Subgroups — C₂×D₁₃×#543 #16769639

Order 3328.

9310 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719188	C₂	1	3	yes	maximal, complemented
2	C₂×#11719339	C₂	1	12	yes	maximal, complemented
3	D₁₃×#543	C₂	1	32	yes	maximal, direct product factor
4	C₂×D₁₃×C₂²⋊Q₈	C₂	1	12	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₁₃	C₂	1	3	yes	maximal
6	C₂×C₁₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×D₁₃	C₂²	1	4	yes	complemented
9	C₂³×C₄×D₁₃	C₂²	1	6	yes	—
10	C₁₃×#543	C₂²	1	16	yes	complemented
11	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	12	yes	complemented
12	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
14	C₂×#88200	C₂²	1	12	yes	complemented
15	C₂²×#6103	C₂²	1	6	yes	—
16	C₂×#88178	C₂²	1	12	yes	complemented
17	#11719339	C₂²	1	192	yes	complemented
18	D₁₃×C₂²⋊Q₈	C₂²	1	192	yes	complemented
19	C₂×#88151	C₂²	1	12	yes	complemented
20	C₂×#88150	C₂²	1	24	yes	complemented
21	C₂×C₄⋊C₄×D₁₃	C₂²	1	12	yes	complemented
22	C₂×C₂²⋊C₄×D₁₃	C₂²	1	48	yes	—
23	C₂×C₂²⋊C₄×D₁₃	C₂²	1	12	yes	—
24	C₂×C₂²⋊C₄×D₁₃	C₂²	1	12	yes	—
25	#11719188	C₂²	1	48	yes	complemented
26	C₂×#88132	C₂²	1	24	yes	complemented
27	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
28	C₂⁴×D₁₃	—	4	8	no	—
29	C₂⁴×D₁₃	Q₈	1	16	yes	complemented
30	C₂⁴×D₁₃	C₂³	1	3	yes	—
31	C₂⁴×D₁₃	C₂³	1	8	yes	—
32	C₂⁴×D₁₃	C₂³	1	3	yes	—
33	C₂²×Q₈×C₁₃	C₂³	1	4	yes	complemented
34	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
35	C₂³×Dic₁₃	C₂³	1	6	yes	—
36	C₂×Q₈×D₁₃	—	4	16	no	—
37	C₂×Q₈×D₁₃	C₂³	1	32	yes	complemented
38	C₂²×C₄×D₁₃	—	4	24	no	—
39	C₂²×C₄×D₁₃	C₂³	1	12	yes	—
40	C₂²×C₄×D₁₃	C₂³	1	12	yes	—
41	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
42	C₂²×Dic₂₆	C₂³	1	12	yes	complemented
43	C₁₃×C₂²⋊Q₈	C₂³	1	96	yes	complemented
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	complemented
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
46	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
47	C₂×C₂²⋊Dic₁₃	C₂³	1	12	yes	—
48	C₂×C₂²⋊Dic₁₃	C₂³	1	24	yes	—
49	#88200	C₂³	1	96	yes	complemented
50	C₂×#6103	C₂³	1	24	yes	—
51	C₂×#6103	C₂³	1	48	yes	—
52	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	complemented
53	#88178	C₂³	1	96	yes	complemented
54	C₂×#6101	C₂³	1	24	yes	complemented
55	#88151	C₂³	1	96	yes	complemented
56	#88150	C₂³	1	192	yes	complemented
57	C₄⋊C₄×D₁₃	C₂³	1	96	yes	complemented
58	C₂²⋊C₄×D₁₃	C₂³	1	96	yes	—
