Subgroups — #16780321

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#88111	C₂	1	2	yes	maximal
2	C₂×#11718251	C₂	1	2	yes	maximal, complemented
3	#13129658	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88111	C₄	1	4	yes	—
9	C₂×C₄×#1057	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1055	C₄	1	1	yes	characteristic
14	C₄×C₈×D₁₃	—	2	4	no	—
15	#11718290	C₄	1	4	yes	—
16	#11718251	—	2	4	no	—
17	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
19	C₂²×#1057	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
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	C₂×C₈×D₁₃	—	2	2	no	—
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	—	2	2	no	—
31	C₂×C₈×D₁₃	C₈	1	4	yes	complemented
32	C₂×C₈×D₁₃	C₈	1	4	yes	complemented
33	C₄²×D₁₃	—	2	2	no	—
34	C₄²×D₁₃	—	2	2	no	—
35	#88111	—	2	8	no	—
36	C₄×#1057	—	2	4	no	—
37	C₄×#1057	—	2	4	no	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₈×Dic₁₃	D₄	1	4	yes	—
41	C₄×#1055	—	2	2	no	—
42	#8144	—	13	1	no	maximal, Sylow
43	C₂³×D₁₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×D₁₃	—	2	1	no	—
49	C₂×C₄×D₁₃	—	2	1	no	—
50	C₂×C₄×D₁₃	—	2	1	no	—
51	C₂×C₄×D₁₃	—	2	1	no	—
52	C₂×C₄×D₁₃	—	2	4	no	—
53	C₂×C₄×D₁₃	—	2	4	no	—
54	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
56	C₂×C₄×D₁₃	M₁₆	1	8	yes	—
57	C₂×#1057	—	2	8	no	—
58	C₂×#1057	—	2	8	no	—
59	C₂×#1057	—	2	8	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
66	C₄²×C₁₃	—	2	1	no	—
67	C₄²×C₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	2	4	no	—
69	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
70	C₄×Dic₁₃	#41	1	2	yes	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	—	2	1	no	—
73	C₂×#1055	C₂²⋊C₄	1	2	yes	—
74	C₂×#1055	C₂×C₈	1	2	yes	—
75	C₂×#1055	C₂×C₈	1	2	yes	—
76	C₂×#1055	M₁₆	1	2	yes	—
77	C₈×D₁₃	—	2	8	no	—
78	C₈×D₁₃	—	2	8	no	—
79	C₂²×C₄×C₈	—	13	1	no	—
80	C₄×C₂²⋊C₈	—	13	2	no	—
81	C₂×#322	—	13	2	no	—
82	#1273	—	13	2	no	—
83	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	1	no	—
85	C₂²×D₁₃	—	2	1	no	—
86	C₂²×D₁₃	—	2	2	no	—
87	C₂²×D₁₃	C₄×C₈	1	2	yes	—
88	C₂²×D₁₃	#97	1	4	yes	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	#97	1	4	yes	—
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×Dic₁₃	—	2	1	no	—
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
101	C₂×Dic₁₃	C₄×D₄	1	4	yes	—
102	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
103	C₂×Dic₁₃	#131	1	4	yes	—
104	C₄×D₁₃	—	2	2	no	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	2	16	no	—
109	#1057	—	4	8	no	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
112	#1055	—	2	2	no	—
113	#1055	C₂²⋊C₈	1	4	yes	complemented
114	C₂³×C₈	—	13	1	no	—
115	C₂³×C₈	—	13	1	no	—
116	C₂²×C₄²	—	13	1	no	—
117	C₂×C₂²⋊C₈	—	13	4	no	—
118	C₂×C₄×C₈	—	13	1	no	—
119	C₂×C₄×C₈	—	13	4	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	—	13	1	no	—
122	C₂×C₄×C₈	—	26	4	no	—
123	C₂×C₄×C₈	#261	1	1	yes	characteristic
124	#336	—	13	4	no	—
125	#322	—	26	4	no	—
126	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×#97	1	2	yes	—
129	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
130	C₂²×C₁₃	#405	1	2	yes	—
131	C₂×D₁₃	—	2	2	no	—
132	C₂×D₁₃	—	2	4	no	—
133	C₂×D₁₃	—	2	2	no	—
134	C₂×D₁₃	—	2	4	no	—
135	C₂×D₁₃	#322	1	4	yes	complemented
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₁₃	#431	1	1	yes	characteristic
140	C₄×C₁₃	#431	1	1	yes	characteristic
141	Dic₁₃	—	2	2	no	—
142	Dic₁₃	#435	1	8	yes	—
143	C₂³×C₄	—	13	2	no	—
144	C₂³×C₄	—	13	1	no	—
145	C₂²×C₈	—	13	1	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	4	no	—
148	C₂²×C₈	—	13	4	no	—
149	C₂²×C₈	—	13	8	no	—
