Subgroups — #16015560

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88052	C₂	1	2	yes	maximal
2	#13116433	C₂	1	8	yes	maximal, complemented
3	C₄×#88052	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	1	yes	characteristic
8	C₁₃×#405	C₄	1	4	yes	—
9	C₁₃×#405	C₂²	1	4	yes	—
10	C₂×C₁₃×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	C₂×#88052	C₄	1	16	yes	complemented
13	C₂×#88052	C₂²	1	4	yes	—
14	C₂×#88052	C₂²	1	4	yes	—
15	C₂²×#6090	C₂²	1	4	yes	—
16	#11718053	C₂²	1	8	yes	complemented
17	C₄×#6090	C₂²	1	8	yes	complemented
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	C₈	1	8	yes	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
27	C₁₃×#117	C₈	1	4	yes	—
28	C₁₃×#117	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₈	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₁₃×#97	C₂×C₄	1	8	yes	—
33	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₁₃	C₂³	1	4	yes	—
35	#88052	C₂×C₄	1	16	yes	—
36	#88052	C₂×C₄	1	16	yes	complemented
37	C₂×#6090	C₂×C₄	1	16	yes	complemented
38	C₂×#6090	C₂³	1	8	yes	—
39	#33324	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
42	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₁₃×M₁₆	C₂×C₈	1	16	yes	—
49	C₂×C₈×C₁₃	C₄²	1	4	yes	—
50	C₂×C₈×C₁₃	C₄²	1	8	yes	—
51	C₂×C₈×C₁₃	C₂×C₈	1	8	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
60	C₁₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
61	C₄²×C₁₃	C₂×C₈	1	4	yes	—
62	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
63	#6090	C₂²×C₄	1	16	yes	—
64	C₂²×M₃₂	—	13	2	no	—
65	C₂×#405	D₁₃	1	1	yes	characteristic
66	#2108	—	13	8	no	—
67	C₄×M₃₂	—	13	4	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	C₄×C₈	1	4	yes	—
70	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
71	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
74	C₂×C₄×C₁₃	C₄×C₈	1	8	yes	—
75	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
78	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₈×C₁₃	C₂×C₄²	1	2	yes	—
84	C₈×C₁₃	C₂×C₄²	1	4	yes	—
85	C₈×C₁₃	C₂²×C₈	1	8	yes	—
86	C₈×C₁₃	C₂³×C₄	1	2	yes	—
87	C₂²×M₁₆	Dic₁₃	1	1	yes	characteristic
88	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
89	C₂×#117	Dic₁₃	1	1	yes	characteristic
90	C₂×M₃₂	—	13	16	no	—
91	C₂×M₃₂	—	13	4	no	—
92	C₂×M₃₂	—	13	4	no	—
93	C₂²×C₁₆	—	13	4	no	—
94	#405	Dic₁₃	1	4	yes	—
95	#405	C₂×D₁₃	1	4	yes	—
96	C₂×#97	Dic₁₃	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
98	#346	—	13	8	no	—
99	C₄×C₁₆	—	13	8	no	—
100	C₂²×C₁₃	—	2	2	no	—
101	C₂²×C₁₃	—	2	4	no	—
102	C₂²×C₁₃	—	4	4	no	—
103	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
104	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
105	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
106	C₄×C₁₃	—	2	8	no	—
107	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
108	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
109	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
110	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂×M₁₆	#1055	1	8	yes	—
113	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
114	C₂²×C₈	C₄×D₁₃	1	8	yes	—
115	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
116	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
118	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
120	#117	#1055	1	4	yes	—
121	#117	C₂×Dic₁₃	1	4	yes	—
122	C₂×C₄⋊C₄	#1055	1	2	yes	—
123	C₂×C₂²⋊C₄	#1055	1	2	yes	—
124	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
125	M₃₂	—	13	16	no	—
126	M₃₂	—	13	16	no	—
127	C₂×C₁₆	—	13	16	no	—
128	C₂×C₁₆	—	13	8	no	—
129	#97	C₂×Dic₁₃	1	8	yes	—
130	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
131	C₄×C₈	C₂²×D₁₃	1	4	yes	—
132	C₂×C₁₃	—	2	2	no	—
133	C₂×C₁₃	—	2	4	no	—
134	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₁₃	#3407	1	2	yes	—
136	C₂⁴	C₂×#1055	1	1	yes	characteristic
137	C₂²×C₄	C₂×#1055	1	4	yes	—
138	C₂²×C₄	C₂×#1055	1	4	yes	—
139	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
140	C₂²×C₄	C₂×#1055	1	2	yes	—
141	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
142	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
144	M₁₆	C₂×#1055	1	16	yes	—
145	C₂×C₈	C₂×#1055	1	8	yes	—
146	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
147	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
148	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
149	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
150	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
151	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
152	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
154	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
155	C₄⋊C₄	C₂×#1055	1	8	yes	—
156	C₂²⋊C₄	C₂×#1055	1	8	yes	—
157	C₄²	C₂×#1055	1	4	yes	—
158	C₄²	C₂²×Dic₁₃	1	4	yes	—
159	C₁₆	—	13	16	no	—
160	C₁₃	#33324	1	1	yes	characteristic, complemented, Sylow
161	C₂³	—	4	2	no	—
162	C₂³	C₄×#1055	1	4	yes	—
163	C₂³	C₂²×#1055	1	2	yes	—
164	C₂³	C₂²×#1055	1	1	yes	characteristic
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	C₄×#1055	1	4	yes	—
167	C₂×C₄	C₄×#1055	1	8	yes	—
168	C₂×C₄	C₂²×#1055	1	8	yes	—
169	C₂×C₄	C₂²×#1055	1	2	yes	—
170	C₂×C₄	C₂²×#1055	1	4	yes	—
171	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
172	C₂×C₄	C₂²×#1055	1	2	yes	—
173	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
174	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
175	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
176	C₈	C₂²×#1055	1	8	yes	—
177	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
178	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
179	C₈	C₂²×C₄×D₁₃	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	4	4	no	—
183	C₂²	C₂×C₄×#1055	1	2	yes	—
184	C₂²	C₂×C₄×#1055	1	4	yes	—
185	C₂²	C₂³×#1055	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	C₂×C₄×#1055	1	2	yes	—
188	C₄	C₂×C₄×#1055	1	4	yes	—
189	C₄	C₂³×#1055	1	1	yes	characteristic
190	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	4	no	—
193	C₂	C₂²×C₄×#1055	1	1	yes	characteristic
194	C₂	#13127939	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#33324
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13116433	C₂	semidirect
3. C₄×#88052	C₂	semidirect
12. C₂×#88052	C₄	semidirect
16. #11718053	C₂²	semidirect
17. C₄×#6090	C₂²	semidirect
36. #88052	C₂×C₄	semidirect
37. C₂×#6090	C₂×C₄	semidirect
160. C₁₃	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
#13116433	2
C₄×#88052	2
C₂×C₁₃×#405	2
#33324	13