Subgroups — #16015491

Order 3328.

830 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718344	C₂	1	4	yes	maximal
2	#13123199	C₂	1	2	yes	maximal
3	#13115000	C₂	1	4	yes	maximal
4	#13114996	C₂	1	2	yes	maximal
5	#13114993	C₂	1	2	yes	maximal
6	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#88124	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₂²	1	8	yes	—
14	C₂×#88057	C₂²	1	8	yes	—
15	#11718597	C₂²	1	4	yes	—
16	#11718596	C₂²	1	4	yes	—
17	#11718589	—	2	8	no	—
18	#11718344	—	2	16	no	—
19	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	4	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₃	—	2	8	no	—
26	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
27	C₂×#6101	—	2	8	no	—
28	#88124	—	2	16	no	—
29	C₄×C₈×C₁₃	D₄	1	8	yes	—
30	C₄×C₈×C₁₃	Q₈	1	4	yes	—
31	C₄×C₈×C₁₃	Q₈	1	4	yes	—
32	C₄×C₈×C₁₃	Q₈	1	8	yes	—
33	#88071	—	2	8	no	—
34	#88058	—	2	16	no	—
35	#88057	—	2	16	no	—
36	#33311	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₁₃	—	2	8	no	—
42	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
43	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
44	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
45	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
46	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
47	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
48	C₄²×C₁₃	—	2	8	no	—
49	C₄²×C₁₃	C₂×D₄	1	4	yes	—
50	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
51	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
52	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
53	C₄⋊Dic₁₃	—	2	16	no	—
54	C₄⋊Dic₁₃	—	4	16	no	—
55	#6101	—	4	16	no	—
56	C₂×#499	—	13	4	no	—
57	#2859	—	13	2	no	—
58	#1846	—	13	4	no	—
59	#1845	—	13	2	no	—
60	#1844	—	13	2	no	—
61	C₄²×C₈	D₁₃	1	1	yes	characteristic
62	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	4	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	#127	1	4	yes	—
66	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
67	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
68	C₂×C₄×C₁₃	#135	1	8	yes	—
69	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
70	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
73	C₂×Dic₁₃	—	4	8	no	—
74	C₂×Dic₁₃	—	4	16	no	—
75	C₈×C₁₃	C₄⋊Q₈	1	8	yes	—
76	C₈×C₁₃	C₄⋊Q₈	1	8	yes	—
77	C₂×#125	—	13	4	no	—
78	#499	—	26	16	no	—
79	C₂×#107	—	13	8	no	—
80	C₂×#106	—	13	8	no	—
81	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
84	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
85	#389	—	26	8	no	—
86	#384	—	13	4	no	—
87	#382	—	13	4	no	—
88	C₄³	C₂×D₁₃	1	1	yes	characteristic
89	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
90	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂²×C₁₃	C₂×#135	1	4	yes	—
93	C₄×C₁₃	—	2	4	no	—
94	C₄×C₁₃	—	2	4	no	—
95	C₄×C₁₃	#499	1	8	yes	complemented
96	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
97	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
98	Dic₁₃	—	8	8	no	—
99	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
100	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
101	#125	—	26	16	no	—
102	C₂×C₄⋊C₄	—	13	8	no	—
103	C₂×C₄⋊C₄	—	26	8	no	—
104	C₂×C₄⋊C₄	—	26	8	no	—
105	C₂×C₄²	D₅₂	1	4	yes	—
106	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
107	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
109	#107	—	26	16	no	—
110	#106	—	26	16	no	—
111	C₄×C₈	Dic₂₆	1	4	yes	—
112	C₄×C₈	Dic₂₆	1	4	yes	—
113	C₄×C₈	Dic₂₆	1	8	yes	—
114	C₄×C₈	D₅₂	1	8	yes	—
115	#95	—	26	8	no	—
116	C₂×C₁₃	#2868	1	1	yes	characteristic
117	C₂×C₁₃	#3147	1	2	yes	—
118	C₂×C₁₃	C₂×#499	1	2	yes	—
119	C₂×C₁₃	#3160	1	2	yes	—
120	C₂²×C₄	—	26	8	no	—
121	C₂²×C₄	C₂×D₅₂	1	2	yes	—
122	C₂²×C₄	C₂×D₅₂	1	2	yes	—
123	C₂²×C₄	C₂×D₅₂	1	2	yes	—
124	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
125	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
126	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
127	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
128	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
129	C₂×C₈	C₂×D₅₂	1	4	yes	—
130	C₂×C₈	C₂×D₅₂	1	4	yes	—
131	C₄⋊C₄	—	26	16	no	—
132	C₄⋊C₄	—	52	16	no	—
133	C₄⋊C₄	—	52	16	no	—
134	C₄²	—	2	8	no	—
135	C₄²	C₂×Dic₂₆	1	4	yes	—
136	C₄²	C₂×Dic₂₆	1	2	yes	—
137	C₄²	C₂×Dic₂₆	1	2	yes	—
138	C₄²	C₂×D₅₂	1	4	yes	—
139	C₁₃	#33311	1	1	yes	characteristic, complemented, Sylow
140	C₂³	C₂²×D₅₂	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	52	8	no	—
144	C₂×C₄	—	52	16	no	—
145	C₂×C₄	#88123	1	4	yes	—
146	C₂×C₄	#88123	1	4	yes	—
147	C₂×C₄	#88128	1	4	yes	—
148	C₂×C₄	#88158	1	8	yes	—
149	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
151	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
152	C₂×C₄	C₂²×D₅₂	1	2	yes	—
153	C₈	#88123	1	8	yes	—
154	C₈	#88123	1	8	yes	—
155	C₂²	C₂×#88123	1	1	yes	characteristic
156	C₂²	C₂×#88123	1	1	yes	characteristic
157	C₂²	C₂×#88128	1	1	yes	characteristic
158	C₂²	C₂×#88158	1	4	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	2	4	no	—
161	C₄	—	104	8	no	—
162	C₄	#11718344	1	8	yes	complemented
163	C₄	C₂×#88123	1	2	yes	—
164	C₄	C₂×#88123	1	2	yes	—
165	C₂	#13123234	1	1	yes	characteristic
166	C₂	#13125264	1	2	yes	—
167	C₂	C₂×#11718344	1	2	yes	—
168	C₂	#13125350	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#33311
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
95. C₄×C₁₃	#499	semidirect
139. C₁₃	#33311	semidirect
162. C₄	#11718344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718344	2
#13123199	2
#13115000	2
#13114996	2
#13114993	2
C₄²×C₈×C₁₃	2
#33311	13