Subgroups — C₂×#13130549 #16796832

Order 3328.

1438 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×#261	C₂	1	1	yes	maximal, characteristic
3	#13130549	C₂	1	8	yes	maximal, direct product factor
4	C₂×(C₂×#261)⋊C₈	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×#261	C₄	1	2	yes	—
7	C₂²×C₄×#261	C₄	1	2	yes	—
8	C₂²×C₄×#261	C₂²	1	2	yes	—
9	C₂²×D₁₃⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
11	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1057	C₄	1	2	yes	complemented
13	C₂×C₄×#1057	C₄	1	2	yes	complemented
14	C₄²×#261	C₄	1	4	yes	—
15	C₄²×#261	C₂²	1	4	yes	—
16	C₄×D₁₃⋊C₈	C₄	1	4	yes	complemented
17	C₄×D₁₃⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
20	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
22	C₄×C₈×D₁₃	C₂²	1	4	yes	—
23	(C₂×#261)⋊C₈	C₂²	1	16	yes	—
24	C₂³×#261	C₂×C₄	1	4	yes	—
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1057	C₂×C₄	1	4	yes	—
29	C₂×C₄×#261	—	2	8	no	—
30	C₂×C₄×#261	—	2	8	no	—
31	C₂×C₄×#261	C₈	1	16	yes	complemented
32	C₂×C₄×#261	C₂×C₄	1	4	yes	—
33	C₂×C₄×#261	C₂×C₄	1	8	yes	—
34	C₂×C₄×#261	C₂×C₄	1	8	yes	—
35	C₂×C₄×#261	C₂³	1	4	yes	—
36	C₂×D₁₃⋊C₈	—	2	8	no	—
37	C₂×D₁₃⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₁₃	—	2	8	no	—
46	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
47	C₂×C₈×D₁₃	C₂³	1	4	yes	—
48	C₄²×D₁₃	C₂×C₄	1	2	yes	—
49	C₄²×D₁₃	C₂×C₄	1	2	yes	—
50	C₄²×D₁₃	C₂×C₄	1	2	yes	—
51	C₄²×D₁₃	C₂³	1	2	yes	—
52	C₄×#1057	C₂×C₄	1	8	yes	complemented
53	C₄×#1057	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
55	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
56	C₄×#1055	C₂×C₄	1	4	yes	complemented
57	C₂×#1744	—	13	1	no	maximal, Sylow
58	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×#261	—	2	8	no	—
60	C₂²×#261	—	2	8	no	—
61	C₂²×#261	C₂×C₈	1	16	yes	—
62	C₂²×#261	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₄²	1	2	yes	—
70	C₂×C₄×D₁₃	C₄²	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	8	yes	—
72	C₂×C₄×D₁₃	M₁₆	1	8	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₁₃	#41	1	8	yes	—
79	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
80	C₂×#1057	—	2	8	no	—
81	C₂×#1057	C₂²×C₄	1	8	yes	—
82	C₄×#261	—	2	8	no	—
83	C₄×#261	—	2	16	no	—
84	C₄×#261	—	2	8	no	—
85	C₄×#261	C₂×C₈	1	32	yes	complemented
86	D₁₃⋊C₈	—	2	8	no	—
87	D₁₃⋊C₈	—	2	8	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₃	C₄²	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₁₃	C₄²	1	2	yes	—
93	C₄×Dic₁₃	C₄²	1	2	yes	—
94	C₄×Dic₁₃	C₄²	1	2	yes	—
95	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	C₂²×C₄	1	4	yes	—
100	C₈×D₁₃	—	2	8	no	—
101	C₈×D₁₃	—	2	8	no	—
102	C₂²×C₄×C₈	—	13	1	no	—
103	C₂²×C₄×C₈	—	13	1	no	—
104	C₂×C₄³	—	13	1	no	—
105	#1744	—	13	8	no	—
106	C₂×#336	—	13	4	no	—
107	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂²×D₁₃	C₂×C₄²	1	1	yes	characteristic
109	C₂²×D₁₃	#117	1	4	yes	—
110	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
111	C₂²×D₁₃	C₂×M₁₆	1	4	yes	—
112	C₂²×D₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×#261	—	2	8	no	—
114	C₂×#261	—	2	8	no	—
115	C₂×#261	—	2	32	no	—
116	C₂×#261	C₂²×C₈	1	16	yes	—
117	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
118	C₂×C₄×C₁₃	#97	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₁₃	#117	1	4	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