59	C₂²⋊C₄×D₁₃	C₂³	1	96	yes	—
60	#88132	C₂³	1	192	yes	complemented
61	C₂²×#543	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	—	4	4	no	—
63	C₂⁴×C₁₃	C₂×Q₈	1	8	yes	complemented
64	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
65	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
66	C₂³×D₁₃	—	2	24	no	—
67	C₂³×D₁₃	—	4	64	no	—
68	C₂³×D₁₃	—	4	32	no	—
69	C₂³×D₁₃	—	4	12	no	—
70	C₂³×D₁₃	C₂×Q₈	1	24	yes	—
71	C₂³×D₁₃	C₂×Q₈	1	64	yes	complemented
72	C₂³×D₁₃	C₂×Q₈	1	24	yes	complemented
73	C₂³×D₁₃	C₂⁴	1	12	yes	—
74	C₂³×D₁₃	C₂⁴	1	3	yes	—
75	C₂³×D₁₃	C₂⁴	1	12	yes	—
76	C₂³×D₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₁₃	—	4	8	no	—
78	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	complemented
79	C₂²×C₄×C₁₃	—	4	12	no	—
80	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
82	C₂²×Dic₁₃	—	4	12	no	—
83	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
84	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
85	Q₈×D₁₃	—	4	64	no	—
86	C₂×C₄×D₁₃	—	2	48	no	—
87	C₂×C₄×D₁₃	—	4	96	no	—
88	C₂×C₄×D₁₃	—	4	24	no	—
89	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
90	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
91	C₂×Dic₂₆	—	4	24	no	—
92	C₂×Dic₂₆	C₂⁴	1	48	yes	complemented
93	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	complemented
94	C₁₃×C₂²⋊C₄	C₂⁴	1	48	yes	—
95	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	—
96	#6103	C₂⁴	1	96	yes	—
97	C₄⋊Dic₁₃	C₂⁴	1	48	yes	complemented
98	#6101	C₂⁴	1	96	yes	complemented
99	C₂×#543	—	13	32	no	—
100	C₂×#543	—	13	12	no	—
101	C₂×#543	—	13	3	no	—
102	C₂×#543	D₁₃	1	1	yes	characteristic, direct product factor
103	C₂²×C₂²⋊Q₈	—	13	12	no	—
104	C₂³×C₂²⋊C₄	—	13	3	no	—
105	C₂³×C₁₃	—	2	12	no	—
106	C₂³×C₁₃	—	4	16	no	—
107	C₂³×C₁₃	—	4	8	no	—
108	C₂³×C₁₃	C₂²×Q₈	1	12	yes	—
109	C₂³×C₁₃	C₂²×Q₈	1	16	yes	complemented
110	C₂³×C₁₃	C₂⁵	1	6	yes	—
111	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	12	no	—
113	C₂²×D₁₃	—	2	96	no	—
114	C₂²×D₁₃	—	2	48	no	—
115	C₂²×D₁₃	—	4	128	no	—
116	C₂²×D₁₃	—	4	48	no	—
117	C₂²×D₁₃	—	4	24	no	—
118	C₂²×D₁₃	C₂²×Q₈	1	12	yes	—
119	C₂²×D₁₃	C₂²×Q₈	1	48	yes	—
120	C₂²×D₁₃	C₂²×Q₈	1	4	yes	—
121	C₂²×D₁₃	C₂²×Q₈	1	48	yes	complemented
122	C₂²×D₁₃	H(5,𝔽₂)	1	8	yes	—
123	C₂²×D₁₃	C₂⁵	1	6	yes	—
124	C₂²×D₁₃	C₂⁵	1	2	yes	—
125	Q₈×C₁₃	—	4	16	no	—
126	C₂×C₄×C₁₃	—	2	24	no	—
127	C₂×C₄×C₁₃	—	4	24	no	—
128	C₂×C₄×C₁₃	C₂⁵	1	24	yes	—
