Subgroups — #16015492

Order 3328.

1006 conjugacy classes of subgroups.

The subgroup lattice has 214 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×#11718739	C₂	1	2	yes	maximal
3	#13123232	C₂	1	2	yes	maximal
4	#13114994	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×#88198	—	2	4	no	—
9	C₂×#88123	C₂²	1	2	yes	—
10	C₁₃×#500	—	2	4	no	—
11	C₂×C₁₃×#107	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	C₂²	1	4	yes	—
14	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
15	#11718739	—	2	8	no	—
16	#11718637	—	2	4	no	—
17	#11718637	C₂²	1	8	yes	—
18	C₂×#88047	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	#88198	—	4	16	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
33	C₂×#6101	—	2	16	no	—
34	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
35	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
37	C₂²×#1055	C₂³	1	2	yes	—
38	#88123	—	2	4	no	—
39	#88123	—	2	4	no	—
40	C₁₃×#107	—	2	8	no	—
41	C₄×C₈×C₁₃	D₄	1	4	yes	—
42	#88058	—	2	8	no	—
43	C₈×Dic₁₃	Q₈	1	16	yes	complemented
44	#88047	—	2	16	no	—
45	C₄×#1055	D₄	1	4	yes	—
46	#33312	—	13	1	no	maximal, Sylow
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	4	8	no	—
49	C₂²×C₄×C₁₃	—	2	4	no	—
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂²×Dic₁₃	—	2	4	no	—
54	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
55	C₂×Dic₂₆	—	4	4	no	—
56	C₂×Dic₂₆	—	4	8	no	—
57	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
59	C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₁₃×C₄⋊C₄	—	2	4	no	—
61	C₁₃×C₄⋊C₄	—	4	4	no	—
62	C₄²×C₁₃	C₂×D₄	1	2	yes	—
63	C₄²×C₁₃	C₂×D₄	1	2	yes	—
64	C₄⋊Dic₁₃	—	2	4	no	—
65	C₄⋊Dic₁₃	—	2	4	no	—
66	C₄⋊Dic₁₃	—	4	4	no	—
67	#6101	—	4	32	no	—
68	C₄×Dic₁₃	D₈	1	8	yes	—
69	C₄×Dic₁₃	Q₁₆	1	8	yes	—
70	C₄×Dic₁₃	C₂×Q₈	1	8	yes	—
71	C₂×#1055	C₂×D₄	1	4	yes	—
72	C₂×#1055	C₂×Q₈	1	8	yes	—
73	C₂×#500	—	13	1	no	—
74	C₂×#500	—	13	2	no	—
75	C₂×#500	D₁₃	1	1	yes	characteristic
76	#2868	—	13	2	no	—
77	#1845	—	13	8	no	—
78	C₄²×C₈	—	13	1	no	—
79	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
80	Q₈×C₁₃	—	8	8	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	4	8	no	—
83	C₂×C₄×C₁₃	#127	1	2	yes	—
84	C₂×C₄×C₁₃	#127	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
86	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
90	C₂×Dic₁₃	—	4	4	no	—
91	C₂×Dic₁₃	—	4	8	no	—
92	C₂×Dic₁₃	C₄⋊Q₈	1	8	yes	—
93	C₂×Dic₁₃	C₂×D₈	1	8	yes	—
94	C₂×Dic₁₃	C₂×Q₁₆	1	8	yes	—
95	Dic₂₆	—	8	8	no	—
96	C₈×C₁₃	C₄⋊Q₈	1	8	yes	complemented
97	#1055	C₄⋊Q₈	1	8	yes	complemented
98	C₂×C₄⋊Q₈	—	13	2	no	—
99	C₂×C₄⋊Q₈	—	26	4	no	—
100	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
101	#500	—	2	4	no	—
102	#500	—	26	8	no	—
103	#500	—	26	4	no	—
104	C₂×#107	—	13	4	no	—
105	C₂×#107	—	13	8	no	—
106	C₂×#107	C₂×D₁₃	1	4	yes	—
107	C₂×C₄×C₈	—	13	4	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
110	#384	—	13	8	no	—
111	#384	—	26	4	no	—
112	C₄³	—	13	1	no	—
113	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
115	C₂²×C₁₃	C₂²×D₈	1	2	yes	—
116	C₂²×C₁₃	C₂²×Q₁₆	1	2	yes	—
117	C₄×C₁₃	—	8	4	no	—
118	C₄×C₁₃	#493	1	4	yes	—
