Subgroups — C₂×D₁₃⋊#364 #16807008

Order 3328.

782 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃⋊M₃₂	C₂	1	2	yes	maximal
2	C₂×#11718233	C₂	1	2	yes	maximal, complemented
3	D₁₃⋊#364	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718261	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	D₁₃⋊M₃₂	—	2	4	no	—
11	D₁₃⋊M₃₂	C₂²	1	8	yes	—
12	C₂×#88098	—	2	2	no	—
13	C₂×D₁₃⋊C₁₆	—	2	2	no	—
14	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₄×C₈×D₁₃	C₄	1	4	yes	—
19	C₄×C₈×D₁₃	C₂²	1	4	yes	—
20	#11718261	C₂²	1	8	yes	complemented
21	#11718233	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃	—	2	1	no	—
23	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	1	no	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1055	C₂³	1	1	yes	characteristic
31	C₂×C₈×D₁₃	C₈	1	2	yes	—
32	C₂×C₈×D₁₃	C₈	1	4	yes	—
33	C₂×C₈×D₁₃	C₈	1	2	yes	—
34	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
37	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
38	C₂×C₈×D₁₃	D₄	1	2	yes	—
39	C₂×C₈×D₁₃	Q₈	1	2	yes	—
40	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₁₃	Q₈	1	1	yes	characteristic
42	C₂×C₈×D₁₃	C₂³	1	2	yes	—
43	C₄²×D₁₃	—	2	2	no	—
44	C₄²×D₁₃	C₂×C₄	1	4	yes	—
45	#88106	—	2	4	no	—
46	#88106	C₂³	1	8	yes	—
47	C₂×#6092	—	2	4	no	—
48	#88098	—	2	8	no	—
49	D₁₃⋊C₁₆	—	2	8	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
51	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
52	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₁₃	C₂³	1	2	yes	—
54	C₄×#1055	C₂×C₄	1	2	yes	—
55	C₄×#1055	C₂³	1	2	yes	—
56	C₂²×#364	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	2	1	no	—
61	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
62	C₂×C₄×D₁₃	—	2	8	no	—
63	C₂×C₄×D₁₃	—	2	4	no	—
64	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
71	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄×Dic₁₃	—	2	1	no	—
82	C₄×Dic₁₃	—	2	4	no	—
83	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×#1055	C₂×C₈	1	2	yes	—
85	C₂×#1055	C₂×C₈	1	2	yes	—
86	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
87	C₂×#1055	C₂²×C₄	1	2	yes	—
88	C₂×#1055	C₂×D₄	1	1	yes	characteristic
89	C₂×#1055	C₂×D₄	1	1	yes	characteristic
90	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
91	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
92	C₂×#1055	C₂⁴	1	1	yes	characteristic
93	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
95	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
96	C₈×D₁₃	C₄⋊C₄	1	2	yes	—
97	C₈×D₁₃	C₂×C₈	1	8	yes	—
98	C₈×D₁₃	C₂×C₈	1	8	yes	—
99	C₈×D₁₃	C₂×D₄	1	2	yes	—
100	C₈×D₁₃	C₂×D₄	1	2	yes	—
101	C₈×D₁₃	C₂×Q₈	1	2	yes	—
102	C₈×D₁₃	C₂×Q₈	1	2	yes	—
103	#6092	—	2	8	no	—
104	C₂²×M₃₂	—	13	2	no	—
105	C₂²×C₄×C₈	—	13	1	no	—
106	C₂×#364	—	13	2	no	—
107	C₂×#364	—	13	8	no	—
108	C₂×#364	—	13	2	no	—
109	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×D₁₃	—	2	4	no	—
111	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
112	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
113	C₂²×D₁₃	C₄⋊C₈	1	1	yes	characteristic
114	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	—	2	4	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
126	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
127	C₂×Dic₁₃	C₄⋊C₈	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
129	C₄×D₁₃	—	2	16	no	—
130	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
131	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
132	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
133	C₄×D₁₃	C₄⋊C₈	1	2	yes	—
134	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₄×D₁₃	C₂²×C₈	1	2	yes	—
137	C₄×D₁₃	C₂²×C₈	1	2	yes	—
138	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₁₃	C₂²×C₈	1	4	yes	—
141	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
142	#1055	C₂×C₄⋊C₄	1	1	yes	characteristic
143	#1055	C₂²×C₈	1	4	yes	—
144	#1055	C₂²×D₄	1	1	yes	characteristic
145	#1055	C₂²×Q₈	1	1	yes	characteristic
146	C₂³×C₈	—	13	1	no	—
147	C₂³×C₈	—	13	1	no	—
148	C₂²×C₄²	—	13	1	no	—
149	C₂×M₃₂	—	13	8	no	—
150	C₂×M₃₂	—	13	4	no	—
151	C₂×M₃₂	—	26	2	no	—
152	C₂×M₃₂	—	26	4	no	—
153	C₂²×C₁₆	—	26	2	no	—
154	C₂×C₄×C₈	—	13	1	no	—
155	C₂×C₄×C₈	—	13	1	no	—
156	C₂×C₄×C₈	—	13	1	no	—
157	C₂×C₄×C₈	—	13	4	no	—
158	C₂×C₄×C₈	—	13	4	no	—
159	C₂×C₄×C₈	#261	1	1	yes	characteristic
160	#364	—	13	8	no	—
161	#364	—	13	8	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
166	C₂×D₁₃	—	2	2	no	—
167	C₂×D₁₃	—	2	4	no	—
168	C₂×D₁₃	—	2	4	no	—
