Subgroups — #15778298

Order 3328.

308 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116609	C₂	1	2	yes	maximal
2	#13114715	C₂	1	2	yes	maximal
3	#13112940	C₂	1	1	yes	maximal, characteristic
4	#13112938	C₂	1	1	yes	maximal, characteristic
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₂²	1	2	yes	—
10	#11718767	C₂²	1	2	yes	—
11	#11718762	—	2	2	no	—
12	C₁₃×#346	C₄	1	2	yes	complemented
13	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
14	#11718115	—	2	2	no	—
15	#11718105	C₄	1	8	yes	complemented
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	#88180	—	2	2	no	—
20	C₂×#6099	—	2	2	no	—
21	C₁₃×M₃₂	—	2	2	no	—
22	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
23	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
29	#88059	—	2	2	no	—
30	#88059	C₂×C₄	1	4	yes	—
31	#88058	—	2	2	no	—
32	#88057	C₂×C₄	1	4	yes	—
33	#88055	—	4	2	no	—
34	C₈×Dic₁₃	—	4	2	no	—
35	#12668	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	—	2	1	no	—
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₁₃	#41	1	2	yes	—
46	C₄²×C₁₃	—	2	1	no	—
47	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	2	4	no	—
51	C₄×Dic₁₃	—	4	2	no	—
52	#6099	—	2	4	no	—
53	#6099	—	4	2	no	—
54	C₂×#1055	—	4	2	no	—
55	C₁₃×C₁₆	—	2	2	no	—
56	C₁₃×C₁₆	C₄²	1	4	yes	—
57	#2155	—	13	2	no	—
58	C₄×M₃₂	D₁₃	1	1	yes	characteristic
59	#1772	—	13	2	no	—
60	#1385	—	13	1	no	—
61	#1384	—	13	1	no	—
62	C₂³×C₁₃	#135	1	1	yes	characteristic
63	C₂×C₄×C₁₃	—	2	1	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	4	2	no	—
66	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
70	C₂×Dic₁₃	—	4	4	no	—
71	C₈×C₁₃	#106	1	2	yes	—
72	C₈×C₁₃	#106	1	1	yes	characteristic
73	C₈×C₁₃	#106	1	1	yes	characteristic
74	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₁₃	#117	1	1	yes	characteristic
76	C₈×C₁₃	C₄×Q₈	1	2	yes	—
77	#1055	—	4	4	no	—
78	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
79	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
80	C₂×#108	—	13	2	no	—
81	#427	—	13	2	no	—
82	#405	—	26	2	no	—
83	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
84	#365	—	13	8	no	—
85	#346	Dic₁₃	1	2	yes	complemented
86	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
87	#339	—	26	2	no	—
88	C₂²×C₁₃	—	4	1	no	—
89	C₂²×C₁₃	#427	1	1	yes	characteristic
90	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	4	1	no	—
93	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
95	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
96	Dic₁₃	—	8	4	no	—
97	C₂×M₁₆	—	26	2	no	—
98	C₂²×C₈	D₅₂	1	1	yes	characteristic
99	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	#117	—	26	2	no	—
101	C₂×C₄²	D₅₂	1	1	yes	characteristic
102	M₃₂	—	2	2	no	—
103	M₃₂	C₄×D₁₃	1	4	yes	—
104	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
105	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
106	#108	—	13	4	no	—
107	#108	—	26	2	no	—
108	#107	—	26	2	no	—
109	#106	—	13	4	no	—
110	#97	—	52	2	no	—
111	C₄×C₈	—	52	2	no	—
112	C₄×C₈	Dic₂₆	1	1	yes	characteristic
113	C₄×C₈	Dic₂₆	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
115	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
116	C₂×C₁₃	—	2	2	no	—
117	C₂×C₁₃	—	2	1	no	—
118	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
121	M₁₆	—	26	4	no	—
122	M₁₆	—	52	2	no	—
123	C₂×C₈	—	52	2	no	—
124	C₂×C₈	#6095	1	2	yes	—
125	C₂×C₈	#6095	1	2	yes	—
126	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
128	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
130	C₂×C₈	#6127	1	2	yes	—
131	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
132	C₄⋊C₄	—	26	4	no	—
133	C₂²⋊C₄	—	52	2	no	—
134	C₄²	—	2	1	no	—
135	C₄²	—	52	2	no	—
136	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
137	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
138	C₁₆	—	2	2	no	—
139	C₁₆	C₄×Dic₁₃	1	4	yes	—
140	C₁₃	#12668	1	1	yes	characteristic, complemented, Sylow
141	C₂³	#88158	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	52	4	no	—
146	C₂×C₄	C₄×D₅₂	1	2	yes	—
147	C₂×C₄	C₂×#6095	1	2	yes	—
148	C₂×C₄	#88158	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
150	C₈	—	52	4	no	—
151	C₈	#88057	1	1	yes	characteristic
152	C₈	#88057	1	2	yes	—
153	C₈	#88057	1	1	yes	characteristic
154	C₈	C₄×Dic₂₆	1	2	yes	—
155	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
156	C₈	#88180	1	1	yes	characteristic
157	C₂²	—	4	1	no	—
158	C₂²	C₄×#6095	1	2	yes	—
159	C₂²	#11718767	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	104	4	no	—
163	C₄	C₄×#6095	1	2	yes	—
164	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
165	C₄	C₂×#88057	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	C₄×#88057	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#88057
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂×C₁₃	C₄×#106
prime radical	C₂³×C₁₃	#135
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#12668
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
12. C₁₃×#346	C₄	semidirect
13. C₄×C₁₃×C₁₆	C₄	semidirect
15. #11718105	C₄	semidirect
85. #346	Dic₁₃	semidirect
86. C₄×C₁₆	Dic₁₃	semidirect
140. C₁₃	#12668	semidirect

Maximal subgroups

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

subgroup	index
#13116609	2
#13114715	2
#13112940	2
#13112938	2
C₄×C₁₃×M₃₂	2
#12668	13