Subgroups — #13123920

Order 1664.

356 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#432	C₂	1	1	yes	maximal, characteristic
3	#11718587	C₂	1	1	yes	maximal, characteristic
4	#11718489	C₂	1	4	yes	maximal, complemented
5	#11718380	C₂	1	2	yes	maximal
6	#11718379	C₂	1	2	yes	maximal
7	#11718362	C₂	1	4	yes	maximal, complemented
8	C₂²×Dic₂₆	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×#6101	C₄	1	1	yes	characteristic
12	C₂×#6101	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₆	C₄	1	2	yes	—
16	C₄×Dic₂₆	C₄	1	2	yes	—
17	C₄×Dic₂₆	C₂²	1	2	yes	—
18	C₄×Dic₂₆	C₂²	1	2	yes	—
19	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₁₃×#97	C₂²	1	2	yes	—
22	C₂²⋊#1055	C₂²	1	2	yes	—
23	#88055	C₂²	1	4	yes	—
24	#88054	C₂²	1	4	yes	—
25	#88054	C₂²	1	4	yes	—
26	C₈×Dic₁₃	C₂²	1	4	yes	—
27	C₄⋊#1055	C₂²	1	2	yes	—
28	#88043	C₂²	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₂₆	—	2	4	no	—
37	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₁₃	C₂³	1	4	yes	—
40	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
46	#6101	C₂×C₄	1	2	yes	—
47	#6101	C₂×C₄	1	2	yes	—
48	#6101	C₂×C₄	1	2	yes	—
49	#6101	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₄×Dic₁₃	C₂³	1	2	yes	—
54	C₂×#1055	C₂³	1	4	yes	—
55	#2983	—	13	1	no	maximal, Sylow
56	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	2	2	no	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	4	1	no	—
60	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₁₃	—	2	2	no	—
67	C₂×Dic₁₃	—	4	1	no	—
68	C₂×Dic₁₃	—	4	1	no	—
69	C₂×Dic₁₃	M₁₆	1	4	yes	—
70	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
71	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
72	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
74	Dic₂₆	—	2	4	no	—
75	Dic₂₆	—	2	4	no	—
76	C₈×C₁₃	—	2	4	no	—
77	#1055	—	2	4	no	—
78	C₂×C₄×Q₈	—	13	1	no	—
79	#446	—	13	4	no	—
80	#446	—	13	4	no	—
81	#432	—	13	1	no	—
82	#432	—	13	2	no	—
83	#432	D₁₃	1	1	yes	characteristic
84	#431	—	13	2	no	—
85	C₂²×C₁₃	—	2	1	no	—
86	C₂²×C₁₃	—	2	1	no	—
87	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
88	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	1	no	—
91	C₄×C₁₃	—	2	1	no	—
92	C₄×C₁₃	—	2	1	no	—
93	C₄×C₁₃	—	2	1	no	—
94	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
95	Dic₁₃	—	2	2	no	—
96	Dic₁₃	—	2	2	no	—
97	Dic₁₃	—	2	2	no	—
98	Dic₁₃	C₂×M₁₆	1	4	yes	—
99	C₂²×Q₈	—	13	1	no	—
100	C₄×Q₈	—	13	2	no	—
101	C₄×Q₈	—	13	2	no	—
102	C₄×Q₈	—	13	2	no	—
103	C₄×Q₈	—	13	2	no	—
104	C₂×C₄⋊C₄	—	13	1	no	—
105	C₂×C₄⋊C₄	—	13	1	no	—
106	C₂×C₄⋊C₄	—	13	1	no	—
107	C₂×C₄²	—	13	1	no	—
108	C₂×C₄²	—	13	1	no	—
109	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	C₄⋊C₈	—	13	4	no	—
111	C₄⋊C₈	—	13	4	no	—
112	C₄⋊C₈	—	13	2	no	—
113	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
114	C₂²⋊C₈	—	13	2	no	—
115	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
116	#97	—	13	4	no	—
117	#97	—	13	2	no	—
118	#97	C₂×D₁₃	1	2	yes	—
119	C₄×C₈	—	13	4	no	—
120	C₂×C₁₃	—	2	2	no	—
121	C₂×C₁₃	#519	1	1	yes	characteristic
122	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₂×C₁₃	#568	1	1	yes	characteristic
124	C₂×Q₈	—	13	2	no	—
125	C₂×Q₈	—	13	2	no	—
126	C₂×Q₈	—	26	4	no	—
127	C₂²×C₄	—	13	1	no	—
128	C₂²×C₄	—	13	1	no	—
129	C₂²×C₄	—	13	1	no	—
130	C₂²×C₄	—	13	1	no	—
131	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
132	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₈	—	13	4	no	—
135	C₂×C₈	C₂²×D₁₃	1	4	yes	—
136	C₄⋊C₄	—	13	2	no	—
137	C₄⋊C₄	—	13	2	no	—
138	C₄⋊C₄	—	13	2	no	—
139	C₄⋊C₄	—	13	2	no	—
140	C₄⋊C₄	—	13	2	no	—
141	C₄⋊C₄	—	13	2	no	—
142	C₄²	—	13	2	no	—
143	C₄²	—	13	2	no	—
144	C₄²	—	13	2	no	—
145	C₄²	—	13	2	no	—
146	C₄²	C₄×D₁₃	1	1	yes	characteristic
147	C₄²	C₄×D₁₃	1	1	yes	characteristic
148	C₄²	C₂²×D₁₃	1	1	yes	characteristic
149	C₄²	C₂²×D₁₃	1	1	yes	characteristic
150	C₁₃	#2983	1	1	yes	characteristic, complemented, Sylow
151	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
152	Q₈	—	26	4	no	—
153	Q₈	—	26	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	13	4	no	—
158	C₂×C₄	—	13	2	no	—
159	C₂×C₄	—	13	2	no	—
160	C₂×C₄	—	13	2	no	—
161	C₂×C₄	—	13	2	no	—
162	C₂×C₄	—	26	2	no	—
163	C₂×C₄	—	52	1	no	—
164	C₂×C₄	—	52	1	no	—
165	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
168	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
171	C₈	—	2	4	no	—
172	C₈	—	26	4	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	M₁₆×D₁₃	1	2	yes	—
176	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	13	4	no	—
183	C₄	—	26	2	no	—
184	C₄	—	26	2	no	—
185	C₄	—	26	2	no	—
186	C₄	#88257	1	2	yes	—
187	C₂	—	2	2	no	—
188	C₂	#11719221	1	1	yes	characteristic
189	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
190	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#432	C₂
prime radical	C₂³×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#432	D₁₃
hypocenter	C₁₃	#2983
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
4. #11718489	C₂	semidirect
7. #11718362	C₂	semidirect
150. C₁₃	#2983	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₆	2
C₁₃×#432	2
#11718587	2
#11718489	2
#11718380	2
#11718379	2
#11718362	2
#2983	13