Subgroups — #16782359

Order 3328.

382 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13130753	C₂	1	2	yes	maximal
2	#13129666	C₂	1	2	yes	maximal
3	#13129666	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#1057	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₄	1	1	yes	characteristic
13	#11718290	—	2	2	no	—
14	#11718290	—	2	2	no	—
15	C₂²×#1057	—	2	4	no	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₄⋊#1057	—	2	4	no	—
28	C₄⋊#1057	—	2	4	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	—	2	2	no	—
32	C₈×Dic₁₃	—	2	2	no	—
33	C₈×Dic₁₃	C₈	1	4	yes	complemented
34	C₄×#1055	—	2	2	no	—
35	#8785	—	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	4	no	—
44	C₂×#1057	—	4	4	no	—
45	C₂×#1057	—	4	4	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
52	C₄²×C₁₃	—	2	1	no	—
53	C₄²×C₁₃	—	2	1	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	C₄×Dic₁₃	—	2	2	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₄×Dic₁₃	—	2	2	no	—
58	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₃	M₁₆	1	2	yes	—
60	C₄×Dic₁₃	#41	1	2	yes	—
61	C₄×Dic₁₃	#41	1	2	yes	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	C₂×C₈	1	2	yes	—
67	C₂×#1055	M₁₆	1	2	yes	—
68	#1831	—	13	2	no	—
69	C₄²×C₈	—	13	1	no	—
70	#1277	—	13	2	no	—
71	#1277	—	13	2	no	—
72	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	#108	1	2	yes	—
78	C₂×C₄×C₁₃	#108	1	2	yes	—
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
92	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
93	C₂×Dic₁₃	#124	1	2	yes	—
94	C₂×Dic₁₃	#125	1	2	yes	—
95	C₂×Dic₁₃	#135	1	2	yes	—
96	C₂×Dic₁₃	#135	1	2	yes	—
97	#1057	—	8	4	no	—
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	2	no	—
100	#1055	—	2	2	no	—
101	#1055	—	2	2	no	—
102	C₂×C₄⋊C₈	—	13	4	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	—	13	1	no	—
108	C₂×C₄×C₈	#261	1	1	yes	characteristic
109	C₄³	—	13	1	no	—
110	#336	—	26	2	no	—
111	#336	—	26	2	no	—
112	C₂²×C₁₃	#382	1	1	yes	characteristic
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₁₃	#423	1	1	yes	characteristic
115	C₂²×C₁₃	#427	1	1	yes	characteristic
116	C₂²×C₁₃	#427	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	#431	1	1	yes	characteristic
122	C₄×C₁₃	#431	1	1	yes	characteristic
123	C₄×C₁₃	#432	1	1	yes	characteristic
124	C₄×C₁₃	#432	1	1	yes	characteristic
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	#488	1	4	yes	—
129	C₂²×C₈	—	13	1	no	—
130	C₂²×C₈	—	13	1	no	—
131	C₂²×C₈	—	26	4	no	—
132	C₂²×C₈	C₂×#261	1	1	yes	characteristic
133	C₂²×C₈	C₂×#261	1	1	yes	characteristic
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	—	13	1	no	—
140	C₂×C₄²	C₂×#261	1	1	yes	characteristic
141	C₄⋊C₈	—	26	4	no	—
142	C₄⋊C₈	—	26	4	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	13	4	no	—
145	C₄×C₈	—	26	2	no	—
146	C₄×C₈	—	26	2	no	—
147	C₄×C₈	—	26	2	no	—
148	C₄×C₈	—	26	2	no	—
149	C₂×C₁₃	#1564	1	1	yes	characteristic
150	C₂×C₁₃	#1564	1	1	yes	characteristic
151	C₂×C₁₃	#1568	1	1	yes	characteristic
152	C₂×C₁₃	#1568	1	1	yes	characteristic
153	C₂×C₁₃	#1831	1	1	yes	characteristic
154	C₂×C₁₃	#1835	1	1	yes	characteristic
155	C₂×C₁₃	#1837	1	1	yes	characteristic
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	13	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	26	1	no	—
170	C₂×C₈	—	26	1	no	—
171	C₂×C₈	—	26	1	no	—
172	C₂×C₈	—	26	1	no	—
173	C₂×C₈	—	52	4	no	—
174	C₂×C₈	—	52	4	no	—
175	C₂×C₈	—	52	4	no	—
176	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
177	C₂×C₈	#6122	1	2	yes	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	13	2	no	—
181	C₄²	—	13	2	no	—
182	C₄²	—	13	2	no	—
183	C₄²	—	13	2	no	—
184	C₄²	—	26	2	no	—
185	C₄²	—	26	2	no	—
186	C₄²	—	26	2	no	—
187	C₄²	—	26	2	no	—
188	C₁₃	#8785	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	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	#88103	1	2	yes	—
207	C₂×C₄	#88104	1	2	yes	—
208	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
209	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
210	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
211	C₂×C₄	#88237	1	1	yes	characteristic
212	C₂×C₄	#88237	1	1	yes	characteristic
213	C₂×C₄	C₂×#6122	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	26	2	no	—
217	C₈	—	26	2	no	—
218	C₈	—	104	4	no	—
219	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
220	C₂²	#11719060	1	1	yes	characteristic
221	C₂²	#11719092	1	1	yes	characteristic
222	C₂²	#11719092	1	1	yes	characteristic
223	C₂²	C₂×#88103	1	1	yes	characteristic
224	C₂²	C₂×#88104	1	1	yes	characteristic
225	C₂²	#11719129	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	13	4	no	—
229	C₄	—	26	2	no	—
230	C₄	—	26	2	no	—
231	C₄	—	26	2	no	—
232	C₄	#11719049	1	1	yes	characteristic
233	C₄	#11719055	1	1	yes	characteristic
234	C₄	#11719125	1	1	yes	characteristic
235	C₄	#11719128	1	1	yes	characteristic
236	C₂	#13130384	1	1	yes	characteristic
237	C₂	#13130386	1	1	yes	characteristic
238	C₂	#13130396	1	1	yes	characteristic
239	C₂	#13130398	1	1	yes	characteristic
240	C₂	#13130752	1	1	yes	characteristic
241	C₂	#13130758	1	1	yes	characteristic
242	C₂	#13130760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#8785
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
33. C₈×Dic₁₃	C₈	semidirect
188. C₁₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#13130753	2
#13129666	2
C₄×C₈×Dic₁₃	2
#8785	13