Subgroups — #15762749

Order 3328.

526 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115798	C₂	1	2	yes	maximal
2	#13115798	C₂	1	2	yes	maximal
3	#13114964	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718046	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	—	2	2	no	—
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₁₃×#494	—	2	2	no	—
9	C₁₃×#494	C₄	1	4	yes	complemented
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718596	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	#11718143	—	2	2	no	—
15	#11718121	—	2	4	no	—
16	#11718121	—	2	4	no	—
17	#11718046	—	2	2	no	—
18	C₂²×Q₈×C₁₃	—	2	2	no	—
19	C₂×C₁₃×Q₁₆	—	2	8	no	—
20	C₂×C₁₃×Q₁₆	—	4	8	no	—
21	C₂²×C₈×C₁₃	D₄	1	2	yes	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	C₂×C₄⋊Dic₁₃	—	2	2	no	—
31	C₂×C₄⋊Dic₁₃	—	2	2	no	—
32	C₂²×#1055	D₄	1	2	yes	—
33	C₂²×#1055	D₄	1	2	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₃	D₄	1	2	yes	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	C₄×#1055	—	2	1	no	—
39	C₄×#1055	—	2	1	no	—
40	#9835	—	13	1	no	maximal, Sylow
41	C₂×Q₈×C₁₃	—	2	4	no	—
42	C₂×Q₈×C₁₃	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	4	8	no	—
44	C₂²×C₄×C₁₃	—	4	2	no	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	4	2	no	—
49	C₁₃×Q₁₆	—	4	16	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	4	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₁₃×C₄⋊C₄	—	4	2	no	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₃	#41	1	1	yes	characteristic
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₁₃	—	4	4	no	—
61	C₄⋊Dic₁₃	—	4	4	no	—
62	C₄⋊Dic₁₃	—	4	4	no	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	C₂×#1055	—	2	4	no	—
67	C₂×#1055	Q₁₆	1	4	yes	complemented
68	C₂×#494	D₁₃	1	1	yes	characteristic
69	#1990	—	13	2	no	—
70	#1990	—	13	2	no	—
71	#1834	—	13	1	no	—
72	C₂×#322	—	13	1	no	—
73	C₂³×C₁₃	#127	1	1	yes	characteristic
74	Q₈×C₁₃	—	4	8	no	—
75	Q₈×C₁₃	—	8	4	no	—
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	—	8	4	no	—
78	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	#135	1	2	yes	—
86	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
87	C₂×C₄×C₁₃	#137	1	4	yes	—
88	C₂×Dic₁₃	—	8	2	no	—
89	C₂×Dic₁₃	—	8	2	no	—
90	C₂×Dic₁₃	—	8	2	no	—
91	C₈×C₁₃	—	2	4	no	—
92	#1055	—	2	4	no	—
93	#1055	—	4	2	no	—
94	C₂²×Q₁₆	—	2	2	no	—
95	C₂×C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
96	#494	—	2	2	no	—
97	#494	Dic₁₃	1	4	yes	complemented
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	#384	—	13	2	no	—
102	#340	—	26	4	no	—
103	#340	—	26	4	no	—
104	#340	—	26	2	no	—
105	#322	—	26	2	no	—
106	C₂²×C₁₃	#390	1	1	yes	characteristic
107	C₂²×C₁₃	#494	1	1	yes	characteristic
108	C₂²×C₁₃	#495	1	1	yes	characteristic
109	C₂²×C₁₃	#496	1	2	yes	—
110	C₂²×C₁₃	#497	1	2	yes	—
111	C₄×C₁₃	—	8	4	no	—
112	C₄×C₁₃	#418	1	2	yes	—
113	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
114	C₄×C₁₃	#441	1	2	yes	—
115	C₄×C₁₃	#467	1	2	yes	—
116	C₄×C₁₃	#470	1	2	yes	—
117	C₄×C₁₃	#490	1	2	yes	—
118	Dic₁₃	—	16	2	no	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂×Q₁₆	—	2	8	no	—
121	C₂×Q₁₆	—	4	8	no	—
122	C₂²×C₈	—	13	2	no	—
123	C₂²×C₈	—	13	2	no	—
124	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	C₂×Dic₁₃	1	4	yes	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	26	2	no	—
132	C₂×C₄⋊C₄	—	26	2	no	—
133	C₂×C₄⋊C₄	—	26	2	no	—
134	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
135	C₄×C₈	—	26	1	no	—
136	C₄×C₈	—	26	1	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	—	26	1	no	—
139	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
140	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
141	C₂×C₁₃	#1720	1	2	yes	—
142	C₂×C₁₃	#1723	1	2	yes	—
143	C₂×C₁₃	#1961	1	1	yes	characteristic
144	C₂×C₁₃	#1969	1	1	yes	characteristic
145	C₂×C₁₃	#1971	1	1	yes	characteristic
146	C₂×Q₈	—	2	4	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	8	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	52	2	no	—
151	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
152	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	Q₁₆	—	4	16	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	13	4	no	—
157	C₂×C₈	—	26	2	no	—
158	C₂×C₈	—	26	2	no	—
159	C₂×C₈	—	26	4	no	—
160	C₂×C₈	—	26	2	no	—
161	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	52	4	no	—
167	C₄⋊C₄	—	52	4	no	—
168	C₄⋊C₄	—	52	4	no	—
169	C₄²	#6129	1	1	yes	characteristic
170	C₄²	#6129	1	1	yes	characteristic
171	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
172	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
173	C₁₃	#9835	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88194	1	1	yes	characteristic
175	Q₈	—	4	8	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	8	4	no	—
179	C₂×C₄	—	104	2	no	—
180	C₂×C₄	—	104	2	no	—
181	C₂×C₄	—	104	2	no	—
182	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
183	C₂×C₄	#88172	1	4	yes	—
184	C₂×C₄	#88173	1	2	yes	—
185	C₂×C₄	#88186	1	4	yes	—
186	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
187	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
188	C₂×C₄	#88190	1	1	yes	characteristic
189	C₂×C₄	#88192	1	1	yes	characteristic
190	C₂×C₄	#88192	1	1	yes	characteristic
191	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₈	—	2	4	no	—
193	C₈	—	26	4	no	—
194	C₈	—	52	2	no	—
195	C₂²	#11718735	1	2	yes	—
196	C₂²	#11718842	1	1	yes	characteristic
197	C₂²	#11718851	1	2	yes	—
198	C₂²	#11718852	1	1	yes	characteristic
199	C₂²	#11718882	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	208	2	no	—
202	C₄	#11718732	1	2	yes	—
203	C₄	Q₁₆×Dic₁₃	1	2	yes	—
204	C₄	#11718845	1	2	yes	—
205	C₄	#11718849	1	2	yes	—
206	C₄	#11718850	1	2	yes	—
207	C₄	#11718876	1	2	yes	—
208	C₂	#13114499	1	2	yes	—
209	C₂	#13114511	1	2	yes	—
210	C₂	#13115661	1	1	yes	characteristic
211	C₂	#13115698	1	1	yes	characteristic
212	C₂	#13115708	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#494	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#494	D₁₃
hypocenter	C₁₃	#9835
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
9. C₁₃×#494	C₄	semidirect
67. C₂×#1055	Q₁₆	semidirect
97. #494	Dic₁₃	semidirect
173. C₁₃	#9835	semidirect

Maximal subgroups

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

subgroup	index
#13115798	2
#13114964	2
C₂×#11718046	2
C₂×C₁₃×#494	2
#9835	13