Subgroups — #15781545

Order 3328.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88052	C₂	1	1	yes	maximal, characteristic
2	#13115000	C₂	1	1	yes	maximal, characteristic
3	#13112941	C₂	1	4	yes	maximal
4	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#533	—	2	1	no	—
6	C₁₃×#500	C₄	1	2	yes	complemented
7	C₁₃×#499	—	2	1	no	—
8	C₁₃×#499	C₄	1	2	yes	complemented
9	C₁₃×#498	—	2	1	no	—
10	C₁₃×#427	—	2	1	no	—
11	C₁₃×#427	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88058	—	2	1	no	—
14	C₂×#88057	C₂²	1	2	yes	—
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718597	—	2	1	no	—
17	#11718053	—	2	1	no	—
18	C₄×#6090	—	2	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	2	no	—
22	C₁₃×#125	—	2	2	no	—
23	C₁₃×C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈×C₁₃	—	4	2	no	—
25	C₁₃×#117	—	2	2	no	—
26	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₃	—	2	2	no	—
28	C₁₃×#107	—	2	2	no	—
29	C₁₃×#107	C₂×C₄	1	4	yes	—
30	C₁₃×#106	—	2	2	no	—
31	C₁₃×#106	—	2	2	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
36	#88071	—	4	2	no	—
37	#88058	—	4	2	no	—
38	#88057	—	2	4	no	—
39	#88052	—	2	2	no	—
40	#88052	Q₈	1	4	yes	complemented
41	C₂×#6090	—	2	2	no	—
42	#13288	—	13	1	no	maximal, Sylow
43	C₂×Q₈×C₁₃	—	4	2	no	—
44	C₂²×C₄×C₁₃	D₈	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
50	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₃	#41	1	2	yes	—
55	C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₁₃×C₄⋊C₄	—	4	4	no	—
57	C₁₃×C₄⋊C₄	—	4	2	no	—
58	C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₄²×C₁₃	—	4	2	no	—
60	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₃	SD₁₆	1	2	yes	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	#6090	—	2	4	no	—
66	#3160	D₁₃	1	1	yes	characteristic
67	C₄×M₃₂	—	13	1	no	—
68	#1846	—	13	1	no	—
69	#1386	—	13	4	no	—
70	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
71	Q₈×C₁₃	—	8	4	no	—
72	C₂×C₄×C₁₃	—	4	2	no	—
73	C₂×C₄×C₁₃	—	4	4	no	—
74	C₂×C₄×C₁₃	#102	1	2	yes	—
75	C₂×C₄×C₁₃	#102	1	2	yes	—
76	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×Dic₁₃	—	8	4	no	—
83	C₂×Dic₁₃	—	8	2	no	—
84	C₈×C₁₃	#103	1	2	yes	—
85	C₈×C₁₃	#103	1	2	yes	—
86	C₈×C₁₃	C₄×Q₈	1	2	yes	—
87	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
88	#533	—	2	1	no	—
89	C₂×M₃₂	—	13	2	no	—
90	#500	Dic₁₃	1	2	yes	complemented
91	#499	—	2	1	no	—
92	#499	Dic₁₃	1	2	yes	complemented
93	#498	—	2	1	no	—
94	#427	—	2	1	no	—
95	#427	C₂×D₁₃	1	2	yes	—
96	C₂×#107	—	26	1	no	—
97	C₂×#106	—	13	2	no	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	#382	—	26	1	no	—
100	#346	—	26	1	no	—
101	C₄×C₁₆	—	26	1	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
104	C₂²×C₁₃	#486	1	1	yes	characteristic
105	C₂²×C₁₃	#500	1	1	yes	characteristic
106	C₄×C₁₃	—	2	2	no	—
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	—	8	4	no	—
109	C₄×C₁₃	#391	1	1	yes	characteristic
110	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
111	C₄×C₁₃	#487	1	1	yes	characteristic
112	C₄×C₁₃	#498	1	1	yes	characteristic
113	Dic₁₃	—	16	2	no	—
114	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	2	no	—
117	#125	—	2	2	no	—
118	C₂²⋊Q₈	—	4	2	no	—
119	C₄×Q₈	—	4	2	no	—
120	#117	—	2	2	no	—
121	C₂×C₄⋊C₄	—	26	2	no	—
122	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
123	M₃₂	—	13	4	no	—
124	M₃₂	—	26	2	no	—
125	C₂×C₁₆	—	26	2	no	—
126	#107	—	2	2	no	—
127	#107	—	52	2	no	—
128	#107	C₂×Dic₁₃	1	4	yes	—
129	#106	—	2	2	no	—
130	#106	—	2	2	no	—
131	#106	—	26	4	no	—
132	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
136	#95	—	52	2	no	—
137	C₂×C₁₃	—	2	1	no	—
138	C₂×C₁₃	—	2	1	no	—
139	C₂×C₁₃	#1990	1	1	yes	characteristic
140	C₂×C₁₃	#2145	1	2	yes	—
141	C₂×Q₈	—	4	2	no	—
142	C₂²×C₄	—	52	2	no	—
143	C₂²×C₄	#6104	1	2	yes	—
144	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
145	C₂×C₈	#6106	1	2	yes	—
146	C₂×C₈	#6107	1	2	yes	—
147	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₁₃	1	2	yes	—
150	C₂×C₈	#6131	1	2	yes	—
151	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₄⋊C₄	—	52	2	no	—
158	C₂²⋊C₄	—	4	2	no	—
159	C₄²	—	4	2	no	—
160	C₄²	#6105	1	2	yes	—
161	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
162	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
163	C₁₆	—	26	4	no	—
164	C₁₃	#13288	1	1	yes	characteristic, complemented, Sylow
165	C₂³	C₂×#6104	1	1	yes	characteristic
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	104	4	no	—
170	C₂×C₄	—	104	2	no	—
171	C₂×C₄	#88072	1	2	yes	—
172	C₂×C₄	#88072	1	2	yes	—
173	C₂×C₄	C₂×#6105	1	1	yes	characteristic
174	C₂×C₄	C₂×#6106	1	1	yes	characteristic
175	C₂×C₄	C₂×#6107	1	1	yes	characteristic
176	C₂×C₄	#88198	1	1	yes	characteristic
177	C₂×C₄	#88201	1	1	yes	characteristic
178	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
179	C₈	#88075	1	2	yes	—
180	C₈	#88075	1	2	yes	—
181	C₈	Q₈×Dic₁₃	1	2	yes	—
182	C₈	#88200	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	#11718739	1	1	yes	characteristic
185	C₂²	#11718749	1	1	yes	characteristic
186	C₂²	C₂×#88072	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	208	2	no	—
191	C₄	#11718740	1	1	yes	characteristic
192	C₄	#11718754	1	1	yes	characteristic
193	C₄	C₂×#88075	1	1	yes	characteristic
194	C₄	#11718891	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	#13115798	1	1	yes	characteristic
198	C₂	#13116572	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#13288
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
6. C₁₃×#500	C₄	semidirect
8. C₁₃×#499	C₄	semidirect
40. #88052	Q₈	semidirect
90. #500	Dic₁₃	semidirect
92. #499	Dic₁₃	semidirect
164. C₁₃	#13288	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88052	2
#13115000	2
#13112941	2
C₁₃×#3160	2
#13288	13