129	C₂×Dic₁₃	—	2	24	no	—
130	C₂×Dic₁₃	—	4	24	no	—
131	C₂×Dic₁₃	C₂⁵	1	24	yes	—
132	C₄×D₁₃	—	2	96	no	—
133	C₄×D₁₃	—	4	48	no	—
134	Dic₂₆	—	4	48	no	—
135	C₂⁶	—	13	1	no	—
136	C₂³×Q₈	—	13	4	no	—
137	C₂⁴×C₄	—	13	6	no	—
138	#543	—	13	192	no	—
139	#543	—	13	48	no	—
140	#543	C₂×D₁₃	1	16	yes	direct product factor
141	C₂×C₂²⋊Q₈	—	13	12	no	—
142	C₂×C₂²⋊Q₈	—	13	12	no	—
143	C₂×C₂²⋊Q₈	—	13	192	no	—
144	C₂×C₂²⋊Q₈	—	13	24	no	—
145	C₂×C₂²⋊Q₈	—	13	12	no	—
146	C₂×C₂²⋊Q₈	—	13	24	no	—
147	C₂×C₂²⋊Q₈	C₂×D₁₃	1	12	yes	complemented
148	C₂²×C₄⋊C₄	—	13	12	no	—
149	C₂²×C₂²⋊C₄	—	13	3	no	—
150	C₂²×C₂²⋊C₄	—	13	6	no	—
151	C₂²×C₂²⋊C₄	—	13	12	no	—
152	C₂²×C₂²⋊C₄	—	13	12	no	—
153	C₂²×C₂²⋊C₄	—	13	48	no	—
154	C₂²×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
155	C₂²×C₁₃	—	2	12	no	—
156	C₂²×C₁₃	—	2	24	no	—
157	C₂²×C₁₃	—	4	16	no	—
158	C₂²×C₁₃	C₂³×Q₈	1	12	yes	—
159	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
160	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	4	yes	—
161	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
162	C₂×D₁₃	—	2	12	no	—
163	C₂×D₁₃	—	2	24	no	—
164	C₂×D₁₃	—	2	96	no	—
165	C₂×D₁₃	—	4	48	no	—
166	C₂×D₁₃	#543	1	16	yes	direct product factor
167	C₂×D₁₃	C₂³×Q₈	1	12	yes	—
168	C₂×D₁₃	C₂³×Q₈	1	4	yes	—
169	C₂×D₁₃	C₂×H(5,𝔽₂)	1	8	yes	—
170	C₄×C₁₃	—	2	24	no	—
171	Dic₁₃	—	2	24	no	—
172	C₂⁵	—	13	3	no	—
173	C₂⁵	—	13	16	no	—
174	C₂⁵	—	13	8	no	—
175	C₂⁵	—	13	3	no	—
176	C₂⁵	—	52	8	no	—
177	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×Q₈	—	13	32	no	—
179	C₂²×Q₈	—	13	12	no	—
180	C₂²×Q₈	—	52	16	no	—
181	C₂²×Q₈	C₂²×D₁₃	1	4	yes	complemented
182	C₂³×C₄	—	13	6	no	—
183	C₂³×C₄	—	13	12	no	—
184	C₂³×C₄	—	13	12	no	—
185	C₂³×C₄	—	13	48	no	—
186	C₂³×C₄	—	52	24	no	—
187	C₂³×C₄	C₂²×D₁₃	1	6	yes	—
188	C₂²⋊Q₈	—	13	96	no	—
189	C₂²⋊Q₈	—	13	96	no	—
190	C₂²⋊Q₈	—	13	192	no	—
191	C₂²⋊Q₈	—	13	96	no	—
192	C₂²⋊Q₈	—	13	192	no	—
193	C₂²⋊Q₈	C₂²×D₁₃	1	96	yes	complemented
194	C₂×C₄⋊C₄	—	13	24	no	—
195	C₂×C₄⋊C₄	—	13	12	no	—
196	C₂×C₄⋊C₄	—	13	96	no	—
197	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	complemented
198	C₂×C₂²⋊C₄	—	13	12	no	—
199	C₂×C₂²⋊C₄	—	13	24	no	—
200	C₂×C₂²⋊C₄	—	13	24	no	—
201	C₂×C₂²⋊C₄	—	13	48	no	—
202	C₂×C₂²⋊C₄	—	13	96	no	—
203	C₂×C₂²⋊C₄	—	13	96	no	—
204	C₂×C₂²⋊C₄	C₂²×D₁₃	1	12	yes	—
205	C₂×C₂²⋊C₄	C₂²×D₁₃	1	24	yes	—
206	C₂×C₁₃	—	2	12	no	—
207	C₂×C₁₃	C₂×#543	1	4	yes	complemented
