Subgroups — #16016297

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123704	C₂	1	2	yes	maximal
2	#13116707	C₂	1	4	yes	maximal, complemented
3	Q₃₂×Dic₁₃	C₂	1	4	yes	maximal, complemented
4	#13116620	C₂	1	1	yes	maximal, characteristic
5	#13116556	C₂	1	2	yes	maximal
6	#13116432	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	#11719517	—	2	2	no	—
10	C₂×C₁₃×Q₃₂	C₄	1	8	yes	complemented
11	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718886	—	2	2	no	—
15	#11718845	—	2	4	no	—
16	Q₁₆×Dic₁₃	C₂²	1	8	yes	—
17	#11718767	C₂²	1	2	yes	—
18	#11718762	C₂²	1	1	yes	characteristic
19	C₂×#88052	C₂²	1	1	yes	characteristic
20	#11718156	C₂²	1	8	yes	—
21	#11718097	C₂²	1	2	yes	—
22	#11718096	C₂²	1	2	yes	—
23	#11718094	C₂²	1	2	yes	—
24	C₁₆×Dic₁₃	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	2	no	—
27	C₂×C₁₃×Q₁₆	—	2	4	no	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	8	yes	—
29	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	Q₈×Dic₁₃	—	2	8	no	—
32	Q₈×Dic₁₃	—	4	4	no	—
33	#88180	—	2	2	no	—
34	#88180	—	4	4	no	—
35	#88180	D₄	1	1	yes	characteristic
36	C₂×#6099	D₄	1	1	yes	characteristic
37	C₁₃×Q₃₂	C₂×C₄	1	8	yes	—
38	C₁₃×Q₃₂	C₂×C₄	1	8	yes	—
39	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
40	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
42	#88075	—	2	8	no	—
43	#88058	C₂³	1	4	yes	—
44	#88057	—	2	2	no	—
45	#88055	D₄	1	2	yes	—
46	C₈×Dic₁₃	C₂³	1	2	yes	—
47	#88052	—	2	2	no	—
48	C₂×#6090	C₂³	1	2	yes	—
49	#33458	—	13	1	no	maximal, Sylow
50	C₂×Q₈×C₁₃	—	2	8	no	—
51	C₂×Q₈×C₁₃	—	2	4	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×Q₈×C₁₃	—	4	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₁₃×Q₁₆	—	2	8	no	—
57	C₁₃×Q₁₆	—	2	4	no	—
58	C₁₃×Q₁₆	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	#41	1	2	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	—	4	2	no	—
64	C₂²⋊Dic₁₃	D₈	1	2	yes	—
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	4	8	no	—
67	C₄⋊Dic₁₃	D₈	1	2	yes	—
68	C₄×Dic₁₃	—	4	2	no	—
69	C₄×Dic₁₃	—	4	8	no	—
70	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
71	#6099	—	2	2	no	—
72	C₂×#1055	C₂×D₄	1	2	yes	—
73	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
74	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
75	#6090	—	2	2	no	—
76	C₂²×Q₃₂	D₁₃	1	1	yes	characteristic
77	#2950	—	13	2	no	—
78	#2176	—	13	4	no	—
79	C₄×Q₃₂	—	13	4	no	—
80	#2158	—	13	1	no	—
81	#2141	—	13	2	no	—
82	#2108	—	13	1	no	—
83	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	8	no	—
85	Q₈×C₁₃	—	4	4	no	—
86	Q₈×C₁₃	—	4	8	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	4	8	no	—
89	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#135	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₃	—	4	4	no	—
94	C₂×Dic₁₃	C₂×D₈	1	4	yes	—
95	C₈×C₁₃	C₄×D₄	1	2	yes	—
96	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
98	#1055	—	2	2	no	—
99	C₂²×Q₁₆	C₂×D₁₃	1	2	yes	—
100	#519	—	26	2	no	—
101	C₂×Q₃₂	Dic₁₃	1	8	yes	complemented
102	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
103	C₂×Q₃₂	C₂×D₁₃	1	2	yes	—
104	C₂×M₃₂	—	13	1	no	—
105	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
106	#441	—	26	4	no	—
107	C₄×Q₁₆	—	13	8	no	—
108	#427	—	13	2	no	—
109	#419	—	26	2	no	—
