Subgroups — D₄×C₁₃×D₁₆ #15720184

Order 3328.

812 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2184	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	4	yes	complemented
10	D₄²×C₁₃	—	2	2	no	—
11	C₂×C₁₃×D₁₆	—	2	8	no	—
12	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×D₁₆	C₂²	1	2	yes	complemented
14	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
15	C₂×C₁₃×D₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
17	C₁₃×#493	C₂²	1	2	yes	complemented
18	C₁₃×#466	C₂²	1	4	yes	complemented
19	C₁₃×#459	—	2	2	no	—
20	C₁₃×#447	—	2	4	no	—
21	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
22	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#366	C₂²	1	1	yes	characteristic, complemented
24	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₁₃×#357	C₂²	1	2	yes	complemented
26	C₁₃×#357	C₂²	1	4	yes	complemented
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
29	C₂²×D₄×C₁₃	—	2	4	no	—
30	C₂²×D₄×C₁₃	—	4	2	no	—
31	C₂×C₁₃×D₈	—	2	8	no	—
32	C₂×C₁₃×D₈	—	2	4	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	4	4	no	—
35	C₂×C₁₃×D₈	C₂³	1	4	yes	—
36	C₂×C₁₃×D₈	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×#127	—	2	2	no	—
39	C₁₃×#121	—	2	4	no	—
40	C₁₃×#121	—	4	2	no	—
41	C₁₃×#120	—	4	4	no	—
42	C₄×D₄×C₁₃	—	2	2	no	—
43	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×D₁₆	—	2	8	no	—
45	C₁₃×D₁₆	—	2	4	no	—
46	C₁₃×D₁₆	—	4	2	no	—
47	C₁₃×D₁₆	D₄	1	4	yes	direct product factor
48	C₂×C₁₃×C₁₆	—	2	4	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×#102	—	2	2	no	—
55	C₁₃×#102	—	2	4	no	—
56	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	D₄×D₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂⁴×C₁₃	—	4	4	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	4	no	—
62	C₂×D₄×C₁₃	—	4	8	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	8	4	no	—
68	C₂×D₄×C₁₃	—	8	8	no	—
69	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₁₃×D₈	—	2	4	no	—
73	C₁₃×D₈	—	2	2	no	—
74	C₁₃×D₈	—	2	4	no	—
75	C₁₃×D₈	—	4	8	no	—
76	C₁₃×D₈	—	8	2	no	—
77	C₁₃×D₈	C₂×D₄	1	4	yes	complemented
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
85	C₁₃×C₂²⋊C₄	—	4	2	no	—
86	C₁₃×C₂²⋊C₄	—	4	4	no	—
87	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×C₁₆	—	2	2	no	—
90	C₁₃×C₁₆	—	2	1	no	—
91	C₁₃×C₁₆	C₂×D₄	1	2	yes	complemented
92	C₂²×D₁₆	C₂×C₁₃	1	2	yes	complemented
93	D₄×D₈	C₂×C₁₃	1	2	yes	complemented
94	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#2215	C₂×C₁₃	1	2	yes	complemented
96	#2206	C₂×C₁₃	1	2	yes	complemented
97	#2184	C₂×C₁₃	1	4	yes	complemented
98	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
99	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	8	8	no	—
103	C₂³×C₁₃	—	8	8	no	—
104	C₂³×C₁₃	—	16	4	no	—
105	C₂³×C₁₃	C₂×D₈	1	2	yes	—
106	D₄×C₁₃	—	2	4	no	—
107	D₄×C₁₃	—	4	4	no	—
108	D₄×C₁₃	—	4	2	no	—
109	D₄×C₁₃	—	4	4	no	—
110	D₄×C₁₃	—	8	8	no	—
111	D₄×C₁₃	—	8	8	no	—
112	D₄×C₁₃	—	8	4	no	—
113	D₄×C₁₃	—	16	2	no	—
114	D₄×C₁₃	D₁₆	1	4	yes	direct product factor