208	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
209	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
210	D₁₃	—	2	12	no	—
211	D₁₃	C₂×#543	1	8	yes	direct product factor
212	C₂⁴	—	4	4	no	—
213	C₂⁴	—	13	1	no	—
214	C₂⁴	—	13	24	no	—
215	C₂⁴	—	13	12	no	—
216	C₂⁴	—	13	3	no	—
217	C₂⁴	—	13	24	no	—
218	C₂⁴	—	13	64	no	—
219	C₂⁴	—	13	12	no	—
220	C₂⁴	—	26	24	no	—
221	C₂⁴	—	52	12	no	—
222	C₂⁴	—	52	64	no	—
223	C₂⁴	—	52	32	no	—
224	C₂⁴	Q₈×D₁₃	1	8	yes	complemented
225	C₂⁴	C₂³×D₁₃	1	3	yes	—
226	C₂⁴	C₂³×D₁₃	1	4	yes	—
227	C₂×Q₈	—	4	8	no	—
228	C₂×Q₈	—	13	48	no	—
229	C₂×Q₈	—	52	24	no	—
230	C₂×Q₈	—	52	64	no	—
231	C₂×Q₈	C₂³×D₁₃	1	16	yes	complemented
232	C₂²×C₄	—	4	12	no	—
233	C₂²×C₄	—	13	12	no	—
234	C₂²×C₄	—	13	24	no	—
235	C₂²×C₄	—	13	48	no	—
236	C₂²×C₄	—	13	48	no	—
237	C₂²×C₄	—	26	48	no	—
238	C₂²×C₄	—	52	24	no	—
239	C₂²×C₄	—	52	96	no	—
240	C₂²×C₄	—	52	12	no	—
241	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
242	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
243	C₄⋊C₄	—	13	96	no	—
244	C₄⋊C₄	—	13	48	no	—
245	C₄⋊C₄	C₂³×D₁₃	1	48	yes	complemented
246	C₂²⋊C₄	—	13	48	no	—
247	C₂²⋊C₄	—	13	96	no	—
248	C₂²⋊C₄	C₂³×D₁₃	1	48	yes	—
249	C₁₃	C₂²×#543	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	2	12	no	—
251	C₂³	—	4	16	no	—
252	C₂³	—	4	8	no	—
253	C₂³	—	13	4	no	—
254	C₂³	—	13	8	no	—
255	C₂³	—	13	2	no	—
256	C₂³	—	13	48	no	—
257	C₂³	—	13	12	no	—
258	C₂³	—	13	6	no	—
259	C₂³	—	13	48	no	—
260	C₂³	—	26	48	no	—
261	C₂³	—	26	96	no	—
262	C₂³	—	26	12	no	—
263	C₂³	—	52	48	no	—
264	C₂³	—	52	24	no	—
265	C₂³	—	52	128	no	—
266	C₂³	C₂×Q₈×D₁₃	1	12	yes	—
267	C₂³	C₂×Q₈×D₁₃	1	16	yes	complemented
268	C₂³	C₂⁴×D₁₃	1	6	yes	—
269	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
270	Q₈	—	4	16	no	—
271	Q₈	—	52	48	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	4	24	no	—
274	C₂×C₄	—	13	24	no	—
275	C₂×C₄	—	26	96	no	—
276	C₂×C₄	—	26	24	no	—
277	C₂×C₄	—	52	48	no	—
278	C₂×C₄	—	52	24	no	—
279	C₂×C₄	C₂⁴×D₁₃	1	24	yes	—
280	C₂²	—	2	24	no	—
281	C₂²	—	2	12	no	—
282	C₂²	—	4	16	no	—
283	C₂²	—	13	16	no	—
284	C₂²	—	13	4	no	—
285	C₂²	—	13	8	no	—
286	C₂²	—	13	12	no	—
287	C₂²	—	26	96	no	—
288	C₂²	—	26	24	no	—
289	C₂²	—	26	12	no	—
290	C₂²	—	52	48	no	—
291	C₂²	C₂²×Q₈×D₁₃	1	12	yes	—
292	C₂²	C₂²×Q₈×D₁₃	1	2	yes	—
293	C₂²	D₁₃×H(5,𝔽₂)	1	4	yes	—
294	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
295	C₄	—	2	24	no	—