150	C₂²×C₈	—	26	2	no	—
151	C₂²×C₈	—	26	2	no	—
152	C₂²×C₈	—	26	2	no	—
153	C₂²×C₈	—	26	2	no	—
154	C₂²×C₈	C₂×#261	1	1	yes	characteristic
155	C₂²×C₈	C₂×#261	1	1	yes	characteristic
156	C₂×C₄²	—	13	1	no	—
157	C₂×C₄²	—	13	2	no	—
158	C₂×C₄²	—	26	2	no	—
159	C₂×C₄²	—	26	2	no	—
160	C₂×C₄²	C₂×#261	1	1	yes	characteristic
161	C₂²⋊C₈	—	26	8	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	13	4	no	—
164	C₄×C₈	—	26	2	no	—
165	C₄×C₈	—	26	2	no	—
166	C₄×C₈	—	26	4	no	—
167	C₄×C₈	—	26	4	no	—
168	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
169	C₂×C₁₃	#1450	1	1	yes	characteristic
170	C₂×C₁₃	#1451	1	2	yes	—
171	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
172	C₂×C₁₃	#1752	1	2	yes	—
173	D₁₃	—	2	4	no	—
174	C₂⁴	—	13	1	no	—
175	C₂²×C₄	—	13	2	no	—
176	C₂²×C₄	—	13	1	no	—
177	C₂²×C₄	—	13	8	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	26	4	no	—
181	C₂²×C₄	—	26	4	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	—	26	1	no	—
184	C₂²×C₄	—	26	1	no	—
185	C₂²×C₄	—	26	1	no	—
186	C₂²×C₄	C₄×#261	1	2	yes	—
187	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	13	2	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	26	8	no	—
195	C₂×C₈	—	26	1	no	—
196	C₂×C₈	—	26	1	no	—
197	C₂×C₈	—	26	8	no	—
198	C₂×C₈	—	26	8	no	—
199	C₂×C₈	—	26	8	no	—
200	C₂×C₈	—	26	8	no	—
201	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
202	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
203	C₂×C₈	#6118	1	2	yes	—
204	C₂×C₈	C₂²⋊#261	1	2	yes	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	13	2	no	—
208	C₄²	—	13	2	no	—
209	C₄²	—	26	4	no	—
210	C₁₃	#8144	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	13	2	no	—
212	C₂³	—	13	4	no	—
213	C₂³	—	26	1	no	—
214	C₂³	—	26	2	no	—
215	C₂³	—	26	1	no	—
216	C₂³	C₂×C₄×#261	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	13	4	no	—
220	C₂×C₄	—	13	4	no	—
221	C₂×C₄	—	13	4	no	—
222	C₂×C₄	—	13	2	no	—
223	C₂×C₄	—	26	16	no	—
224	C₂×C₄	—	26	1	no	—
225	C₂×C₄	—	26	1	no	—
226	C₂×C₄	—	26	2	no	—
227	C₂×C₄	—	26	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	#88100	1	4	yes	—
231	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
233	C₂×C₄	C₂×#6118	1	1	yes	characteristic
234	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
235	C₂×C₄	#88237	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	13	4	no	—
239	C₈	—	26	2	no	—
240	C₈	—	52	8	no	—
241	C₈	#88111	1	4	yes	complemented
242	C₂²	—	13	4	no	—
243	C₂²	—	26	2	no	—
244	C₂²	—	26	4	no	—
245	C₂²	—	26	2	no	—
246	C₂²	—	26	4	no	—
247	C₂²	C₄×D₁₃⋊C₈	1	1	yes	characteristic
248	C₂²	C₄×#6118	1	1	yes	characteristic
249	C₂²	C₂×#88100	1	2	yes	—
250	C₂²	#11719089	1	2	yes	—
251	C₂²	C₄×C₂²⋊#261	1	1	yes	characteristic
252	C₄	—	2	4	no	—
253	C₄	—	13	8	no	—
254	C₄	—	26	2	no	—
255	C₄	#11719049	1	1	yes	characteristic
256	C₄	#11719051	1	1	yes	characteristic
257	C₄	C₂×#88111	1	1	yes	characteristic
258	C₄	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
259	C₂	—	26	4	no	—
260	C₂	#13130030	1	1	yes	characteristic
261	C₂	#13130040	1	1	yes	characteristic
262	C₂	#13130046	1	2	yes	—
263	C₂	C₄×#88111	1	1	yes	characteristic
264	C₂	#13130572	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#8144
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718251	C₂	semidirect
3. #13129658	C₂	semidirect
31. C₂×C₈×D₁₃	C₈	semidirect
32. C₂×C₈×D₁₃	C₈	semidirect
111. C₈×C₁₃	C₂²⋊C₈	semidirect
113. #1055	C₂²⋊C₈	semidirect
135. C₂×D₁₃	#322	semidirect
210. C₁₃	#8144	semidirect
241. C₈	#88111	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88111	2
C₂×#11718251	2
#13129658	2
C₂×C₄×C₈×D₁₃	2
#8144	13