Subgroups — #16782364

Order 3328.

362 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130769	C₂	1	1	yes	maximal, characteristic
2	#13130768	C₂	1	1	yes	maximal, characteristic
3	#13129666	C₂	1	4	yes	maximal
4	#13114580	C₂	1	1	yes	maximal, characteristic
5	C₂×#88112	—	2	1	no	—
6	C₂×#88112	—	2	1	no	—
7	C₂×C₄⋊#1057	—	2	1	no	—
8	C₂×C₄⋊#1057	—	2	1	no	—
9	C₂×C₄⋊#1057	C₂²	1	2	yes	—
10	C₂×C₄⋊#1057	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
12	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#336	C₄	1	1	yes	characteristic
14	#11718142	C₄	1	2	yes	—
15	#11718116	C₄	1	1	yes	characteristic
16	C₂²×#1057	—	2	2	no	—
17	C₂²×#1057	—	2	2	no	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	#88112	—	4	2	no	—
28	#88112	—	4	2	no	—
29	C₄⋊#1057	—	2	4	no	—
30	C₄⋊#1057	—	2	4	no	—
31	C₄⋊#1057	—	4	2	no	—
32	C₄⋊#1057	—	4	2	no	—
33	C₈×Dic₁₃	—	2	2	no	—
34	C₈×Dic₁₃	—	2	2	no	—
35	#8787	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×#1057	—	4	2	no	—
44	C₂×#1057	—	4	2	no	—
45	C₂×#1057	—	4	2	no	—
46	C₂×#1057	—	4	2	no	—
47	C₂×#1057	—	4	2	no	—
48	C₂×#1057	—	4	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₄²×C₁₃	—	2	2	no	—
53	C₄×Dic₁₃	—	2	1	no	—
54	C₄×Dic₁₃	—	2	1	no	—
55	C₄×Dic₁₃	—	4	1	no	—
56	C₄×Dic₁₃	—	4	1	no	—
57	C₄×Dic₁₃	D₈	1	2	yes	—
58	C₄×Dic₁₃	SD₁₆	1	2	yes	—
59	C₄×Dic₁₃	SD₁₆	1	2	yes	—
60	C₄×Dic₁₃	Q₁₆	1	2	yes	—
61	C₄×Dic₁₃	#41	1	2	yes	—
62	C₄×Dic₁₃	#41	1	2	yes	—
63	C₂×#1055	—	2	2	no	—
64	C₂×#1055	—	2	2	no	—
65	C₂×#1055	—	2	2	no	—
66	#1842	—	13	1	no	—
67	#1842	—	13	1	no	—
68	#1744	—	13	1	no	—
69	#1277	—	13	4	no	—
70	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	#106	1	2	yes	—
76	C₂×C₄×C₁₃	#107	1	2	yes	—
77	C₂×C₄×C₁₃	#117	1	2	yes	—
78	C₂×C₄×C₁₃	#131	1	2	yes	—
79	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	1	no	—
82	C₂×Dic₁₃	—	2	1	no	—
83	C₂×Dic₁₃	—	2	1	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	#106	1	2	yes	—
88	C₂×Dic₁₃	#107	1	2	yes	—
89	C₂×Dic₁₃	#124	1	1	yes	characteristic
90	C₂×Dic₁₃	#124	1	1	yes	characteristic
91	C₂×Dic₁₃	#125	1	2	yes	—
92	C₂×Dic₁₃	C₂×D₈	1	1	yes	characteristic
93	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×Dic₁₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×Dic₁₃	C₂×Q₁₆	1	1	yes	characteristic
96	#1057	—	8	2	no	—
97	#1057	—	8	2	no	—
98	C₈×C₁₃	—	4	2	no	—
99	#1055	—	4	2	no	—
100	C₂×C₄⋊C₈	—	13	2	no	—
101	C₂×C₄⋊C₈	—	13	2	no	—
102	C₂×C₄⋊C₈	—	26	1	no	—
103	C₂×C₄⋊C₈	—	26	1	no	—
104	C₂×C₄⋊C₈	—	26	1	no	—
105	C₂×C₄⋊C₈	—	26	1	no	—
106	C₂×C₄×C₈	—	13	2	no	—
107	C₄³	—	13	1	no	—
108	#336	—	13	2	no	—
109	#336	—	13	1	no	—
110	#336	#261	1	1	yes	characteristic
111	C₂²×C₁₃	#382	1	1	yes	characteristic
112	C₂²×C₁₃	#424	1	1	yes	characteristic
113	C₂²×C₁₃	#424	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
116	C₂²×C₁₃	#430	1	1	yes	characteristic
117	C₂²×C₁₃	#430	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	#433	1	2	yes	—
121	C₄×C₁₃	#433	1	2	yes	—
122	Dic₁₃	—	2	2	no	—
123	Dic₁₃	—	2	2	no	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	#486	1	1	yes	characteristic
126	Dic₁₃	#486	1	1	yes	characteristic
127	Dic₁₃	#487	1	1	yes	characteristic
128	Dic₁₃	#487	1	1	yes	characteristic
