Subgroups — #15781479

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718055	C₂	1	1	yes	maximal, characteristic
2	#13114715	C₂	1	1	yes	maximal, characteristic
3	#13112943	C₂	1	2	yes	maximal
4	#13112934	C₂	1	2	yes	maximal
5	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
7	C₁₃×#498	—	2	2	no	—
8	C₁₃×#498	C₄	1	4	yes	complemented
9	C₂×C₁₃×#106	—	2	1	no	—
10	C₂×C₁₃×#106	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88059	C₂²	1	1	yes	characteristic
13	#11718767	C₂²	1	1	yes	characteristic
14	#11718762	—	2	1	no	—
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718115	—	2	1	no	—
17	#11718055	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	4	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₁₃×C₄⋊Q₈	—	2	4	no	—
22	C₁₃×C₄⋊Q₈	—	4	2	no	—
23	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
25	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
26	#88180	—	2	1	no	—
27	C₂×#6099	—	2	1	no	—
28	C₁₃×#106	—	2	4	no	—
29	C₁₃×#106	—	2	2	no	—
30	C₁₃×#106	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	D₄	1	2	yes	—
33	#88059	—	2	1	no	—
34	#88059	Q₈	1	2	yes	complemented
35	#88058	Q₈	1	2	yes	complemented
36	#88057	—	2	1	no	—
37	#88055	—	4	1	no	—
38	C₈×Dic₁₃	—	4	1	no	—
39	#88052	—	2	2	no	—
40	#88052	—	2	2	no	—
41	C₂×#6090	—	2	2	no	—
42	#13272	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	4	4	no	—
44	C₂×Q₈×C₁₃	—	8	4	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	4	2	no	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₁₃×C₄⋊C₄	—	4	4	no	—
60	C₁₃×C₄⋊C₄	—	4	2	no	—
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²⋊Dic₁₃	—	4	1	no	—
64	C₄⋊Dic₁₃	—	2	2	no	—
65	C₄×Dic₁₃	—	4	1	no	—
66	#6099	—	2	2	no	—
67	#6099	—	4	1	no	—
68	C₂×#1055	—	4	1	no	—
69	#6090	—	4	2	no	—
70	C₂×#498	D₁₃	1	1	yes	characteristic
71	C₂×#364	—	13	1	no	—
72	#1772	—	13	1	no	—
73	#1386	—	13	2	no	—
74	#1383	—	13	2	no	—
75	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Q₈×C₁₃	—	8	8	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	4	4	no	—
80	C₂×C₄×C₁₃	—	8	4	no	—
81	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
87	C₂×Dic₁₃	—	4	2	no	—
88	C₈×C₁₃	#103	1	2	yes	—
89	C₈×C₁₃	#103	1	2	yes	—
90	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
91	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
92	C₈×C₁₃	#124	1	1	yes	characteristic
93	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
94	#1055	—	4	2	no	—
95	C₂×C₄⋊Q₈	—	2	1	no	—
96	C₂×M₃₂	—	13	2	no	—
97	#498	—	2	2	no	—
98	#498	Dic₁₃	1	4	yes	complemented
99	C₂×#108	—	13	1	no	—
100	#427	—	13	1	no	—
101	C₂×#106	—	2	1	no	—
102	C₂×#106	C₂×D₁₃	1	2	yes	—
103	#405	—	26	1	no	—
104	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
105	#364	—	26	2	no	—
106	#339	—	26	1	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	#418	1	1	yes	characteristic
109	C₂²×C₁₃	#476	1	1	yes	characteristic
110	C₂²×C₁₃	#477	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	8	4	no	—
113	C₄×C₁₃	#391	1	1	yes	characteristic
114	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
115	C₄×C₁₃	#476	1	1	yes	characteristic
116	C₄×C₁₃	#477	1	1	yes	characteristic
117	Dic₁₃	—	8	2	no	—
118	C₂²×Q₈	—	4	1	no	—
119	C₂×M₁₆	—	26	1	no	—
120	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
122	C₄⋊Q₈	—	2	4	no	—
123	C₄⋊Q₈	—	4	2	no	—
124	#117	—	26	1	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
128	M₃₂	—	26	2	no	—
129	M₃₂	—	26	2	no	—
130	C₂×C₁₆	—	26	2	no	—
131	#108	—	13	2	no	—
132	#108	—	26	1	no	—
133	#107	—	13	2	no	—
134	#106	—	2	4	no	—
135	#106	—	2	2	no	—
136	#106	—	26	1	no	—
137	#106	C₂×Dic₁₃	1	4	yes	—
138	#97	—	52	1	no	—
139	C₄×C₈	—	52	1	no	—
140	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
141	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
142	C₂×C₁₃	—	2	1	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	#1987	1	1	yes	characteristic
145	C₂×C₁₃	#2230	1	2	yes	—
146	C₂×Q₈	—	4	4	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
151	M₁₆	—	26	2	no	—
152	M₁₆	—	52	1	no	—
153	C₂×C₈	—	52	1	no	—
154	C₂×C₈	#6106	1	1	yes	characteristic
155	C₂×C₈	#6106	1	1	yes	characteristic
156	C₂×C₈	#6107	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
158	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
159	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
160	C₂×C₈	#6131	1	1	yes	characteristic
161	C₂×C₈	#6131	1	1	yes	characteristic
162	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
163	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	26	2	no	—
168	C₂²⋊C₄	—	52	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	52	1	no	—
171	C₄²	C₂²⋊Dic₁₃	1	2	yes	—
172	C₁₆	—	52	2	no	—
173	C₁₃	#13272	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88186	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	—	52	2	no	—
181	C₂×C₄	#88186	1	1	yes	characteristic
182	C₂×C₄	C₂×#6106	1	1	yes	characteristic
183	C₂×C₄	C₂×#6107	1	1	yes	characteristic
184	C₂×C₄	#88200	1	1	yes	characteristic
185	C₂×C₄	#88200	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
187	C₈	—	52	2	no	—
188	C₈	#88075	1	2	yes	—
189	C₈	#88075	1	2	yes	—
190	C₈	#88198	1	1	yes	characteristic
191	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
192	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
193	C₈	#88201	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#11718745	1	1	yes	characteristic
196	C₂²	#11718752	1	1	yes	characteristic
197	C₂²	#11718876	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	—	104	2	no	—
201	C₄	#11718745	1	1	yes	characteristic
202	C₄	#11718752	1	1	yes	characteristic
203	C₄	C₂×#88075	1	1	yes	characteristic
204	C₄	#11718891	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#13115785	1	1	yes	characteristic
208	C₂	#13117004	1	2	yes	—

	subgroup	G/H
center	C₂²	#11718745
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#498	C₂
socle	C₂²×C₁₃	#476
prime radical	C₂³×C₁₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#498	D₁₃
hypocenter	C₁₃	#13272
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
8. C₁₃×#498	C₄	semidirect
34. #88059	Q₈	semidirect
35. #88058	Q₈	semidirect
98. #498	Dic₁₃	semidirect
173. C₁₃	#13272	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718055	2
#13114715	2
#13112943	2
#13112934	2
C₂×C₁₃×#498	2
#13272	13