110	#405	—	13	1	no	—
111	#367	—	13	2	no	—
112	#366	—	13	2	no	—
113	#358	—	13	8	no	—
114	#346	—	13	2	no	—
115	C₄×C₁₆	—	13	2	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	C₄×D₈	1	2	yes	—
118	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
119	C₄×C₁₃	—	4	8	no	—
120	C₄×C₁₃	C₄×D₈	1	2	yes	—
121	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
123	Dic₁₃	—	2	4	no	—
124	Dic₁₃	—	8	4	no	—
125	C₂²×Q₈	—	2	2	no	—
126	C₂×Q₁₆	—	2	2	no	—
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×Q₁₆	C₂×Dic₁₃	1	8	yes	—
129	C₂×Q₁₆	C₂²×D₁₃	1	4	yes	—
130	C₂×M₁₆	—	13	1	no	—
131	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₄×Q₈	—	26	8	no	—
133	C₄×Q₈	—	52	4	no	—
134	#117	—	13	1	no	—
135	#117	—	26	2	no	—
136	#117	—	52	4	no	—
137	Q₃₂	C₂×Dic₁₃	1	8	yes	—
138	Q₃₂	C₂×Dic₁₃	1	8	yes	—
139	M₃₂	—	26	2	no	—
140	C₂×C₁₆	—	13	2	no	—
141	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
142	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
143	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
144	#107	—	13	4	no	—
145	#106	—	26	2	no	—
146	#103	—	26	8	no	—
147	#97	—	13	2	no	—
148	C₄×C₈	—	13	2	no	—
149	C₂×C₁₃	—	2	2	no	—
150	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
151	C₂×C₁₃	#3417	1	2	yes	—
152	C₂×Q₈	—	2	8	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	4	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
158	Q₁₆	—	2	8	no	—
159	Q₁₆	—	2	4	no	—
160	Q₁₆	C₂²×Dic₁₃	1	8	yes	—
161	M₁₆	—	26	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
164	C₂×C₈	#6129	1	2	yes	—
165	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
166	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	13	2	no	—
168	C₄⋊C₄	—	26	4	no	—
169	C₄⋊C₄	—	52	8	no	—
170	C₂²⋊C₄	—	13	2	no	—
171	C₂²⋊C₄	—	52	2	no	—
172	C₄²	—	13	2	no	—
173	C₄²	—	52	8	no	—
174	C₄²	—	52	2	no	—
175	C₁₆	—	26	2	no	—
176	C₁₆	C₂²×Dic₁₃	1	2	yes	—
177	C₁₆	C₂²×Dic₁₃	1	2	yes	—
178	C₁₃	#33458	1	1	yes	characteristic, complemented, Sylow
179	C₂³	D₈×D₁₃	1	1	yes	characteristic
180	Q₈	—	2	8	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	13	4	no	—
186	C₂×C₄	—	52	4	no	—
187	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
188	C₂×C₄	#88166	1	2	yes	—
189	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
191	C₈	—	26	2	no	—
192	C₈	D₄×Dic₁₃	1	2	yes	—
193	C₈	C₂×#6129	1	1	yes	characteristic
194	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
195	C₂²	—	2	2	no	—
196	C₂²	D₈×Dic₁₃	1	2	yes	—
197	C₂²	C₂×D₈×D₁₃	1	1	yes	characteristic
198	C₄	—	4	8	no	—
199	C₄	—	26	4	no	—
200	C₄	—	104	4	no	—
201	C₄	D₈×Dic₁₃	1	2	yes	—
202	C₄	C₂×#88166	1	1	yes	characteristic
203	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
204	C₂	—	2	2	no	—
205	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic
206	C₂	#13127985	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₁₃×Q₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×Q₃₂	D₁₃
hypocenter	C₁₃	#33458
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #13116707	C₂	semidirect
3. Q₃₂×Dic₁₃	C₂	semidirect
10. C₂×C₁₃×Q₃₂	C₄	semidirect
101. C₂×Q₃₂	Dic₁₃	semidirect
178. C₁₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
#13123704	2
#13116707	2
Q₃₂×Dic₁₃	2
#13116620	2
#13116556	2
#13116432	2
C₂²×C₁₃×Q₃₂	2
#33458	13