Subgroups — #16015498

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718343	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718740	C₂	1	2	yes	maximal
3	#13123213	C₂	1	2	yes	maximal
4	#13114999	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#88123	C₂²	1	2	yes	—
9	C₁₃×#500	—	2	4	no	—
10	C₂×C₁₃×#107	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718891	—	2	4	no	—
13	C₂×#88058	C₂²	1	4	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
15	#11718740	—	2	8	no	—
16	#11718638	C₂²	1	8	yes	—
17	#11718635	—	2	4	no	—
18	C₂×#88048	C₂²	1	8	yes	—
19	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	#11718343	—	2	4	no	—
22	C₂²×Q₈×C₁₃	—	2	2	no	—
23	C₂²×Dic₂₆	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
25	C₁₃×C₄⋊Q₈	—	2	4	no	—
26	C₁₃×C₄⋊Q₈	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
35	C₂²×#1055	C₂³	1	2	yes	—
36	#88123	—	2	4	no	—
37	#88123	—	2	4	no	—
38	C₁₃×#107	—	2	8	no	—
39	C₄×C₈×C₁₃	D₄	1	4	yes	—
40	#88071	—	2	16	no	—
41	#88058	—	2	8	no	—
42	C₈×Dic₁₃	Q₈	1	16	yes	complemented
43	#88048	—	2	16	no	—
44	C₄×#1055	D₄	1	4	yes	—
45	#33313	—	13	1	no	maximal, Sylow
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₃	—	4	8	no	—
48	C₂²×C₄×C₁₃	—	2	4	no	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	2	4	no	—
53	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
54	C₂×Dic₂₆	—	4	4	no	—
55	C₂×Dic₂₆	—	4	8	no	—
56	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
57	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	4	4	no	—
61	C₄²×C₁₃	C₂×D₄	1	2	yes	—
62	C₄²×C₁₃	C₂×D₄	1	2	yes	—
63	C₄⋊Dic₁₃	—	2	4	no	—
64	C₄⋊Dic₁₃	—	2	4	no	—
65	C₄⋊Dic₁₃	—	4	4	no	—
66	C₄×Dic₁₃	—	2	4	no	—
67	C₄×Dic₁₃	—	2	4	no	—
68	C₄×Dic₁₃	C₂×Q₈	1	8	yes	—
69	C₂×#1055	C₂×D₄	1	4	yes	—
70	C₂×#1055	C₂×Q₈	1	8	yes	—
71	C₂×#500	—	13	1	no	—
72	C₂×#500	D₁₃	1	1	yes	characteristic
73	C₂×#498	—	13	2	no	—
74	#2862	—	13	2	no	—
75	#1846	—	13	8	no	—
76	C₄²×C₈	—	13	1	no	—
77	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
78	Q₈×C₁₃	—	8	8	no	—
79	C₂×C₄×C₁₃	—	4	4	no	—
80	C₂×C₄×C₁₃	—	4	8	no	—
81	C₂×C₄×C₁₃	#127	1	2	yes	—
82	C₂×C₄×C₁₃	#127	1	2	yes	—
83	C₂×C₄×C₁₃	#135	1	4	yes	—
84	C₂×C₄×C₁₃	#135	1	4	yes	—
85	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
88	C₂×Dic₁₃	—	2	4	no	—
89	C₂×Dic₁₃	—	2	4	no	—
90	C₂×Dic₁₃	—	4	4	no	—
91	C₂×Dic₁₃	—	4	8	no	—
92	C₂×Dic₁₃	C₄⋊Q₈	1	8	yes	—
93	Dic₂₆	—	8	8	no	—
94	C₈×C₁₃	C₄⋊Q₈	1	8	yes	—
95	#1055	C₄⋊Q₈	1	8	yes	—
96	C₂×C₄⋊Q₈	—	13	2	no	—
97	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
98	#500	—	2	4	no	—
99	#500	—	26	4	no	—
100	#498	—	26	8	no	—
101	C₂×#107	—	13	4	no	—
102	C₂×#107	C₂×D₁₃	1	4	yes	—
103	C₂×#106	—	13	8	no	—
104	C₂×C₄×C₈	—	13	4	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
107	#391	—	26	4	no	—
108	#391	—	26	4	no	—
109	#382	—	13	8	no	—
110	C₄³	—	13	1	no	—
111	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
113	C₂²×C₁₃	C₂×#135	1	2	yes	—
114	C₂²×C₁₃	C₂×#135	1	2	yes	—
115	C₄×C₁₃	—	8	4	no	—
116	C₄×C₁₃	#495	1	4	yes	—
