Subgroups — #13123768

Order 1664.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719222	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#405	C₂	1	1	yes	maximal, characteristic
3	#11718780	C₂	1	1	yes	maximal, characteristic
4	#11718779	C₂	1	1	yes	maximal, characteristic
5	#11718762	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718660	C₂	1	1	yes	maximal, characteristic
8	#11718362	C₂	1	4	yes	maximal, complemented
9	C₈×Dic₂₆	C₂	1	4	yes	maximal, complemented
10	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#117	C₂²	1	1	yes	characteristic
13	#88180	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	1	yes	characteristic
16	C₂²×#1055	C₂²	1	1	yes	characteristic
17	#88144	C₄	1	2	yes	—
18	#88142	C₄	1	2	yes	—
19	#88132	C₄	1	4	yes	—
20	C₄×Dic₂₆	C₂²	1	4	yes	—
21	C₁₃×#97	C₂²	1	2	yes	—
22	C₄×C₈×C₁₃	C₂²	1	2	yes	—
23	#88055	C₂²	1	2	yes	—
24	#88054	C₂²	1	4	yes	—
25	#88054	C₂²	1	4	yes	—
26	C₈×Dic₁₃	C₂²	1	2	yes	—
27	C₈×Dic₁₃	C₂²	1	2	yes	—
28	C₈×Dic₁₃	C₂²	1	2	yes	—
29	C₄⋊#1055	C₂²	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
32	C₂×Dic₂₆	C₂×C₄	1	4	yes	—
33	C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₈×C₁₃	Q₈	1	4	yes	complemented
35	C₂×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
40	C₄²×C₁₃	C₂³	1	2	yes	—
41	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
43	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
44	#6101	C₂×C₄	1	4	yes	—
45	#6101	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₁₃	C₂³	1	2	yes	—
49	C₄×Dic₁₃	C₂³	1	2	yes	—
50	#6099	—	2	2	no	—
51	C₂×#1055	—	2	2	no	—
52	C₂×#1055	C₂³	1	2	yes	—
53	C₂×#1055	C₂³	1	2	yes	—
54	#2960	—	13	1	no	maximal, Sylow
55	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
59	C₂×C₄×C₁₃	#41	1	2	yes	—
60	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
64	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
65	Dic₂₆	—	2	8	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	C₂×Q₈	1	2	yes	—
68	C₈×C₁₃	C₂×Q₈	1	2	yes	—
69	#1055	—	2	2	no	—
70	#1055	—	2	2	no	—
71	#533	—	13	1	no	—
72	#446	—	13	4	no	—
73	C₈×Q₈	—	13	4	no	—
74	#424	—	13	1	no	—
75	#424	—	13	1	no	—
76	#423	—	13	1	no	—
77	#405	—	13	1	no	—
78	#405	D₁₃	1	1	yes	characteristic
79	C₂×C₄×C₈	—	13	1	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	C₄×Q₈	1	2	yes	—
82	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₁₃	—	2	4	no	—
84	C₄×C₁₃	C₄×Q₈	1	2	yes	—
85	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
86	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
87	Dic₁₃	—	2	4	no	—
88	Dic₁₃	—	2	2	no	—
89	Dic₁₃	#131	1	4	yes	—
90	C₂×M₁₆	—	13	1	no	—
91	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
92	C₂²×C₈	—	13	1	no	—
93	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₄⋊Q₈	—	13	2	no	—
95	#125	—	13	2	no	—
96	C₂²⋊Q₈	—	13	4	no	—
97	C₄×Q₈	—	13	4	no	—
98	#117	—	13	1	no	—
99	#117	C₂×D₁₃	1	1	yes	characteristic
100	C₂×C₄²	—	13	1	no	—
101	C₄⋊C₈	—	13	4	no	—
102	C₄⋊C₈	—	13	4	no	—
103	C₄⋊C₈	—	13	4	no	—
104	#97	—	13	2	no	—
105	#97	C₂×D₁₃	1	2	yes	—
106	C₄×C₈	—	13	2	no	—
107	C₄×C₈	—	13	2	no	—
108	C₄×C₈	—	13	2	no	—
109	C₄×C₈	C₂×D₁₃	1	2	yes	—
110	C₂×C₁₃	—	2	2	no	—
111	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#131	1	2	yes	—
113	C₂×Q₈	—	13	4	no	—
114	C₂²×C₄	—	13	2	no	—
115	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
116	M₁₆	—	2	2	no	—
117	M₁₆	—	26	2	no	—
118	C₂×C₈	—	13	2	no	—
119	C₂×C₈	—	13	2	no	—
120	C₂×C₈	—	26	2	no	—
121	C₂×C₈	Dic₂₆	1	4	yes	complemented
122	C₂×C₈	C₂²×D₁₃	1	2	yes	—
123	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
124	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₄⋊C₄	—	13	2	no	—
126	C₄⋊C₄	—	13	4	no	—
127	C₄⋊C₄	—	13	4	no	—
128	C₄⋊C₄	—	13	4	no	—
129	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
130	C₂²⋊C₄	—	13	2	no	—
131	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
132	C₄²	—	13	1	no	—
133	C₄²	—	13	1	no	—
134	C₄²	—	13	2	no	—
135	C₄²	—	13	2	no	—
136	C₄²	C₂²×D₁₃	1	2	yes	—
137	C₁₃	#2960	1	1	yes	characteristic, complemented, Sylow
138	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
139	Q₈	—	26	8	no	—
140	C₂×C₄	—	13	2	no	—
141	C₂×C₄	—	13	2	no	—
142	C₂×C₄	—	13	4	no	—
143	C₂×C₄	—	26	4	no	—
144	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₁₃	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
147	C₂×C₄	#6127	1	2	yes	—
148	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	26	2	no	—
151	C₈	—	26	2	no	—
152	C₈	C₂×Dic₂₆	1	2	yes	—
153	C₈	C₂×Dic₂₆	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₂₆	1	2	yes	—
156	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	13	4	no	—
159	C₄	—	26	2	no	—
160	C₄	—	26	4	no	—
161	C₄	C₄×Dic₂₆	1	2	yes	—
162	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
163	C₄	C₂×#6127	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₂₆	1	1	yes	characteristic
166	C₂	D₁₃×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#405	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#405	D₁₃
hypocenter	C₁₃	#2960
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
8. #11718362	C₂	semidirect
9. C₈×Dic₂₆	C₂	semidirect
34. C₂×C₈×C₁₃	Q₈	semidirect
121. C₂×C₈	Dic₂₆	semidirect
137. C₁₃	#2960	semidirect

Maximal subgroups

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

subgroup	index
#11719222	2
C₁₃×#405	2
#11718780	2
#11718779	2
#11718762	2
C₂×C₈×Dic₁₃	2
#11718660	2
#11718362	2
C₈×Dic₂₆	2
#2960	13