Subgroups — #16807438

Order 3328.

462 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	#13131260	C₂	1	6	yes	maximal, complemented
2	#13131259	C₂	1	2	yes	maximal, complemented
3	C₂×#11718214	C₂	1	3	yes	maximal, complemented
4	D₁₃⋊M₆₄	C₂	1	3	yes	maximal, complemented
5	D₁₃×#504	C₂	1	1	yes	maximal, characteristic
6	D₁₃×#131	C₄	1	1	yes	characteristic
7	C₁₃×#504	C₄	1	1	yes	characteristic
8	#11718801	C₂²	1	1	yes	characteristic
9	#11718568	C₂²	1	3	yes	—
10	D₁₃×M₃₂	C₄	1	3	yes	—
11	#11718562	C₄	1	3	yes	—
12	C₂×C₁₆×D₁₃	C₂²	1	3	yes	—
13	#11718249	C₂²	1	6	yes	complemented
14	C₂×#88036	C₂²	1	6	yes	complemented
15	#11718214	C₂²	1	3	yes	—
16	#11718214	C₂²	1	1	yes	characteristic
17	#11718214	C₂²	1	6	yes	complemented
18	D₁₃⋊C₃₂	C₂²	1	6	yes	complemented
19	#41×D₁₃	C₈	1	1	yes	characteristic
20	C₁₃×#131	C₂×C₄	1	1	yes	characteristic
21	#88202	C₈	1	1	yes	characteristic
22	#88161	C₈	1	3	yes	—
23	M₁₆×D₁₃	C₈	1	3	yes	—
24	#88155	C₂×C₄	1	3	yes	—
25	C₂×C₈×D₁₃	C₂×C₄	1	3	yes	—
26	C₁₃×M₃₂	C₂×C₄	1	3	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	3	yes	—
28	#88052	C₂×C₄	1	3	yes	—
29	C₂×#6090	C₂³	1	3	yes	—
30	#88038	C₂×C₄	1	6	yes	—
31	C₁₆×D₁₃	C₂×C₄	1	3	yes	—
32	C₁₆×D₁₃	C₂×C₄	1	3	yes	—
33	C₁₆×D₁₃	C₂³	1	3	yes	—
34	C₁₆×D₁₃	C₂³	1	1	yes	characteristic
35	#88036	C₂³	1	6	yes	—
36	#88036	C₂³	1	2	yes	—
37	#33974	—	13	1	no	maximal, Sylow
38	C₁₃×#41	C₂×C₈	1	1	yes	characteristic
39	#6131	C₁₆	1	1	yes	characteristic
40	Q₈×D₁₃	C₁₆	1	1	yes	characteristic
41	#6129	C₁₆	1	3	yes	—
42	D₄×D₁₃	C₁₆	1	3	yes	—
43	#6127	C₂×C₈	1	3	yes	—
44	C₂×C₄×D₁₃	C₂×C₈	1	3	yes	—
45	C₁₃×M₁₆	C₂×C₈	1	3	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
47	#6099	C₂×C₈	1	3	yes	—
48	C₂×#1055	C₂×C₈	1	3	yes	—
49	#6094	C₂×C₈	1	6	yes	—
50	C₈×D₁₃	C₂×C₈	1	3	yes	—
51	C₈×D₁₃	C₂×C₈	1	3	yes	—
52	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₈×D₁₃	C₂²×C₄	1	3	yes	—
54	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
55	C₁₃×C₁₆	C₂²×C₄	1	3	yes	—
56	#6090	C₂²×C₄	1	3	yes	—
57	#6090	C₂⁴	1	1	yes	characteristic
58	C₂×#504	—	13	1	no	—
59	#2258	—	13	2	no	—
60	#2258	—	13	6	no	—
61	C₂×M₆₄	—	13	3	no	—
62	C₂×M₆₄	—	13	3	no	—
63	C₂²×D₁₃	C₂×C₁₆	1	3	yes	—
64	Q₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
65	D₄×C₁₃	C₂×C₁₆	1	3	yes	—
66	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
67	C₂²⋊D₁₃	C₂×C₁₆	1	6	yes	—
68	C₂×Dic₁₃	C₂×C₁₆	1	3	yes	—
69	D₅₂	C₂×C₁₆	1	3	yes	—
70	C₄×D₁₃	C₂×C₁₆	1	3	yes	—
71	C₄×D₁₃	C₂×C₁₆	1	3	yes	—
72	C₄×D₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₄×D₁₃	C₂²×C₈	1	3	yes	—
74	Dic₂₆	C₂×C₁₆	1	3	yes	—
75	C₈×C₁₃	C₂²×C₈	1	3	yes	—
76	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
77	#1055	C₂²×C₈	1	1	yes	characteristic
78	#1055	C₂²×C₈	1	3	yes	—
79	C₂×#131	—	13	1	no	—
80	#504	—	13	1	no	—
81	#504	—	13	3	no	—
82	#504	—	13	3	no	—
83	#504	#261	1	1	yes	characteristic
84	C₂×M₃₂	—	13	3	no	—
