Subgroups — #15754110

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114691	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
4	#13112670	C₂	1	2	yes	maximal
5	C₂×C₁₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	—	2	2	no	—
9	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
10	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
11	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₁₃×#334	C₄	1	4	yes	complemented
15	#11718142	—	2	2	no	—
16	#11718049	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
25	C₂²×#1055	—	2	2	no	—
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₁₃×C₄⋊C₈	—	2	2	no	—
28	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	#88055	—	4	4	no	—
32	C₈×Dic₁₃	—	2	4	no	—
33	C₈×Dic₁₃	—	2	4	no	—
34	C₄⋊#1055	—	2	2	no	—
35	C₄⋊#1055	C₂×C₄	1	4	yes	—
36	C₄×#1055	—	2	1	no	—
37	C₄×#1055	—	2	1	no	—
38	#8273	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
43	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	4	2	no	—
48	C₂×C₈×C₁₃	C₄²	1	2	yes	—
49	C₂×C₈×C₁₃	C₄²	1	2	yes	—
50	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	1	yes	characteristic
54	C₄²×C₁₃	#41	1	1	yes	characteristic
55	C₄×Dic₁₃	—	2	2	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	2	no	—
58	C₄×Dic₁₃	—	2	4	no	—
59	C₄×Dic₁₃	SD₁₆	1	4	yes	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	—	2	2	no	—
62	C₂×#1055	—	2	4	no	—
63	C₂×#1055	—	4	2	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	M₁₆	1	4	yes	—
66	#1767	—	13	2	no	—
67	C₄²×C₈	—	13	1	no	—
68	C₂×#334	—	13	1	no	—
69	C₂×#334	D₁₃	1	1	yes	characteristic
70	#1277	—	13	2	no	—
71	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	#135	1	2	yes	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	#106	1	4	yes	—
88	C₂×Dic₁₃	#108	1	4	yes	—
89	C₈×C₁₃	—	2	2	no	—
90	C₈×C₁₃	—	4	4	no	—
91	C₈×C₁₃	#97	1	4	yes	complemented
92	#1055	—	2	4	no	—
93	#1055	—	4	4	no	—
94	C₂×C₄⋊C₈	—	13	2	no	—
95	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
96	C₂×#97	—	26	2	no	—
97	C₂×C₄×C₈	—	13	2	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
101	C₄³	—	13	1	no	—
102	#336	—	26	2	no	—
103	#334	—	13	4	no	—
104	#334	Dic₁₃	1	4	yes	complemented
105	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₁₃	#423	1	2	yes	—
107	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
108	C₂²×C₁₃	#427	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×#108	1	2	yes	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
113	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₁₃	#432	1	2	yes	—
115	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
116	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
117	Dic₁₃	—	2	4	no	—
118	Dic₁₃	—	2	4	no	—
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	13	2	no	—
121	C₂²×C₈	—	26	2	no	—
122	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
123	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
124	C₂×C₄²	—	13	1	no	—
125	C₂×C₄²	—	13	2	no	—
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	—	13	2	no	—
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	13	4	no	—
131	C₄⋊C₈	—	26	2	no	—
132	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
133	#97	—	52	4	no	—
134	C₄×C₈	—	26	4	no	—
135	C₄×C₈	—	26	4	no	—
136	C₄×C₈	—	26	1	no	—
137	C₄×C₈	—	26	1	no	—
138	C₄×C₈	C₄×D₁₃	1	2	yes	—
139	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
140	C₂×C₁₃	#1566	1	2	yes	—
141	C₂×C₁₃	#1568	1	2	yes	—
142	C₂×C₁₃	#1767	1	1	yes	characteristic
143	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
144	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
145	C₂²×C₄	—	13	2	no	—
146	C₂²×C₄	—	13	2	no	—
147	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
148	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
149	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	13	4	no	—
155	C₂×C₈	—	26	2	no	—
156	C₂×C₈	—	26	2	no	—
157	C₂×C₈	—	26	4	no	—
158	C₂×C₈	—	52	2	no	—
159	C₂×C₈	—	52	2	no	—
160	C₂×C₈	#6094	1	4	yes	—
161	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
162	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
163	C₄²	—	13	4	no	—
164	C₄²	—	26	4	no	—
165	C₄²	—	26	2	no	—
166	C₄²	—	26	2	no	—
167	C₄²	—	26	2	no	—
168	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
169	C₄²	#6129	1	1	yes	characteristic
170	C₄²	#6131	1	1	yes	characteristic
171	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
172	C₁₃	#8273	1	1	yes	characteristic, complemented, Sylow
173	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	13	4	no	—
176	C₂×C₄	—	13	4	no	—
177	C₂×C₄	—	26	2	no	—
178	C₂×C₄	—	26	2	no	—
179	C₂×C₄	—	26	2	no	—
180	C₂×C₄	—	26	2	no	—
181	C₂×C₄	#88141	1	1	yes	characteristic
182	C₂×C₄	#88146	1	1	yes	characteristic
183	C₂×C₄	#88147	1	1	yes	characteristic
184	C₂×C₄	C₂×#6094	1	2	yes	—
185	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
186	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
187	C₂×C₄	#88170	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
190	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	—	26	4	no	—
194	C₈	—	52	4	no	—
195	C₈	#88055	1	4	yes	complemented
196	C₂²	#11718498	1	2	yes	—
197	C₂²	D₁₃×#106	1	1	yes	characteristic
198	C₂²	#11718521	1	1	yes	characteristic
199	C₂²	D₁₃×#108	1	2	yes	—
200	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	26	4	no	—
204	C₄	—	26	4	no	—
205	C₄	#11718496	1	2	yes	—
206	C₄	#11718513	1	2	yes	—
207	C₄	C₂×#88055	1	1	yes	characteristic
208	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
209	C₄	SD₁₆×Dic₁₃	1	2	yes	—
210	C₂	#13113590	1	2	yes	—
211	C₂	#13113596	1	2	yes	—
212	C₂	#13114691	1	1	yes	characteristic
213	C₂	#106×Dic₁₃	1	1	yes	characteristic
214	C₂	#108×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#334	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#334	D₁₃
hypocenter	C₁₃	#8273
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
14. C₁₃×#334	C₄	semidirect
16. #11718049	C₄	semidirect
91. C₈×C₁₃	#97	semidirect
104. #334	Dic₁₃	semidirect
172. C₁₃	#8273	semidirect
195. C₈	#88055	semidirect

Maximal subgroups

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

subgroup	index
#13114691	2
C₄×C₈×Dic₁₃	2
C₂×#11718049	2
#13112670	2
C₂×C₁₃×#334	2
#8273	13