Subgroups — #16018411

Order 3328.

490 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	#13125350	C₂	1	2	yes	maximal
2	#13117118	C₂	1	2	yes	maximal, complemented
3	#13117115	C₂	1	4	yes	maximal, complemented
4	#13117111	C₂	1	2	yes	maximal, complemented
5	#13116620	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	#11719175	—	2	2	no	—
8	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11718767	—	2	2	no	—
11	#11718767	C₂²	1	4	yes	—
12	#11718344	—	2	2	no	—
13	#11718344	C₂²	1	4	yes	—
14	#11718343	C₂²	1	4	yes	—
15	#11718341	—	2	2	no	—
16	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
18	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
19	#11718097	C₂²	1	8	yes	—
20	#11718096	C₂²	1	8	yes	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	#88180	—	2	4	no	—
25	#88178	—	4	4	no	—
26	#88124	—	2	4	no	—
27	#88123	—	2	4	no	—
28	C₄×Dic₂₆	—	4	4	no	—
29	C₂×C₁₃×C₁₆	Q₈	1	8	yes	complemented
30	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
31	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
32	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	#88058	—	2	4	no	—
37	#88058	C₂³	1	8	yes	—
38	#88057	—	2	4	no	—
39	#88057	—	2	4	no	—
40	#33726	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	D₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₂₆	—	4	4	no	—
44	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
45	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
49	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₄²×C₁₃	D₈	1	2	yes	—
51	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊Dic₁₃	—	4	4	no	—
54	C₄⋊Dic₁₃	—	2	8	no	—
55	C₄⋊Dic₁₃	—	4	4	no	—
56	#6101	—	4	8	no	—
57	C₄×Dic₁₃	—	4	4	no	—
58	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
59	C₁₃×C₁₆	C₂×Q₈	1	4	yes	—
60	#3160	—	13	2	no	—
61	#2254	—	13	2	no	—
62	#2253	—	13	4	no	—
63	#2252	—	13	2	no	—
64	#2158	—	13	4	no	—
65	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×Dic₁₃	—	4	8	no	—
75	Dic₂₆	—	8	8	no	—
76	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
77	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
78	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
79	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
80	#533	—	26	2	no	—
81	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
82	#500	—	13	4	no	—
83	#499	—	13	4	no	—
84	#499	—	26	2	no	—
85	#498	—	26	2	no	—
86	#427	—	13	4	no	—
87	#427	—	26	2	no	—
88	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
89	#367	—	13	8	no	—
90	#366	—	13	8	no	—
91	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
92	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
93	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	#500	1	2	yes	—
96	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	#500	1	2	yes	—
99	C₄×C₁₃	#508	1	4	yes	—
100	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₁₃	—	8	8	no	—
103	C₂²×C₈	D₅₂	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
105	C₄⋊Q₈	—	26	4	no	—
106	#125	—	26	4	no	—
107	C₂²⋊Q₈	—	52	4	no	—
108	C₄×Q₈	—	52	4	no	—
109	#117	—	26	4	no	—
110	C₂×C₄²	D₅₂	1	1	yes	characteristic
111	C₂×C₁₆	Dic₂₆	1	8	yes	complemented
112	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
113	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
114	#107	—	13	8	no	—
115	#107	—	26	4	no	—
116	#106	—	26	4	no	—
117	#106	—	26	4	no	—
118	C₄×C₈	D₅₂	1	1	yes	characteristic
119	C₄×C₈	D₅₂	1	1	yes	characteristic
120	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	C₂×C₁₃	—	2	2	no	—
123	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
124	C₂×C₁₃	C₂×#508	1	2	yes	—
125	C₂×Q₈	—	52	4	no	—
126	C₂²×C₄	D₁₀₄	1	2	yes	—
127	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
128	C₂×C₈	Dic₅₂	1	4	yes	—
129	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
130	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
131	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
132	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
133	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
134	C₄⋊C₄	—	26	8	no	—
135	C₄⋊C₄	—	52	8	no	—
136	C₄⋊C₄	—	52	4	no	—
137	C₂²⋊C₄	—	52	4	no	—
138	C₄²	—	52	4	no	—
139	C₄²	D₁₀₄	1	2	yes	—
140	C₄²	C₂×D₅₂	1	1	yes	characteristic
141	C₄²	C₂×D₅₂	1	1	yes	characteristic
142	C₁₆	C₂×Dic₂₆	1	4	yes	—
143	C₁₆	C₂×Dic₂₆	1	4	yes	—
144	C₁₃	#33726	1	1	yes	characteristic, complemented, Sylow
145	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
146	Q₈	—	104	8	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	52	8	no	—
149	C₂×C₄	#88123	1	2	yes	—
150	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
151	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
152	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
153	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
154	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
155	C₈	#88123	1	2	yes	—
156	C₈	C₂×Dic₅₂	1	2	yes	—
157	C₈	C₂×Dic₅₂	1	2	yes	—
158	C₈	C₂²×Dic₂₆	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#11718343	1	2	yes	—
161	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	104	8	no	—
164	C₄	#11718343	1	2	yes	—
165	C₄	#11718565	1	4	yes	—
166	C₄	C₂×#88123	1	1	yes	characteristic
167	C₄	C₂²×Dic₅₂	1	1	yes	characteristic
168	C₂	—	2	2	no	—
169	C₂	C₂×#11718343	1	1	yes	characteristic
170	C₂	C₂×#11718565	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#33726
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13117118	C₂	semidirect
3. #13117115	C₂	semidirect
4. #13117111	C₂	semidirect
29. C₂×C₁₃×C₁₆	Q₈	semidirect
111. C₂×C₁₆	Dic₂₆	semidirect
144. C₁₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13117118	2
#13117115	2
#13117111	2
#13116620	2
C₂×C₄×C₁₃×C₁₆	2
#33726	13