117	C₄×C₁₃	#495	1	4	yes	—
118	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
119	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
120	Dic₁₃	—	2	8	no	—
121	Dic₁₃	—	8	4	no	—
122	C₂²×Q₈	—	2	2	no	—
123	C₂²×Q₈	—	26	2	no	—
124	C₂²×C₈	—	13	2	no	—
125	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	26	4	no	—
129	C₄⋊Q₈	—	26	4	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	13	4	no	—
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
134	C₂×C₄²	—	13	2	no	—
135	C₂×C₄²	—	13	2	no	—
136	C₂×C₄²	—	13	2	no	—
137	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
138	#107	—	2	8	no	—
139	#107	—	26	8	no	—
140	#106	—	26	16	no	—
141	C₄×C₈	—	13	16	no	—
142	C₄×C₈	—	13	4	no	—
143	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
144	#95	—	26	16	no	—
145	C₂×C₁₃	#2868	1	1	yes	characteristic
146	C₂×C₁₃	C₂×#495	1	1	yes	characteristic
147	C₂×C₁₃	C₂×#495	1	1	yes	characteristic
148	C₂×C₁₃	#3160	1	4	yes	—
149	C₂×Q₈	—	4	8	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	52	8	no	—
152	C₂×Q₈	—	52	4	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	13	4	no	—
155	C₂²×C₄	—	26	4	no	—
156	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
159	C₂×C₈	—	13	4	no	—
160	C₂×C₈	—	13	8	no	—
161	C₂×C₈	Q₈×D₁₃	1	8	yes	—
162	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	26	4	no	—
167	C₄⋊C₄	—	26	4	no	—
168	C₄⋊C₄	—	52	4	no	—
169	C₄²	—	13	8	no	—
170	C₄²	—	26	4	no	—
171	C₄²	—	26	4	no	—
172	C₄²	D₄×D₁₃	1	2	yes	—
173	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
174	C₁₃	#33313	1	1	yes	characteristic, complemented, Sylow
175	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
176	Q₈	—	8	8	no	—
177	Q₈	—	104	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	13	8	no	—
181	C₂×C₄	—	26	4	no	—
182	C₂×C₄	—	26	4	no	—
183	C₂×C₄	—	52	8	no	—
184	C₂×C₄	—	52	4	no	—
185	C₂×C₄	#88166	1	4	yes	—
186	C₂×C₄	#88173	1	4	yes	—
187	C₂×C₄	#88194	1	2	yes	—
188	C₂×C₄	#88194	1	2	yes	—
189	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
191	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
192	C₈	—	13	8	no	—
193	C₈	#88198	1	8	yes	—
194	C₂²	D₁₃×C₄⋊Q₈	1	2	yes	—
195	C₂²	C₂×#88166	1	2	yes	—
196	C₂²	C₂×#88173	1	2	yes	—
197	C₂²	C₂×#88194	1	1	yes	characteristic
198	C₄	—	8	4	no	—
199	C₄	—	26	8	no	—
200	C₄	—	104	4	no	—
201	C₄	#11718816	1	4	yes	—
202	C₄	#11718852	1	4	yes	—
203	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
204	C₄	C₂×#88198	1	2	yes	—
205	C₂	#13123232	1	1	yes	characteristic
206	C₂	C₂×#11718816	1	1	yes	characteristic
207	C₂	C₂×#11718852	1	1	yes	characteristic
208	C₂	#13125347	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#500	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₂×#500	D₁₃
hypocenter	C₁₃	#33313
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
42. C₈×Dic₁₃	Q₈	semidirect
174. C₁₃	#33313	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718343	2
C₂×#11718740	2
#13123213	2
#13114999	2
C₄×C₈×Dic₁₃	2
C₂×C₁₃×#500	2
#33313	13