Subgroups — #16789638

Order 3328.

296 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×#11718261	C₂	1	1	yes	maximal, characteristic
2	#13131057	C₂	1	2	yes	maximal
3	#13131042	C₂	1	1	yes	maximal, characteristic
4	#13129886	C₂	1	2	yes	maximal
5	#13114717	C₂	1	1	yes	maximal, characteristic
6	C₂×#88106	C₂²	1	2	yes	—
7	C₂×#88106	C₂²	1	1	yes	characteristic
8	C₂×#88106	C₂²	1	1	yes	characteristic
9	C₁₃×#427	C₄	1	1	yes	characteristic
10	C₂×#88059	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	#11718658	C₂²	1	2	yes	—
13	#11718261	C₄	1	2	yes	complemented
14	#11718261	C₄	1	2	yes	complemented
15	#11718258	C₄	1	2	yes	complemented
16	#11718257	C₄	1	2	yes	complemented
17	#11718232	—	2	4	no	—
18	#11718124	C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₁₃×#117	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#6099	C₂×C₄	1	2	yes	—
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	#88106	—	2	2	no	—
25	#88106	—	2	1	no	—
26	#88106	—	2	1	no	—
27	#88106	C₂×C₄	1	4	yes	—
28	#88106	D₄	1	2	yes	—
29	#88106	Q₈	1	2	yes	—
30	C₂×#6092	—	2	2	no	—
31	C₂×#6092	C₂×C₄	1	2	yes	—
32	C₂×#6092	C₂×C₄	1	2	yes	—
33	C₁₃×#107	—	2	1	no	—
34	C₁₃×#106	—	2	1	no	—
35	#88059	—	2	1	no	—
36	#88059	—	2	1	no	—
37	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	#88043	—	2	2	no	—
42	C₄×#1055	—	2	2	no	—
43	#12667	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₄²×C₁₃	—	2	2	no	—
55	C₄×Dic₁₃	—	2	1	no	—
56	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
57	C₄×Dic₁₃	C₄²	1	1	yes	characteristic
58	#6099	—	2	2	no	—
59	#6099	—	2	2	no	—
60	C₂×#1055	—	2	2	no	—
61	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
62	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
63	C₂×#1055	C₂×D₄	1	1	yes	characteristic
64	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
65	C₂×#1055	#41	1	1	yes	characteristic
66	C₂×#1055	#41	1	1	yes	characteristic
67	#6092	—	2	2	no	—
68	#6092	—	2	4	no	—
69	#6092	C₄⋊C₄	1	4	yes	—
70	C₂×#364	—	13	1	no	—
71	#2110	—	13	2	no	—
72	C₄×M₃₂	—	13	1	no	—
73	#1772	—	13	1	no	—
74	#1378	—	13	2	no	—
75	C₂³×C₁₃	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
83	C₂×Dic₁₃	—	2	1	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	4	2	no	—
86	C₈×C₁₃	#97	1	1	yes	characteristic
87	C₈×C₁₃	#97	1	1	yes	characteristic
88	C₈×C₁₃	#97	1	1	yes	characteristic
89	C₈×C₁₃	#97	1	1	yes	characteristic
90	#1055	—	4	2	no	—
91	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
92	#1055	#117	1	1	yes	characteristic
93	#1055	C₄×D₄	1	1	yes	characteristic
94	#1055	C₄×Q₈	1	1	yes	characteristic
95	C₂×M₃₂	—	13	1	no	—
96	C₂×M₃₂	—	13	1	no	—
97	C₂×M₃₂	—	13	2	no	—
98	C₂×#108	—	13	1	no	—
99	#427	#261	1	1	yes	characteristic
100	#405	—	13	2	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	#364	—	13	2	no	—
103	#364	—	13	2	no	—
104	#347	—	26	4	no	—
105	#346	—	13	2	no	—
106	C₄×C₁₆	—	13	2	no	—
107	#339	—	13	2	no	—
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	#405	1	1	yes	characteristic
110	C₂²×C₁₃	#436	1	1	yes	characteristic
111	C₂²×C₁₃	#446	1	1	yes	characteristic
112	C₄×C₁₃	—	8	2	no	—
113	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