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	26	2	no	—
131	C₂²×C₈	—	26	2	no	—
132	C₂²×C₈	C₂×#261	1	2	yes	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	—	13	1	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	—	13	1	no	—
138	C₂×C₄²	—	13	1	no	—
139	C₂×C₄²	C₂×#261	1	1	yes	characteristic
140	C₄⋊C₈	—	26	4	no	—
141	C₄⋊C₈	—	26	4	no	—
142	C₄⋊C₈	—	52	2	no	—
143	C₄⋊C₈	—	52	2	no	—
144	C₄⋊C₈	—	52	2	no	—
145	C₄⋊C₈	—	52	2	no	—
146	C₄×C₈	—	26	2	no	—
147	C₄×C₈	—	26	2	no	—
148	C₂×C₁₃	#1570	1	1	yes	characteristic
149	C₂×C₁₃	#1570	1	1	yes	characteristic
150	C₂×C₁₃	#1571	1	1	yes	characteristic
151	C₂×C₁₃	#1571	1	1	yes	characteristic
152	C₂×C₁₃	#1832	1	1	yes	characteristic
153	C₂×C₁₃	#1834	1	1	yes	characteristic
154	C₂×C₁₃	#1838	1	1	yes	characteristic
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
161	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	26	2	no	—
166	C₂×C₈	—	26	2	no	—
167	C₂×C₈	—	26	2	no	—
168	C₂×C₈	—	52	2	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	—	52	2	no	—
171	C₂×C₈	—	52	2	no	—
172	C₂×C₈	—	52	2	no	—
173	C₂×C₈	—	52	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	13	2	no	—
176	C₄²	—	13	2	no	—
177	C₄²	—	13	2	no	—
178	C₄²	—	13	2	no	—
179	C₄²	—	13	2	no	—
180	C₄²	—	13	2	no	—
181	C₄²	—	26	1	no	—
182	C₄²	—	26	1	no	—
183	C₄²	—	52	1	no	—
184	C₄²	—	52	1	no	—
185	C₁₃	#8787	1	1	yes	characteristic, complemented, Sylow
186	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	13	1	no	—
192	C₂×C₄	—	13	1	no	—
193	C₂×C₄	—	13	1	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	1	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	13	2	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	26	1	no	—
202	C₂×C₄	—	26	1	no	—
203	C₂×C₄	—	26	1	no	—
204	C₂×C₄	—	26	1	no	—
205	C₂×C₄	—	26	1	no	—
206	C₂×C₄	#88101	1	2	yes	—
207	C₂×C₄	#88102	1	2	yes	—
208	C₂×C₄	#88237	1	2	yes	—
209	C₂×C₄	#88238	1	1	yes	characteristic
210	C₂×C₄	#88238	1	2	yes	—
211	C₂×C₄	#88240	1	1	yes	characteristic
212	C₈	—	4	2	no	—
213	C₈	—	52	2	no	—
214	C₈	—	104	2	no	—
215	C₈	—	104	2	no	—
216	C₂²	#11719060	1	1	yes	characteristic
217	C₂²	#11719067	1	1	yes	characteristic
218	C₂²	#11719079	1	1	yes	characteristic
219	C₂²	C₂×#88101	1	1	yes	characteristic
220	C₂²	C₂×#88102	1	1	yes	characteristic
221	C₂²	#11719100	1	1	yes	characteristic
222	C₂²	#11719100	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	13	1	no	—
226	C₄	—	13	1	no	—
227	C₄	—	13	1	no	—
228	C₄	—	13	1	no	—
229	C₄	—	26	2	no	—
230	C₄	—	26	2	no	—
231	C₄	—	26	2	no	—
232	C₄	#11719068	1	2	yes	—
233	C₄	#11719077	1	2	yes	—
234	C₂	#13130403	1	1	yes	characteristic
235	C₂	#13130404	1	1	yes	characteristic
236	C₂	#13130405	1	1	yes	characteristic
237	C₂	#13130406	1	1	yes	characteristic
238	C₂	#13130754	1	1	yes	characteristic
239	C₂	#13130757	1	1	yes	characteristic
240	C₂	#13130762	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#336	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	#336	#261
hypocenter	C₁₃	#8787
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
185. C₁₃	#8787	semidirect

Maximal subgroups

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

subgroup	index
#13130769	2
#13130768	2
#13129666	2
#13114580	2
#8787	13