Subgroups — #16792593

Order 3328.

890 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#1057	C₂	1	2	yes	maximal
2	#13130039	C₂	1	3	yes	maximal, complemented
3	M₁₆×#1057	C₂	1	6	yes	maximal, complemented
4	C₂×#11718251	C₂	1	3	yes	maximal, complemented
5	#131×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	#41×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88202	C₄	1	1	yes	characteristic
9	Q₈×#1057	C₄	1	4	yes	—
10	D₄×#1057	C₄	1	12	yes	—
11	#11719119	C₂²	1	6	yes	—
12	C₂×C₄×#1057	C₂²	1	6	yes	—
13	#11718783	C₄	1	3	yes	—
14	M₁₆×Dic₁₃	C₂²	1	3	yes	—
15	#11718762	C₄	1	3	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	3	yes	—
17	#11718251	C₂²	1	1	yes	characteristic
18	#11718251	C₂²	1	6	yes	—
19	#11718251	C₂²	1	3	yes	—
20	C₈×#1057	C₂²	1	6	yes	—
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1057	C₂×C₄	1	12	yes	—
23	C₁₃×#131	C₈	1	8	yes	complemented
24	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
26	#88202	C₈	1	8	yes	complemented
27	Q₈×Dic₁₃	C₂×C₄	1	2	yes	—
28	D₄×Dic₁₃	C₂×C₄	1	6	yes	—
29	#88180	C₂×C₄	1	3	yes	—
30	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
31	C₂×#6099	C₂×C₄	1	3	yes	—
32	C₂²×#1055	C₂×C₄	1	3	yes	—
33	C₂²⋊#1057	C₂×C₄	1	12	yes	—
34	C₄⋊#1057	C₂×C₄	1	12	yes	—
35	C₄×#1057	C₂×C₄	1	12	yes	—
36	C₄×#1057	C₂³	1	6	yes	—
37	C₄×#1057	C₂³	1	2	yes	—
38	#88055	C₂×C₄	1	6	yes	—
39	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
40	C₈×Dic₁₃	C₂×C₄	1	3	yes	—
41	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₈×Dic₁₃	C₂³	1	3	yes	—
43	#17316	—	13	1	no	maximal, Sylow
44	C₁₃×#41	C₂×C₈	1	4	yes	—
45	C₁₃×#41	C₂×C₈	1	4	yes	—
46	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₄²	1	3	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
49	C₂²×Dic₁₃	C₂²×C₄	1	6	yes	—
50	C₂×#1057	—	4	12	no	—
51	C₂×#1057	C₂²×C₄	1	4	yes	—
52	C₂×#1057	C₂²×C₄	1	12	yes	—
53	C₁₃×M₁₆	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	complemented
55	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂²⋊Dic₁₃	C₄²	1	6	yes	—
58	C₄⋊Dic₁₃	C₄²	1	6	yes	—
59	C₄×Dic₁₃	C₂²×C₄	1	3	yes	—
60	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
61	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
62	#6099	C₂×C₈	1	12	yes	complemented
63	C₂×#1055	C₂×C₈	1	12	yes	complemented
64	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
65	C₂×#1055	C₂²×C₄	1	3	yes	—
66	C₈×#41	—	13	2	no	—
67	C₄×#131	—	13	1	no	—
68	#1450	—	13	3	no	—
69	C₈×M₁₆	—	13	6	no	—
70	C₂×#322	—	13	3	no	—
71	C₂³×C₁₃	C₂×C₄²	1	3	yes	—
72	Q₈×C₁₃	C₄×C₈	1	4	yes	—
73	D₄×C₁₃	C₄×C₈	1	12	yes	—
74	C₂×C₄×C₁₃	C₂×C₄²	1	3	yes	—
75	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×Dic₁₃	—	2	12	no	—
79	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
80	C₂×Dic₁₃	C₂³×C₄	1	2	yes	—
81	#1057	—	2	4	no	—
82	#1057	—	2	12	no	—
83	C₈×C₁₃	C₂²×C₈	1	2	yes	—
84	C₈×C₁₃	C₂²×C₈	1	6	yes	—
85	#1055	C₂²×C₈	1	2	yes	—
86	#1055	C₂²×C₈	1	6	yes	—
87	C₂×#131	—	13	1	no	—
88	C₂×#131	#261	1	1	yes	characteristic
89	C₄×#41	—	13	1	no	—
90	C₈×Q₈	—	13	4	no	—
91	C₈×D₄	—	13	12	no	—
92	#431	—	13	6	no	—
93	#405	—	13	3	no	—
94	#405	—	13	3	no	—
95	C₄×M₁₆	—	13	3	no	—
96	C₂×C₄×C₈	—	13	6	no	—
97	C₂×C₄×C₈	—	13	3	no	—
98	#322	—	13	1	no	—
