Subgroups — #15861667

Order 3328.

1026 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	2	yes	maximal
2	#13114705	C₂	1	8	yes	maximal, complemented
3	#13114598	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
9	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₁₃×#95	C₄	1	2	yes	—
11	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
12	#11718586	C₄	1	16	yes	complemented
13	C₂×C₄⋊#1055	C₂²	1	4	yes	—
14	C₂×#88043	C₂²	1	8	yes	—
15	C₂×C₄×#1055	C₂²	1	4	yes	—
16	#11718116	C₂²	1	8	yes	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
22	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₁₃	—	4	4	no	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
29	C₂²×#1055	C₂³	1	8	yes	—
30	C₁₃×#95	C₂×C₄	1	8	yes	—
31	C₂²⋊#1055	C₂×C₄	1	16	yes	—
32	C₄⋊#1055	C₂×C₄	1	16	yes	complemented
33	#88043	—	4	8	no	—
34	C₄×#1055	C₂×C₄	1	16	yes	complemented
35	#20536	—	13	1	no	maximal, Sylow
36	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	—	2	2	no	—
39	C₂²×C₄×C₁₃	—	2	8	no	—
40	C₂²×C₄×C₁₃	—	4	4	no	—
41	C₂²×C₄×C₁₃	—	4	2	no	—
42	C₂²×C₄×C₁₃	—	4	4	no	—
43	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
44	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
45	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	16	no	—
53	C₄²×C₁₃	—	2	8	no	—
54	C₄²×C₁₃	—	4	4	no	—
55	C₄²×C₁₃	C₄²	1	8	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
57	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×#1055	—	2	8	no	—
59	C₂×#1055	—	2	8	no	—
60	C₂×#1055	C₂²×C₄	1	16	yes	—
61	C₂×#431	—	13	2	no	—
62	C₂×C₄×C₄⋊C₄	D₁₃	1	1	yes	characteristic
63	#1770	—	13	8	no	—
64	#1749	—	13	4	no	—
65	C₂³×C₁₃	—	2	1	no	—
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
69	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
70	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	8	no	—
75	C₂×C₄×C₁₃	—	2	8	no	—
76	C₂×C₄×C₁₃	—	2	8	no	—
77	C₂×C₄×C₁₃	—	2	8	no	—
78	C₂×C₄×C₁₃	—	4	4	no	—
79	C₂×C₄×C₁₃	—	4	16	no	—
80	C₂×C₄×C₁₃	#97	1	16	yes	complemented
81	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
85	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
88	#1055	—	2	16	no	—
89	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
90	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
91	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
92	#431	—	13	16	no	—
93	C₂×C₄⋊C₈	—	13	4	no	—
94	C₂×C₂²⋊C₈	—	13	4	no	—
95	C₂×#97	—	13	8	no	—
96	C₂×C₄×C₈	—	13	4	no	—
97	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
98	C₄×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
99	C₂×#95	Dic₁₃	1	2	yes	—
100	#336	—	13	8	no	—
101	C₂²×C₁₃	—	2	2	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	2	4	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	C₂×#97	1	4	yes	—
106	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
107	C₂²×C₁₃	#518	1	1	yes	characteristic
108	C₂²×C₁₃	#519	1	1	yes	characteristic
109	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
110	C₂²×C₁₃	#568	1	2	yes	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	4	8	no	—
114	C₄×C₁₃	C₂×#97	1	8	yes	—
115	C₄×C₁₃	#520	1	4	yes	—
116	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
117	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₂²×C₈	—	13	8	no	—
121	C₂×C₄⋊C₄	—	2	8	no	—
122	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
123	C₂×C₄²	—	4	4	no	—
124	C₂×C₄²	C₄×D₁₃	1	8	yes	—
125	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
126	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
128	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
129	C₄⋊C₈	—	13	16	no	—
130	C₂²⋊C₈	—	13	16	no	—
131	#97	—	52	8	no	—
132	C₄×C₈	—	13	16	no	—
133	#95	C₂×Dic₁₃	1	8	yes	—
134	C₂×C₁₃	—	2	4	no	—
135	C₂×C₁₃	#2274	1	1	yes	characteristic
136	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
137	C₂×C₁₃	#2873	1	1	yes	characteristic
138	C₂×C₁₃	#2975	1	2	yes	—
139	C₂×C₁₃	#2983	1	2	yes	—
140	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	4	4	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	#6099	1	8	yes	—
148	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
149	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
150	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
151	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
152	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
155	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
156	C₂×C₈	—	13	16	no	—
157	C₂×C₈	—	26	8	no	—
158	C₂×C₈	—	26	8	no	—
159	C₄⋊C₄	—	2	16	no	—
160	C₄²	—	2	8	no	—
161	C₄²	—	4	4	no	—
162	C₄²	C₄×Dic₁₃	1	8	yes	—
163	C₄²	C₂×C₄×D₁₃	1	4	yes	—
164	C₄²	C₂×C₄×D₁₃	1	4	yes	—
165	C₁₃	#20536	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×#6099	1	4	yes	—
170	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
171	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	16	no	—
181	C₂×C₄	#88043	1	16	yes	complemented
182	C₂×C₄	C₂×#6099	1	8	yes	—
183	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
184	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
185	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
186	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
187	C₂×C₄	#88251	1	2	yes	—
188	C₂×C₄	#88254	1	2	yes	—
189	C₈	—	26	16	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	2	no	—
194	C₂²	C₂×#88043	1	4	yes	—
195	C₂²	C₂²×#6099	1	2	yes	—
196	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
197	C₂²	#11719503	1	1	yes	characteristic
198	C₂²	#11719517	1	1	yes	characteristic
199	C₂²	#11719524	1	2	yes	—
200	C₄	—	2	2	no	—
201	C₄	—	2	4	no	—
202	C₄	—	4	8	no	—
203	C₄	C₂×#88043	1	8	yes	—
204	C₄	#11719209	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	#13117193	1	1	yes	characteristic
207	C₂	C₂²×#88043	1	1	yes	characteristic
208	C₂	#13123255	1	1	yes	characteristic
209	C₂	#13123877	1	2	yes	—
210	C₂	#13123921	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₂×C₄×C₁₃×C₄⋊C₄	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#20536
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #13114705	C₂	semidirect
12. #11718586	C₄	semidirect
32. C₄⋊#1055	C₂×C₄	semidirect
34. C₄×#1055	C₂×C₄	semidirect
80. C₂×C₄×C₁₃	#97	semidirect
165. C₁₃	#20536	semidirect
181. C₂×C₄	#88043	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
#13114705	2
#13114598	2
C₂×C₄×C₁₃×C₄⋊C₄	2
#20536	13