Subgroups — #15756683

Order 3328.

382 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	#13114988	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718050	C₂	1	2	yes	maximal
4	C₂×#11718049	C₂	1	2	yes	maximal
5	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#1055	—	2	2	no	—
9	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₁₃×#336	C₄	1	2	yes	—
13	C₁₃×#336	C₄	1	2	yes	—
14	#11718116	—	2	2	no	—
15	#11718050	—	2	4	no	—
16	#11718049	—	2	4	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	—	2	1	no	—
25	C₂²×#1055	—	2	1	no	—
26	C₄×C₈×C₁₃	D₄	1	4	yes	—
27	C₄×C₈×C₁₃	Q₈	1	4	yes	—
28	C₄⋊#1055	—	2	2	no	—
29	C₄⋊#1055	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	—	4	4	no	—
33	#8812	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃	—	2	4	no	—
40	C₂×C₈×C₁₃	—	2	4	no	—
41	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
42	C₄²×C₁₃	—	2	4	no	—
43	C₄²×C₁₃	—	2	2	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	C₂×D₄	1	2	yes	—
48	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
49	C₂×#1055	—	4	2	no	—
50	C₂×#1055	—	4	1	no	—
51	C₂×#1055	—	4	1	no	—
52	C₂×#1055	—	4	2	no	—
53	C₂×#1055	—	4	4	no	—
54	C₂×#1055	—	4	2	no	—
55	#1842	—	13	1	no	—
56	#1831	—	13	1	no	—
57	#1744	D₁₃	1	1	yes	characteristic
58	C₂×#335	—	13	2	no	—
59	C₂×#334	—	13	2	no	—
60	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	#108	1	4	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
71	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#131	1	2	yes	—
73	C₂×C₄×C₁₃	#131	1	2	yes	—
74	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₁₃	#137	1	2	yes	—
76	C₈×C₁₃	—	2	8	no	—
77	#1055	—	8	1	no	—
78	#1055	—	8	2	no	—
79	#1055	—	8	1	no	—
80	C₂×C₄⋊C₈	—	13	1	no	—
81	C₂×C₄⋊C₈	—	13	2	no	—
82	C₂×C₄⋊C₈	—	13	1	no	—
83	C₂×C₄⋊C₈	—	26	2	no	—
84	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
85	C₄³	C₂×D₁₃	1	1	yes	characteristic
86	#336	—	26	2	no	—
87	#336	Dic₁₃	1	2	yes	—
88	#336	Dic₁₃	1	2	yes	—
89	#335	—	26	4	no	—
90	#334	—	26	4	no	—
91	C₂²×C₁₃	#384	1	1	yes	characteristic
92	C₂²×C₁₃	#424	1	1	yes	characteristic
93	C₂²×C₁₃	#424	1	1	yes	characteristic
94	C₂²×C₁₃	#428	1	2	yes	—
95	C₂²×C₁₃	C₂×#108	1	2	yes	—
96	C₄×C₁₃	—	2	4	no	—
97	C₄×C₁₃	—	2	2	no	—
98	C₄×C₁₃	—	4	2	no	—
99	C₄×C₁₃	#424	1	4	yes	—
100	C₄×C₁₃	#496	1	1	yes	characteristic
101	C₄×C₁₃	#497	1	1	yes	characteristic
102	C₄×C₁₃	#501	1	2	yes	—
103	C₂²×C₈	—	26	1	no	—
104	C₂²×C₈	—	26	1	no	—
105	C₂²×C₈	—	26	2	no	—
106	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
107	C₂×C₄²	Dic₂₆	1	2	yes	—
108	C₂×C₄²	D₅₂	1	2	yes	—
109	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
111	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
112	C₄⋊C₈	—	26	2	no	—
113	C₄⋊C₈	—	26	2	no	—
114	C₄⋊C₈	—	26	2	no	—
115	C₄⋊C₈	—	26	2	no	—
116	C₄⋊C₈	—	52	4	no	—
117	C₄×C₈	Dic₂₆	1	4	yes	—
118	C₄×C₈	D₅₂	1	4	yes	—
119	C₂×C₁₃	#1573	1	2	yes	—
120	C₂×C₁₃	#1574	1	2	yes	—
121	C₂×C₁₃	#1843	1	1	yes	characteristic
122	C₂×C₁₃	#1847	1	1	yes	characteristic
123	C₂×C₁₃	#1848	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
125	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
126	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
128	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	52	2	no	—
132	C₂×C₈	—	52	1	no	—
133	C₂×C₈	—	52	1	no	—
134	C₂×C₈	—	52	4	no	—
135	C₂×C₈	—	52	2	no	—
136	C₂×C₈	—	52	2	no	—
137	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
138	C₄²	—	2	2	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	—	2	4	no	—
143	C₄²	C₂×Dic₂₆	1	2	yes	—
144	C₄²	C₂×D₅₂	1	2	yes	—
145	C₁₃	#8812	1	1	yes	characteristic, complemented, Sylow
146	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	#88059	1	4	yes	—
154	C₂×C₄	#88123	1	2	yes	—
155	C₂×C₄	#88123	1	1	yes	characteristic
156	C₂×C₄	#88128	1	1	yes	characteristic
157	C₂×C₄	#88162	1	2	yes	—
158	C₂×C₄	#88163	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
160	C₂×C₄	#88202	1	2	yes	—
161	C₂×C₄	#88202	1	2	yes	—
162	C₈	—	2	8	no	—
163	C₈	—	104	2	no	—
164	C₈	—	104	1	no	—
165	C₈	—	104	1	no	—
166	C₂²	#11718596	1	1	yes	characteristic
167	C₂²	#11718660	1	1	yes	characteristic
168	C₂²	#11718660	1	1	yes	characteristic
169	C₂²	C₂×#88059	1	2	yes	—
170	C₂²	#11718782	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	2	no	—
173	C₄	—	4	2	no	—
174	C₄	#11718363	1	2	yes	—
175	C₄	#11718373	1	1	yes	characteristic
176	C₄	#11718376	1	1	yes	characteristic
177	C₄	#11718660	1	4	yes	—
178	C₂	#13113621	1	2	yes	—
179	C₂	#13113625	1	2	yes	—
180	C₂	#13114990	1	1	yes	characteristic
181	C₂	#13115004	1	1	yes	characteristic
182	C₂	#13115008	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#8812
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
145. C₁₃	#8812	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13114953	2
C₂×#11718050	2
C₂×#11718049	2
C₁₃×#1744	2
#8812	13