99	#322	—	13	6	no	—
100	#322	—	13	3	no	—
101	C₈²	—	13	6	no	—
102	C₂²×C₁₃	—	2	6	no	—
103	C₂²×C₁₃	C₂×C₄×C₈	1	6	yes	—
104	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
105	C₄×C₁₃	C₂×C₄×C₈	1	6	yes	—
106	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
108	Dic₁₃	—	2	6	no	—
109	Dic₁₃	#568	1	4	yes	—
110	C₂×#41	C₂×#261	1	1	yes	characteristic
111	#131	—	13	8	no	—
112	#131	#1057	1	8	yes	complemented
113	C₂×M₁₆	—	13	3	no	—
114	C₂×M₁₆	C₂×#261	1	3	yes	—
115	C₂²×C₈	—	13	3	no	—
116	C₂²×C₈	—	13	12	no	—
117	C₂²×C₈	C₂×#261	1	3	yes	—
118	C₄×Q₈	—	13	2	no	—
119	C₄×D₄	—	13	6	no	—
120	#117	—	13	3	no	—
121	C₂×C₄²	—	13	3	no	—
122	C₄⋊C₈	—	13	12	no	—
123	C₂²⋊C₈	—	13	12	no	—
124	#97	—	13	6	no	—
125	C₄×C₈	—	13	12	no	—
126	C₄×C₈	—	13	6	no	—
127	C₄×C₈	—	13	2	no	—
128	C₄×C₈	—	13	3	no	—
129	C₄×C₈	—	13	1	no	—
130	C₄×C₈	—	13	3	no	—
131	C₄×C₈	—	13	3	no	—
132	C₂×C₁₃	—	2	6	no	—
133	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
134	C₂×C₁₃	#2875	1	2	yes	—
135	#41	D₁₃⋊C₈	1	4	yes	—
136	#41	C₂×#1057	1	4	yes	—
137	C₂×Q₈	C₄×#261	1	1	yes	characteristic
138	C₂×D₄	C₄×#261	1	3	yes	—
139	C₂²×C₄	—	13	6	no	—
140	C₂²×C₄	C₂²×#261	1	3	yes	—
141	M₁₆	—	13	12	no	—
142	M₁₆	C₂×#1057	1	12	yes	complemented
143	C₂×C₈	—	13	12	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	3	no	—
146	C₂×C₈	—	13	4	no	—
147	C₂×C₈	—	13	12	no	—
148	C₂×C₈	—	52	12	no	—
149	C₂×C₈	C₂×#1057	1	12	yes	complemented
150	C₂×C₈	C₂²×#261	1	1	yes	characteristic
151	C₂×C₈	C₂²×#261	1	3	yes	—
152	C₄⋊C₄	—	13	6	no	—
153	C₂²⋊C₄	—	13	6	no	—
154	C₄²	—	13	6	no	—
155	C₄²	—	13	1	no	—
156	C₄²	—	13	3	no	—
157	C₁₃	#17316	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂×C₄×#261	1	3	yes	—
159	Q₈	C₄×#1057	1	4	yes	—
160	D₄	C₄×#1057	1	12	yes	—
161	C₂×C₄	—	13	2	no	—
162	C₂×C₄	—	13	6	no	—
163	C₂×C₄	—	26	12	no	—
164	C₂×C₄	C₂×D₁₃⋊C₈	1	6	yes	—
165	C₂×C₄	C₂×C₄×#261	1	3	yes	—
166	C₂×C₄	C₂²×#1057	1	6	yes	—
167	C₂×C₄	C₂³×#261	1	1	yes	characteristic
168	C₈	—	13	2	no	—
169	C₈	—	13	6	no	—
170	C₈	—	26	4	no	—
171	C₈	—	26	12	no	—
172	C₈	C₂²×#1057	1	2	yes	—
173	C₈	C₂²×#1057	1	6	yes	—
174	C₂²	—	2	6	no	—
175	C₂²	C₂×C₄×#1057	1	6	yes	—
176	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
177	C₄	—	13	4	no	—
178	C₄	—	26	6	no	—
179	C₄	C₂×C₄×#1057	1	6	yes	—
180	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
181	C₄	C₂³×#1057	1	1	yes	characteristic
182	C₂	—	2	6	no	—
183	C₂	C₂²×C₄×#1057	1	1	yes	characteristic
184	C₂	#13131406	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₂×C₁₃×#131	C₄
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×#131	#261
hypocenter	C₁₃	#17316
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #13130039	C₂	semidirect
3. M₁₆×#1057	C₂	semidirect
4. C₂×#11718251	C₂	semidirect
23. C₁₃×#131	C₈	semidirect
26. #88202	C₈	semidirect
53. C₁₃×M₁₆	C₂×C₈	semidirect
54. C₂×C₈×C₁₃	C₂×C₈	semidirect
62. #6099	C₂×C₈	semidirect
63. C₂×#1055	C₂×C₈	semidirect
112. #131	#1057	semidirect
142. M₁₆	C₂×#1057	semidirect
149. C₂×C₈	C₂×#1057	semidirect
157. C₁₃	#17316	semidirect

Maximal subgroups

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

subgroup	index
#41×#1057	2
#13130039	2
M₁₆×#1057	2
C₂×#11718251	2
#131×Dic₁₃	2
#17316	13