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	8	4	no	—
119	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×D₈	C₂²×C₁₃	1	4	yes	complemented
128	D₄²	—	2	2	no	—
129	C₂×D₁₆	—	2	8	no	—
130	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
132	C₂×D₁₆	C₂²×C₁₃	1	2	yes	complemented
133	C₂×D₁₆	C₂²×C₁₃	1	2	yes	complemented
134	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	complemented
135	#493	C₂²×C₁₃	1	2	yes	complemented
136	#466	C₂²×C₁₃	1	4	yes	complemented
137	#459	—	2	2	no	—
138	#447	—	2	4	no	—
139	C₄×D₈	C₂²×C₁₃	1	2	yes	complemented
140	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
141	#366	C₂²×C₁₃	1	1	yes	characteristic, complemented
142	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
143	#357	C₂²×C₁₃	1	2	yes	complemented
144	#357	C₂²×C₁₃	1	4	yes	complemented
145	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
146	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	—	8	2	no	—
151	C₂²×C₁₃	—	8	2	no	—
152	C₂²×C₁₃	—	8	4	no	—
153	C₂²×C₁₃	—	8	4	no	—
154	C₂²×C₁₃	—	8	4	no	—
155	C₂²×C₁₃	—	16	16	no	—
156	C₂²×C₁₃	C₂×D₁₆	1	4	yes	complemented
157	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	8	2	no	—
161	C₄×C₁₃	C₂×D₁₆	1	2	yes	complemented
162	C₄×C₁₃	D₄²	1	1	yes	characteristic
163	C₄×C₁₃	#576	1	1	yes	characteristic
164	C₂²×D₄	—	2	4	no	—
165	C₂²×D₄	—	4	2	no	—
166	C₂×D₈	—	2	8	no	—
167	C₂×D₈	—	2	4	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	—	4	4	no	—
170	C₂×D₈	C₂³×C₁₃	1	4	yes	—
171	C₂×D₈	C₂³×C₁₃	1	2	yes	—
172	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
173	#127	—	2	2	no	—
174	#121	—	2	4	no	—
175	#121	—	4	2	no	—
176	#120	—	4	4	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
179	D₁₆	—	2	8	no	—
180	D₁₆	—	2	4	no	—
181	D₁₆	—	4	2	no	—
182	D₁₆	D₄×C₁₃	1	4	yes	direct product factor
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
186	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
187	#107	C₂³×C₁₃	1	2	yes	—
188	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
189	#102	—	2	2	no	—
190	#102	—	2	4	no	—
191	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
192	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₁₃	—	2	4	no	—
194	C₂×C₁₃	—	8	4	no	—
195	C₂×C₁₃	—	16	4	no	—
196	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
197	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
198	C₂×C₁₃	#3415	1	1	yes	characteristic
199	C₂⁴	—	4	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	4	8	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	8	4	no	—
208	C₂×D₄	—	8	8	no	—
209	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
212	D₈	—	2	4	no	—
213	D₈	—	2	2	no	—
214	D₈	—	2	4	no	—
215	D₈	—	4	8	no	—
216	D₈	—	8	2	no	—
217	D₈	C₂×D₄×C₁₃	1	4	yes	complemented
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
222	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
228	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₁₆	—	2	2	no	—
230	C₁₆	—	2	1	no	—
231	C₁₆	C₂×D₄×C₁₃	1	2	yes	complemented
232	C₁₃	D₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	8	8	no	—
