C₁₃	#60113
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
2. #13123447	C₂	semidirect
3. C₂×#11718786	C₂	semidirect
14. #11718786	C₂²	semidirect
15. C₂×#88059	C₂²	semidirect
147. C₁₃	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719524	2
#13123447	2
C₂×#11718786	2
C₂×C₁₃×#568	2
#60113	13