85	C₂²×C₁₆	—	13	3	no	—
86	M₆₄	—	13	3	no	—
87	M₆₄	—	13	1	no	—
88	M₆₄	—	13	6	no	—
89	M₆₄	—	13	6	no	—
90	C₂×C₃₂	—	13	6	no	—
91	C₂×C₃₂	—	13	6	no	—
92	C₂²×C₁₃	C₂²×C₁₆	1	3	yes	—
93	C₂×D₁₃	—	4	3	no	—
94	C₂×D₁₃	C₂²×C₁₆	1	1	yes	characteristic
95	C₂×D₁₃	C₂²×C₁₆	1	3	yes	—
96	C₄×C₁₃	C₂²×C₁₆	1	3	yes	—
97	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
98	Dic₁₃	C₂²×C₁₆	1	3	yes	—
99	Dic₁₃	C₂²×C₁₆	1	1	yes	characteristic
100	C₂×#41	—	13	1	no	—
101	#131	—	13	1	no	—
102	#131	—	13	3	no	—
103	#131	—	13	3	no	—
104	#131	C₂×#261	1	1	yes	characteristic
105	C₂×M₁₆	—	13	3	no	—
106	C₂²×C₈	—	13	3	no	—
107	M₃₂	—	13	3	no	—
108	M₃₂	—	13	6	no	—
109	M₃₂	C₂×#261	1	3	yes	—
110	C₂×C₁₆	—	13	3	no	—
111	C₂×C₁₆	—	13	3	no	—
112	C₂×C₁₆	—	13	1	no	—
113	C₂×C₁₆	—	13	3	no	—
114	C₂×C₁₆	—	13	3	no	—
115	C₂×C₁₆	C₂×#261	1	3	yes	—
116	C₃₂	—	13	6	no	—
117	C₃₂	—	13	2	no	—
118	C₂×C₁₃	—	2	3	no	—
119	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
120	D₁₃	—	2	1	no	—
121	D₁₃	—	2	3	no	—
122	#41	—	13	1	no	—
123	#41	—	13	3	no	—
124	#41	—	13	3	no	—
125	#41	D₁₃⋊C₈	1	1	yes	characteristic
126	C₂×Q₈	—	13	1	no	—
127	C₂×D₄	—	13	3	no	—
128	C₂²×C₄	—	13	3	no	—
129	M₁₆	—	13	3	no	—
130	M₁₆	—	13	6	no	—
131	M₁₆	D₁₃⋊C₈	1	3	yes	—
132	C₂×C₈	—	13	3	no	—
133	C₂×C₈	—	13	3	no	—
134	C₂×C₈	—	13	3	no	—
135	C₂×C₈	—	13	1	no	—
136	C₂×C₈	—	13	3	no	—
137	C₂×C₈	C₂²×#261	1	3	yes	—
138	C₁₆	—	13	1	no	—
139	C₁₆	—	13	3	no	—
140	C₁₆	C₂²×#261	1	1	yes	characteristic
141	C₁₆	C₂²×#261	1	3	yes	—
142	C₁₃	#33974	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	13	3	no	—
144	Q₈	—	13	3	no	—
145	Q₈	D₁₃⋊C₁₆	1	1	yes	characteristic
146	D₄	—	13	6	no	—
147	D₄	—	13	3	no	—
148	D₄	D₁₃⋊C₁₆	1	3	yes	—
149	C₂×C₄	—	13	3	no	—
150	C₂×C₄	—	13	3	no	—
151	C₂×C₄	—	13	3	no	—
152	C₂×C₄	—	13	1	no	—
153	C₂×C₄	—	13	3	no	—
154	C₂×C₄	C₂×D₁₃⋊C₈	1	3	yes	—
155	C₈	—	13	1	no	—
156	C₈	—	13	3	no	—
157	C₈	C₂×D₁₃⋊C₈	1	3	yes	—
158	C₈	C₂³×#261	1	1	yes	characteristic
159	C₂²	—	13	1	no	—
160	C₂²	—	13	3	no	—
161	C₂²	—	52	3	no	—
162	C₂²	C₂×D₁₃⋊C₁₆	1	3	yes	—
163	C₄	—	13	1	no	—
164	C₄	—	13	3	no	—
165	C₄	C₂×D₁₃⋊C₁₆	1	3	yes	—
166	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	26	1	no	—
169	C₂	—	26	3	no	—
170	C₂	C₂²×D₁₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#261
commutator, socle	C₂×C₁₃	C₂³×C₁₆
Fitting	C₁₃×#504	C₄
prime radical	#41×D₁₃	C₈
cosocle	#6090	C₂⁴
hypercenter	#504	#261
hypocenter	C₁₃	#33974
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #13131260	C₂	semidirect
2. #13131259	C₂	semidirect
3. C₂×#11718214	C₂	semidirect
4. D₁₃⋊M₆₄	C₂	semidirect
13. #11718249	C₂²	semidirect
14. C₂×#88036	C₂²	semidirect
17. #11718214	C₂²	semidirect
18. D₁₃⋊C₃₂	C₂²	semidirect
142. C₁₃	#33974	semidirect

Maximal subgroups

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

subgroup	index
#13131260	2
#13131259	2
C₂×#11718214	2
D₁₃⋊M₆₄	2
D₁₃×#504	2
#33974	13