236	C₂³	—	8	8	no	—
237	C₂³	—	16	4	no	—
238	C₂³	C₂×C₁₃×D₈	1	2	yes	—
239	D₄	—	2	4	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	4	no	—
246	D₄	—	16	2	no	—
247	D₄	C₁₃×D₁₆	1	4	yes	direct product factor
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
254	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	1	no	—
258	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
259	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	16	no	—
269	C₂²	C₂×C₁₃×D₁₆	1	4	yes	complemented
270	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×C₁₃×D₁₆	1	2	yes	complemented
275	C₄	D₄²×C₁₃	1	1	yes	characteristic
276	C₄	C₁₃×#576	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	8	4	no	—
279	C₂	—	16	4	no	—
280	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic
281	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
282	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×D₁₆	C₂	semidirect
2. D₄×C₁₃×D₈	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2215	C₂	semidirect
5. C₁₃×#2206	C₂	semidirect
6. C₁₃×#2184	C₂	semidirect
7. C₄×C₁₃×D₁₆	C₂	semidirect
8. D₄×C₁₃×C₁₆	C₂	semidirect
9. C₂²×C₁₃×D₈	C₂²	semidirect
13. C₂×C₁₃×D₁₆	C₂²	semidirect
14. C₂×C₁₃×D₁₆	C₂²	semidirect
15. C₂×C₁₃×D₁₆	C₂²	semidirect
16. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₁₃×#493	C₂²	semidirect
18. C₁₃×#466	C₂²	semidirect
21. C₄×C₁₃×D₈	C₂²	semidirect
23. C₁₃×#366	C₂²	semidirect
24. C₁₃×C₄⋊C₁₆	C₂²	semidirect
25. C₁₃×#357	C₂²	semidirect
26. C₁₃×#357	C₂²	semidirect
27. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
28. C₄×C₁₃×C₁₆	C₂²	semidirect
47. C₁₃×D₁₆	D₄	direct (N × H)
58. D₄×D₁₆	C₁₃	direct (N × H)
77. C₁₃×D₈	C₂×D₄	semidirect
91. C₁₃×C₁₆	C₂×D₄	semidirect
92. C₂²×D₁₆	C₂×C₁₃	semidirect
93. D₄×D₈	C₂×C₁₃	semidirect
94. #2246	C₂×C₁₃	semidirect
95. #2215	C₂×C₁₃	semidirect
96. #2206	C₂×C₁₃	semidirect
97. #2184	C₂×C₁₃	semidirect
98. C₄×D₁₆	C₂×C₁₃	semidirect
99. D₄×C₁₆	C₂×C₁₃	semidirect
114. D₄×C₁₃	D₁₆	direct (N × H)
127. C₂²×D₈	C₂²×C₁₃	semidirect
131. C₂×D₁₆	C₂²×C₁₃	semidirect
132. C₂×D₁₆	C₂²×C₁₃	semidirect
133. C₂×D₁₆	C₂²×C₁₃	semidirect
134. C₂²×C₁₆	C₂²×C₁₃	semidirect
135. #493	C₂²×C₁₃	semidirect
136. #466	C₂²×C₁₃	semidirect
139. C₄×D₈	C₂²×C₁₃	semidirect
141. #366	C₂²×C₁₃	semidirect
142. C₄⋊C₁₆	C₂²×C₁₃	semidirect
143. #357	C₂²×C₁₃	semidirect
144. #357	C₂²×C₁₃	semidirect
145. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
146. C₄×C₁₆	C₂²×C₁₃	semidirect
156. C₂²×C₁₃	C₂×D₁₆	semidirect
161. C₄×C₁₃	C₂×D₁₆	semidirect
182. D₁₆	D₄×C₁₃	direct (N × H)
217. D₈	C₂×D₄×C₁₃	semidirect
231. C₁₆	C₂×D₄×C₁₃	semidirect
232. C₁₃	D₄×D₁₆	direct (N × H)
247. D₄	C₁₃×D₁₆	direct (N × H)
269. C₂²	C₂×C₁₃×D₁₆	semidirect
274. C₄	C₂×C₁₃×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×D₁₆	2
D₄×C₁₃×D₈	2
C₁₃×#2246	2
C₁₃×#2215	2
C₁₃×#2206	2
C₁₃×#2184	2
C₄×C₁₃×D₁₆	2
D₄×C₁₃×C₁₆	2
D₄×D₁₆	13

Subgroups — D4×C13×D16 #15720184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₁₃×D₁₆ #15720184