119	C₄×C₁₃	#494	1	4	yes	—
120	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
121	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
122	Dic₁₃	—	8	4	no	—
123	Dic₁₃	#500	1	16	yes	complemented
124	C₂²×Q₈	—	2	2	no	—
125	C₂²×Q₈	—	26	2	no	—
126	C₂²×C₈	—	13	2	no	—
127	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
128	C₄⋊Q₈	—	2	4	no	—
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	26	4	no	—
131	C₄⋊Q₈	—	26	4	no	—
132	C₄⋊Q₈	—	52	16	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	13	4	no	—
135	C₂×C₄⋊C₄	—	26	16	no	—
136	C₂×C₄⋊C₄	—	26	2	no	—
137	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
138	C₂×C₄²	—	13	2	no	—
139	C₂×C₄²	—	13	2	no	—
140	C₂×C₄²	—	13	2	no	—
141	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
142	#107	—	2	8	no	—
143	#107	—	26	16	no	—
144	#107	—	26	8	no	—
145	C₄×C₈	—	13	16	no	—
146	C₄×C₈	—	13	4	no	—
147	C₄×C₈	C₂²⋊D₁₃	1	4	yes	—
148	C₂×C₁₃	#2868	1	1	yes	characteristic
149	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
150	C₂×C₁₃	C₂×#494	1	1	yes	characteristic
151	C₂×C₁₃	C₂×#500	1	4	yes	—
152	C₂×Q₈	—	4	8	no	—
153	C₂×Q₈	—	4	4	no	—
154	C₂×Q₈	—	52	8	no	—
155	C₂×Q₈	—	52	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	13	4	no	—
158	C₂²×C₄	—	26	4	no	—
159	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
160	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
162	C₂×C₈	—	13	4	no	—
163	C₂×C₈	—	13	8	no	—
164	C₂×C₈	Q₈×D₁₃	1	8	yes	—
165	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	26	4	no	—
170	C₄⋊C₄	—	26	4	no	—
171	C₄⋊C₄	—	52	32	no	—
172	C₄⋊C₄	—	52	4	no	—
173	C₄²	—	13	8	no	—
174	C₄²	—	13	8	no	—
175	C₄²	—	13	8	no	—
176	C₄²	D₄×D₁₃	1	2	yes	—
177	C₄²	C₂×C₂²⋊D₁₃	1	2	yes	—
178	C₁₃	#33312	1	1	yes	characteristic, complemented, Sylow
179	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	Q₈	—	104	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	13	8	no	—
185	C₂×C₄	—	13	8	no	—
186	C₂×C₄	—	13	8	no	—
187	C₂×C₄	—	52	8	no	—
188	C₂×C₄	—	52	4	no	—
189	C₂×C₄	D₈×D₁₃	1	4	yes	—
190	C₂×C₄	Q₁₆×D₁₃	1	4	yes	—
191	C₂×C₄	#88194	1	2	yes	—
192	C₂×C₄	#88194	1	2	yes	—
193	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₈	—	13	8	no	—
197	C₈	#88198	1	8	yes	complemented
198	C₂²	D₁₃×C₄⋊Q₈	1	2	yes	—
199	C₂²	C₂×D₈×D₁₃	1	2	yes	—
200	C₂²	C₂×Q₁₆×D₁₃	1	2	yes	—
201	C₂²	C₂×#88194	1	1	yes	characteristic
202	C₄	—	8	4	no	—
203	C₄	—	13	16	no	—
204	C₄	—	104	4	no	—
205	C₄	#11718812	1	4	yes	—
206	C₄	#11718842	1	4	yes	—
207	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
208	C₄	C₂×#88198	1	2	yes	—
209	C₂	#13123232	1	1	yes	characteristic
210	C₂	C₂×#11718812	1	1	yes	characteristic
211	C₂	C₂×#11718842	1	1	yes	characteristic
212	C₂	D₁₃×#500	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	#33312
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
43. C₈×Dic₁₃	Q₈	semidirect
96. C₈×C₁₃	C₄⋊Q₈	semidirect
97. #1055	C₄⋊Q₈	semidirect
123. Dic₁₃	#500	semidirect
178. C₁₃	#33312	semidirect
197. C₈	#88198	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×#11718739	2
#13123232	2
#13114994	2
C₄×C₈×Dic₁₃	2
C₂×C₁₃×#500	2
#33312	13