Subgroups — #15778310

Order 3328.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718098	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13114717	C₂	1	2	yes	maximal
4	#13112931	C₂	1	2	yes	maximal
5	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₁₃×#427	C₂²	1	2	yes	—
8	C₂×#88059	C₂²	1	2	yes	—
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	#11718658	—	2	2	no	—
12	C₁₃×#367	C₄	1	2	yes	complemented
13	C₁₃×#366	C₄	1	2	yes	complemented
14	#11718124	—	2	2	no	—
15	#11718098	C₄	1	4	yes	complemented
16	C₁₆×Dic₁₃	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₁₃×#117	—	2	2	no	—
19	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂×#6099	—	2	2	no	—
21	C₂²×#1055	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₁₃×#107	C₂×C₄	1	4	yes	—
27	C₁₃×#106	—	2	2	no	—
28	#88059	—	2	2	no	—
29	#88059	C₂×C₄	1	4	yes	—
30	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
31	C₈×Dic₁₃	Q₈	1	2	yes	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	#88043	—	4	2	no	—
36	C₄×#1055	—	4	2	no	—
37	#12670	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	—	2	1	no	—
40	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₂²⋊C₄	—	4	2	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄×Dic₁₃	—	2	1	no	—
50	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
51	#6099	—	2	4	no	—
52	#6099	—	4	2	no	—
53	C₂×#1055	—	4	2	no	—
54	C₂×#1055	C₄⋊C₄	1	2	yes	—
55	C₂×#1055	D₈	1	1	yes	characteristic
56	C₂×#1055	D₈	1	1	yes	characteristic
57	C₂×#1055	Q₁₆	1	2	yes	—
58	C₁₃×C₁₆	C₄²	1	2	yes	—
59	C₁₃×C₁₆	C₄²	1	2	yes	—
60	#6090	—	2	2	no	—
61	C₂×#368	—	13	1	no	—
62	#2158	D₁₃	1	1	yes	characteristic
63	C₂×C₄×C₁₆	—	13	1	no	—
64	#1772	—	13	2	no	—
65	#1382	—	13	2	no	—
66	C₂³×C₁₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	4	4	no	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
74	C₂×Dic₁₃	—	2	4	no	—
75	C₂×Dic₁₃	—	2	2	no	—
76	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
77	C₈×C₁₃	#117	1	1	yes	characteristic
78	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	#1055	—	4	4	no	—
81	#1055	#107	1	2	yes	—
82	#1055	#107	1	2	yes	—
83	C₂²×C₁₆	—	13	1	no	—
84	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
85	C₂×#108	—	13	2	no	—
86	#427	C₂×D₁₃	1	2	yes	—
87	#405	—	26	2	no	—
88	C₂×C₄×C₈	—	13	1	no	—
89	#368	—	13	4	no	—
90	#367	Dic₁₃	1	2	yes	complemented
91	#366	Dic₁₃	1	2	yes	complemented
92	C₄×C₁₆	—	13	4	no	—
93	#339	—	26	2	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	#427	1	1	yes	characteristic
96	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
98	C₄×C₁₃	—	8	4	no	—
99	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
101	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
102	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
103	Dic₁₃	—	2	4	no	—
104	C₂×M₁₆	—	26	2	no	—
105	C₂²×C₈	—	13	1	no	—
106	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
107	#117	—	2	2	no	—
108	C₂×C₄²	—	13	1	no	—
109	C₂×C₁₆	—	13	2	no	—
110	C₂×C₁₆	—	26	1	no	—
111	C₂×C₁₆	—	26	1	no	—
112	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
113	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
114	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
115	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
116	#108	—	13	4	no	—
117	#108	—	26	2	no	—
118	#107	C₂×Dic₁₃	1	4	yes	—
119	#106	—	2	2	no	—
120	#97	—	52	2	no	—
121	C₄×C₈	—	13	2	no	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	—	52	2	no	—
124	C₂×C₁₃	—	2	1	no	—
125	C₂×C₁₃	—	2	1	no	—
126	C₂×C₁₃	C₄×#107	1	1	yes	characteristic
127	C₂×C₁₃	#2178	1	2	yes	—
128	C₂²×C₄	—	26	1	no	—
129	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
130	M₁₆	—	26	4	no	—
131	M₁₆	—	52	2	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	1	no	—
135	C₂×C₈	—	13	2	no	—
136	C₂×C₈	—	52	2	no	—
137	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
138	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
139	C₂×C₈	#6129	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
141	C₂×C₈	#6131	1	1	yes	characteristic
142	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	4	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	—	13	2	no	—
147	C₄²	—	26	1	no	—
148	C₁₆	—	26	2	no	—
149	C₁₆	C₄×Dic₁₃	1	2	yes	—
150	C₁₆	C₄×Dic₁₃	1	2	yes	—
151	C₁₃	#12670	1	1	yes	characteristic, complemented, Sylow
152	C₂³	#88166	1	1	yes	characteristic
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	26	4	no	—
155	C₂×C₄	—	26	2	no	—
156	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
157	C₂×C₄	#88147	1	1	yes	characteristic
158	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
159	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
160	C₂×C₄	#88173	1	1	yes	characteristic
161	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
162	C₈	—	13	2	no	—
163	C₈	—	13	2	no	—
164	C₈	—	52	4	no	—
165	C₈	#88141	1	1	yes	characteristic
166	C₈	#88146	1	1	yes	characteristic
167	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
168	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	#11718534	1	1	yes	characteristic
171	C₂²	#11718541	1	1	yes	characteristic
172	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
173	C₄	—	8	4	no	—
174	C₄	—	26	4	no	—
175	C₄	#11718533	1	1	yes	characteristic
176	C₄	D₁₃×#107	1	1	yes	characteristic
177	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
178	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#107×Dic₁₃	1	1	yes	characteristic
182	C₂	#13116717	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#2158	C₂
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#2158	D₁₃
hypocenter	C₁₃	#12670
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
12. C₁₃×#367	C₄	semidirect
13. C₁₃×#366	C₄	semidirect
15. #11718098	C₄	semidirect
16. C₁₆×Dic₁₃	C₄	semidirect
90. #367	Dic₁₃	semidirect
91. #366	Dic₁₃	semidirect
151. C₁₃	#12670	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718098	2
C₂×C₁₆×Dic₁₃	2
#13114717	2
#13112931	2
C₁₃×#2158	2
#12670	13