296	C₄	—	26	24	no	—
297	C₂	—	2	12	no	—
298	C₂	—	13	8	no	—
299	C₂	—	26	12	no	—
300	C₂	D₁₃×#543	1	4	yes	direct product factor
301	C₂	C₂³×Q₈×D₁₃	1	1	yes	characteristic
302	C₂	C₂×D₁₃×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂×C₁₃×#543	C₂
socle	C₂³×C₁₃	C₂⁵
prime radical	C₂⁵×D₁₃	C₂²
hypercenter	C₂×#543	D₁₃
hypocenter	C₁₃	C₂²×#543
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. C₂×#11719188	C₂	semidirect
2. C₂×#11719339	C₂	semidirect
3. D₁₃×#543	C₂	direct (N × H)
4. C₂×D₁₃×C₂²⋊Q₈	C₂	semidirect
6. C₂×C₁₃×#543	C₂	semidirect
8. C₂²×Q₈×D₁₃	C₂²	semidirect
10. C₁₃×#543	C₂²	semidirect
11. C₂×C₁₃×C₂²⋊Q₈	C₂²	semidirect
14. C₂×#88200	C₂²	semidirect
16. C₂×#88178	C₂²	semidirect
17. #11719339	C₂²	semidirect
18. D₁₃×C₂²⋊Q₈	C₂²	semidirect
19. C₂×#88151	C₂²	semidirect
20. C₂×#88150	C₂²	semidirect
21. C₂×C₄⋊C₄×D₁₃	C₂²	semidirect
25. #11719188	C₂²	semidirect
26. C₂×#88132	C₂²	semidirect
29. C₂⁴×D₁₃	Q₈	semidirect
33. C₂²×Q₈×C₁₃	C₂³	semidirect
37. C₂×Q₈×D₁₃	C₂³	semidirect
42. C₂²×Dic₂₆	C₂³	semidirect
43. C₁₃×C₂²⋊Q₈	C₂³	semidirect
44. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
49. #88200	C₂³	semidirect
52. C₂×C₄⋊Dic₁₃	C₂³	semidirect
53. #88178	C₂³	semidirect
54. C₂×#6101	C₂³	semidirect
55. #88151	C₂³	semidirect
56. #88150	C₂³	semidirect
57. C₄⋊C₄×D₁₃	C₂³	semidirect
60. #88132	C₂³	semidirect
63. C₂⁴×C₁₃	C₂×Q₈	semidirect
71. C₂³×D₁₃	C₂×Q₈	semidirect
72. C₂³×D₁₃	C₂×Q₈	semidirect
78. C₂×Q₈×C₁₃	C₂⁴	semidirect
92. C₂×Dic₂₆	C₂⁴	semidirect
93. C₁₃×C₄⋊C₄	C₂⁴	semidirect
97. C₄⋊Dic₁₃	C₂⁴	semidirect
98. #6101	C₂⁴	semidirect
102. C₂×#543	D₁₃	direct (N × H)
109. C₂³×C₁₃	C₂²×Q₈	semidirect
121. C₂²×D₁₃	C₂²×Q₈	semidirect
140. #543	C₂×D₁₃	direct (N × H)
147. C₂×C₂²⋊Q₈	C₂×D₁₃	semidirect
166. C₂×D₁₃	#543	direct (N × H)
181. C₂²×Q₈	C₂²×D₁₃	semidirect
193. C₂²⋊Q₈	C₂²×D₁₃	semidirect
197. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
207. C₂×C₁₃	C₂×#543	semidirect
211. D₁₃	C₂×#543	direct (N × H)
224. C₂⁴	Q₈×D₁₃	semidirect
231. C₂×Q₈	C₂³×D₁₃	semidirect
245. C₄⋊C₄	C₂³×D₁₃	semidirect
249. C₁₃	C₂²×#543	semidirect
267. C₂³	C₂×Q₈×D₁₃	semidirect
300. C₂	D₁₃×#543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719188	2
C₂×#11719339	2
D₁₃×#543	2
C₂×D₁₃×C₂²⋊Q₈	2
C₂²×C₂²⋊C₄×D₁₃	2
C₂×C₁₃×#543	2
C₂²×#543	13

Subgroups — C2×D13×#543 #16769639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×#543 #16769639