114	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
115	C₄×C₁₃	#436	1	1	yes	characteristic
116	C₄×C₁₃	#446	1	1	yes	characteristic
117	Dic₁₃	—	2	2	no	—
118	Dic₁₃	—	4	1	no	—
119	C₂×M₁₆	—	13	2	no	—
120	C₂²×C₈	—	13	1	no	—
121	C₂²×C₈	C₂×#261	1	1	yes	characteristic
122	#117	C₂×#261	1	2	yes	—
123	C₂×C₄²	—	13	1	no	—
124	M₃₂	—	13	2	no	—
125	M₃₂	—	13	2	no	—
126	M₃₂	—	13	4	no	—
127	M₃₂	—	26	2	no	—
128	M₃₂	—	26	1	no	—
129	M₃₂	—	26	1	no	—
130	C₂×C₁₆	—	13	2	no	—
131	C₂×C₁₆	—	13	2	no	—
132	C₂×C₁₆	—	26	2	no	—
133	#108	—	26	1	no	—
134	#108	—	26	1	no	—
135	#107	—	2	1	no	—
136	#106	—	2	1	no	—
137	#97	—	26	2	no	—
138	C₄×C₈	—	13	1	no	—
139	C₄×C₈	—	13	1	no	—
140	C₄×C₈	—	13	1	no	—
141	C₄×C₈	—	13	1	no	—
142	C₄×C₈	—	26	2	no	—
143	C₂×C₁₃	—	2	1	no	—
144	C₂×C₁₃	—	2	1	no	—
145	C₂×C₁₃	—	2	1	no	—
146	C₂×C₁₃	#1770	1	1	yes	characteristic
147	C₂²×C₄	—	26	1	no	—
148	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
149	M₁₆	—	26	2	no	—
150	M₁₆	—	26	2	no	—
151	C₂×C₈	—	13	1	no	—
152	C₂×C₈	—	13	1	no	—
153	C₂×C₈	—	13	1	no	—
154	C₂×C₈	—	13	1	no	—
155	C₂×C₈	—	13	1	no	—
156	C₂×C₈	—	13	1	no	—
157	C₂×C₈	—	26	2	no	—
158	C₂×C₈	#6118	1	1	yes	characteristic
159	C₂×C₈	#6118	1	1	yes	characteristic
160	C₂×C₈	C₄×#261	1	1	yes	characteristic
161	C₂×C₈	C₄×#261	1	1	yes	characteristic
162	C₂×C₈	#6122	1	1	yes	characteristic
163	C₂×C₈	#6122	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	13	1	no	—
168	C₄²	—	13	1	no	—
169	C₄²	—	26	1	no	—
170	C₁₆	—	13	4	no	—
171	C₁₆	—	26	2	no	—
172	C₁₆	—	26	4	no	—
173	C₁₃	#12667	1	1	yes	characteristic, complemented, Sylow
174	C₂³	#88238	1	1	yes	characteristic
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	26	1	no	—
177	C₂×C₄	—	26	1	no	—
178	C₂×C₄	—	52	2	no	—
179	C₂×C₄	C₂×#6118	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
181	C₂×C₄	D₄×#261	1	1	yes	characteristic
182	C₂×C₄	#88240	1	1	yes	characteristic
183	C₂×C₄	Q₈×#261	1	1	yes	characteristic
184	C₂×C₄	C₂×#6122	1	1	yes	characteristic
185	C₈	—	13	1	no	—
186	C₈	—	13	1	no	—
187	C₈	—	13	1	no	—
188	C₈	—	13	1	no	—
189	C₈	—	52	2	no	—
190	C₈	#88110	1	1	yes	characteristic
191	C₈	#88110	1	1	yes	characteristic
192	C₈	#88110	1	1	yes	characteristic
193	C₈	#88110	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#11719072	1	1	yes	characteristic
196	C₂²	#11719080	1	1	yes	characteristic
197	C₂²	#11719145	1	1	yes	characteristic
198	C₄	—	8	2	no	—
199	C₄	—	26	2	no	—
200	C₄	—	52	1	no	—
201	C₄	#11719076	1	1	yes	characteristic
202	C₄	C₄⋊C₄×#261	1	1	yes	characteristic
203	C₄	C₂×#88110	1	1	yes	characteristic
204	C₄	#11719158	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#13130632	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊C₄×#261
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#427	C₄
socle	C₂×C₁₃	#1770
prime radical	C₂³×C₁₃	#131
cosocle	C₂²×#1055	C₂³
hypercenter	#427	#261
hypocenter	C₁₃	#12667
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
13. #11718261	C₄	semidirect
14. #11718261	C₄	semidirect
15. #11718258	C₄	semidirect
16. #11718257	C₄	semidirect
173. C₁₃	#12667	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718261	2
#13131057	2
#13131042	2
#13129886	2
#13114717	2
#12667	13