Subgroups — #15756565

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	2	yes	maximal
2	#13112681	C₂	1	2	yes	maximal
3	#13112665	C₂	1	2	yes	maximal
4	C₁₃×#1726	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#376	C₂²	1	1	yes	characteristic
12	#11718586	C₂²	1	4	yes	—
13	#11718050	—	2	2	no	—
14	#11718049	—	2	2	no	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₁₃×#97	—	2	2	no	—
23	C₁₃×#97	—	2	2	no	—
24	C₁₃×#97	—	2	2	no	—
25	C₄×C₈×C₁₃	—	2	2	no	—
26	C₄×C₈×C₁₃	C₈	1	4	yes	complemented
27	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₁₃×#95	—	2	2	no	—
32	C₂²⋊#1055	—	2	4	no	—
33	C₄⋊#1055	—	2	4	no	—
34	C₄×#1055	—	2	4	no	—
35	#8780	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	—	2	1	no	—
37	C₂²×C₄×C₁₃	—	2	1	no	—
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	#41	1	2	yes	—
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
53	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
54	C₄²×C₁₃	—	2	1	no	—
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	C₂×C₈	1	2	yes	—
57	C₄²×C₁₃	M₁₆	1	2	yes	—
58	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₃	#41	1	2	yes	—
61	C₂×#1055	—	4	4	no	—
62	#1726	D₁₃	1	1	yes	characteristic
63	#1564	—	13	2	no	—
64	#1281	—	13	2	no	—
65	#1274	—	13	2	no	—
66	C₂³×C₁₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	4	2	no	—
70	C₂×C₄×C₁₃	—	4	2	no	—
71	C₂×C₄×C₁₃	—	4	2	no	—
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#124	1	2	yes	—
75	C₂×C₄×C₁₃	#125	1	2	yes	—
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
80	C₈×C₁₃	—	2	1	no	—
81	C₈×C₁₃	—	2	1	no	—
82	C₈×C₁₃	—	2	2	no	—
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
85	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
86	#1055	—	8	4	no	—
87	#431	—	13	4	no	—
88	C₂×#97	Dic₁₃	1	1	yes	characteristic
89	C₂×#97	Dic₁₃	1	1	yes	characteristic
90	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
91	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
92	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	#376	C₂×D₁₃	1	1	yes	characteristic
95	#335	—	26	2	no	—
96	#334	—	26	2	no	—
97	C₂²×C₁₃	—	2	2	no	—
98	C₂²×C₁₃	#382	1	1	yes	characteristic
99	C₂²×C₁₃	#431	1	1	yes	characteristic
100	C₂²×C₁₃	#432	1	1	yes	characteristic
101	C₄×C₁₃	—	2	2	no	—
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	4	2	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₁₃	#423	1	1	yes	characteristic
107	C₄×C₁₃	#431	1	1	yes	characteristic
108	C₄×C₁₃	#432	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
111	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
113	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
116	C₄⋊C₈	—	26	4	no	—
117	C₂²⋊C₈	—	26	4	no	—
118	#97	—	2	2	no	—
119	#97	—	2	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	26	4	no	—
123	C₄×C₈	#1055	1	4	yes	complemented
124	C₄×C₈	Dic₂₆	1	1	yes	characteristic
125	C₄×C₈	Dic₂₆	1	1	yes	characteristic
126	C₄×C₈	D₅₂	1	1	yes	characteristic
127	C₄×C₈	D₅₂	1	1	yes	characteristic
128	#95	—	2	2	no	—
129	C₂×C₁₃	—	2	1	no	—
130	C₂×C₁₃	—	2	1	no	—
131	C₂×C₁₃	#1831	1	1	yes	characteristic
132	C₂×C₁₃	#1833	1	2	yes	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	#6127	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	52	4	no	—
144	C₂×C₈	C₂×#1055	1	2	yes	—
145	C₂×C₈	C₂×#1055	1	2	yes	—
146	C₂×C₈	#6099	1	2	yes	—
147	C₂×C₈	#6099	1	2	yes	—
148	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
150	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
151	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
152	C₄²	—	2	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	C₂×#1055	1	2	yes	—
155	C₄²	#6099	1	2	yes	—
156	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
157	C₄²	C₂×D₅₂	1	1	yes	characteristic
158	C₄²	#6127	1	2	yes	—
159	C₁₃	#8780	1	1	yes	characteristic, complemented, Sylow
160	C₂³	#88180	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	#88124	1	2	yes	—
167	C₂×C₄	#88129	1	2	yes	—
168	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
169	C₂×C₄	C₂×#6099	1	1	yes	characteristic
170	C₂×C₄	C₂×#6099	1	1	yes	characteristic
171	C₂×C₄	C₂×#6099	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
173	C₂×C₄	#88180	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	—	104	4	no	—
179	C₈	C₄⋊#1055	1	2	yes	—
180	C₈	C₄⋊#1055	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#11718586	1	1	yes	characteristic
183	C₂²	#11718587	1	1	yes	characteristic
184	C₂²	#11718597	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
190	C₄	#11718585	1	1	yes	characteristic
191	C₄	#11718586	1	1	yes	characteristic
192	C₄	#11718587	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#13114953	1	1	yes	characteristic
196	C₂	#13114960	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88180
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1726	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1726	D₁₃
hypocenter	C₁₃	#8780
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
26. C₄×C₈×C₁₃	C₈	semidirect
123. C₄×C₈	#1055	semidirect
159. C₁₃	#8780	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
#13112681	2
#13112665	2
C₁₃×#1726	2
#8780	13