169	C₂×D₁₃	#364	1	4	yes	complemented
170	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
171	C₂×D₁₃	C₂×C₄⋊C₈	1	2	yes	—
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
177	Dic₁₃	—	2	4	no	—
178	Dic₁₃	C₂×C₄⋊C₈	1	2	yes	—
179	Dic₁₃	C₂×C₄⋊C₈	1	2	yes	—
180	C₂³×C₄	—	13	1	no	—
181	C₂³×C₄	—	26	1	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	13	1	no	—
193	C₂²×C₈	—	13	4	no	—
194	C₂²×C₈	—	13	2	no	—
195	C₂²×C₈	—	13	2	no	—
196	C₂²×C₈	C₂×#261	1	1	yes	characteristic
197	C₂²×C₈	C₂×#261	1	1	yes	characteristic
198	C₂×C₄²	—	13	1	no	—
199	C₂×C₄²	—	13	4	no	—
200	C₂×C₄²	—	26	2	no	—
201	C₂×C₄²	—	26	1	no	—
202	C₂×C₄²	C₂×#261	1	1	yes	characteristic
203	M₃₂	—	13	8	no	—
204	M₃₂	—	26	8	no	—
205	M₃₂	—	26	4	no	—
206	C₂×C₁₆	—	26	8	no	—
207	C₂×C₁₆	—	26	4	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	4	no	—
211	C₄×C₈	—	13	2	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	C₂×#261	1	4	yes	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	C₂×#364	1	2	yes	complemented
217	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
218	D₁₃	—	2	2	no	—
219	D₁₃	C₂×#364	1	4	yes	complemented
220	C₂⁴	—	13	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	1	no	—
226	C₂²×C₄	—	13	2	no	—
227	C₂²×C₄	—	13	2	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	26	4	no	—
232	C₂²×C₄	—	26	8	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	1	no	—
238	C₂×C₈	—	13	1	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	13	1	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	2	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	2	no	—
251	C₂×C₈	—	13	2	no	—
252	C₂×C₈	—	13	2	no	—
253	C₂×C₈	—	13	2	no	—
254	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
255	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
256	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
257	C₂×C₈	C₄⋊#261	1	2	yes	—
258	C₂×C₈	C₄⋊#261	1	1	yes	characteristic
259	C₂×C₈	C₂²×#261	1	2	yes	—
260	C₂×C₈	C₂²×#261	1	2	yes	—
261	C₄²	—	2	1	no	—
262	C₄²	—	13	2	no	—
263	C₄²	—	26	4	no	—
264	C₄²	—	26	1	no	—
265	C₄²	C₂²×#261	1	2	yes	—
266	C₁₆	—	26	8	no	—
267	C₁₃	C₂²×#364	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	13	1	no	—
269	C₂³	—	13	2	no	—
270	C₂³	—	13	1	no	—
271	C₂³	—	13	2	no	—
272	C₂³	—	26	4	no	—
273	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	13	2	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	2	no	—
280	C₂×C₄	—	13	2	no	—
281	C₂×C₄	—	13	2	no	—
282	C₂×C₄	—	13	2	no	—
283	C₂×C₄	—	13	2	no	—
284	C₂×C₄	—	13	2	no	—
285	C₂×C₄	—	13	2	no	—
286	C₂×C₄	—	13	2	no	—
287	C₂×C₄	—	13	2	no	—
288	C₂×C₄	—	26	16	no	—
289	C₂×C₄	—	26	4	no	—
290	C₂×C₄	—	26	2	no	—
291	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
292	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
293	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
295	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
296	C₂×C₄	C₂³×#261	1	1	yes	characteristic
297	C₈	—	13	4	no	—
298	C₈	—	13	1	no	—
299	C₈	—	13	1	no	—
300	C₈	—	13	1	no	—
301	C₈	—	13	1	no	—
302	C₈	C₂×D₁₃⋊C₈	1	4	yes	—
303	C₈	C₂×C₄⋊#261	1	2	yes	—
304	C₈	C₂×C₄⋊#261	1	2	yes	—
305	C₂²	—	2	2	no	—
306	C₂²	—	13	4	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	13	2	no	—
309	C₂²	—	26	4	no	—
310	C₂²	—	26	4	no	—
311	C₂²	—	26	2	no	—
312	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
313	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
314	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	13	2	no	—
317	C₄	—	13	2	no	—
318	C₄	—	26	4	no	—
319	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
320	C₄	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
321	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
322	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	13	4	no	—
326	C₂	—	26	2	no	—
327	C₂	D₁₃⋊#364	1	2	yes	direct product factor
328	C₂	C₂×D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂²×#364
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718233	C₂	semidirect
3. D₁₃⋊#364	C₂	direct (N × H)
4. C₂×#11718261	C₂	semidirect
20. #11718261	C₂²	semidirect
21. #11718233	C₂²	semidirect
169. C₂×D₁₃	#364	semidirect
216. C₂×C₁₃	C₂×#364	semidirect
219. D₁₃	C₂×#364	semidirect
267. C₁₃	C₂²×#364	semidirect
327. C₂	D₁₃⋊#364	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×D₁₃⋊M₃₂	2
C₂×#11718233	2
D₁₃⋊#364	2
C₂×#11718261	2
C₂×C₄×C₈×D₁₃	2
C₂²×#364	13

Subgroups — C2×D13⋊#364 #16807008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃⋊#364 #16807008