125	C₂×Dic₁₃	#97	1	4	yes	—
126	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
128	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
129	C₂×Dic₁₃	#117	1	4	yes	—
130	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₄×D₁₃	C₄×C₈	1	8	yes	—
132	C₄×D₁₃	#97	1	8	yes	—
133	C₄×D₁₃	#117	1	4	yes	—
134	C₄×D₁₃	#117	1	4	yes	—
135	C₄×D₁₃	#117	1	4	yes	—
136	C₄×D₁₃	C₂²×C₈	1	8	yes	—
137	C₄×D₁₃	C₂×M₁₆	1	8	yes	—
138	C₄×D₁₃	C₂×#41	1	4	yes	—
139	#1057	—	2	8	no	—
140	C₈×C₁₃	—	2	4	no	—
141	#1055	—	2	4	no	—
142	C₂³×C₈	—	13	2	no	—
143	C₂³×C₈	—	13	2	no	—
144	C₂²×C₄²	—	13	2	no	—
145	C₂²×C₄²	—	13	1	no	—
146	C₂²×C₄²	—	13	2	no	—
147	C₂²×C₄²	—	13	2	no	—
148	C₂×C₄×C₈	—	13	4	no	—
149	C₂×C₄×C₈	—	13	4	no	—
150	C₂×C₄×C₈	—	13	2	no	—
151	C₂×C₄×C₈	—	13	2	no	—
152	C₂×C₄×C₈	—	13	2	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	4	no	—
155	C₂×C₄×C₈	—	13	4	no	—
156	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
157	C₄³	—	13	4	no	—
158	C₄³	—	13	4	no	—
159	#336	—	13	16	no	—
160	C₂²×C₁₃	#376	1	2	yes	—
161	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
162	C₂²×C₁₃	C₂×#97	1	2	yes	—
163	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
164	C₂×D₁₃	#376	1	2	yes	—
165	C₂×D₁₃	C₂×C₄×C₈	1	2	yes	—
166	C₂×D₁₃	C₂×#97	1	2	yes	—
167	C₂×D₁₃	#431	1	16	yes	—
168	C₂×D₁₃	C₂×#117	1	2	yes	—
169	C₂×D₁₃	C₂³×C₈	1	2	yes	—
170	C₂×D₁₃	C₂²×M₁₆	1	2	yes	—
171	#261	—	2	16	no	—
172	C₄×C₁₃	#376	1	2	yes	—
173	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
174	C₄×C₁₃	C₂×#97	1	4	yes	—
175	C₄×C₁₃	C₂×#117	1	2	yes	—
176	Dic₁₃	#376	1	2	yes	—
177	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
178	Dic₁₃	C₂×#97	1	4	yes	—
179	Dic₁₃	C₂×#117	1	2	yes	—
180	C₂³×C₄	—	13	1	no	—
181	C₂³×C₄	—	13	1	no	—
182	C₂³×C₄	—	13	1	no	—
183	C₂³×C₄	—	13	4	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	4	no	—
189	C₂²×C₈	—	13	4	no	—
190	C₂²×C₈	—	26	8	no	—
191	C₂²×C₈	—	26	8	no	—
192	C₂²×C₈	C₂×#261	1	2	yes	—
193	C₂×C₄²	—	13	8	no	—
194	C₂×C₄²	—	13	16	no	—
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₄²	—	13	1	no	—
197	C₂×C₄²	—	13	1	no	—
198	C₂×C₄²	—	13	2	no	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	13	4	no	—
203	C₂×C₄²	—	13	4	no	—
204	C₂×C₄²	—	13	8	no	—
205	C₂×C₄²	—	26	8	no	—
206	C₂×C₄²	—	26	8	no	—
207	C₂×C₄²	C₂×#261	1	1	yes	characteristic
208	C₄×C₈	—	13	8	no	—
209	C₄×C₈	—	13	8	no	—
210	C₄×C₈	—	13	8	no	—
211	C₄×C₈	—	13	4	no	—
212	C₄×C₈	C₂×#261	1	4	yes	complemented
213	C₂×C₁₃	#1744	1	4	yes	complemented
214	C₂×C₁₃	C₂×#376	1	1	yes	characteristic
215	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
216	C₂×C₁₃	C₂²×#97	1	1	yes	characteristic
217	D₁₃	#1744	1	4	yes	complemented
218	D₁₃	C₂×#431	1	4	yes	—
219	C₂⁴	—	13	1	no	—
220	C₂²×C₄	—	13	1	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	8	no	—
224	C₂²×C₄	—	13	8	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	2	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	8	no	—
232	C₂²×C₄	—	13	1	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	8	no	—
235	C₂²×C₄	—	13	16	no	—
236	C₂²×C₄	—	26	8	no	—
237	C₂²×C₄	—	26	8	no	—
238	C₂²×C₄	C₄×#261	1	1	yes	characteristic
239	C₂²×C₄	C₄×#261	1	1	yes	characteristic
240	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	13	8	no	—
243	C₂×C₈	—	13	4	no	—
244	C₂×C₈	—	26	8	no	—
245	C₂×C₈	—	26	8	no	—
246	C₂×C₈	—	26	8	no	—
247	C₂×C₈	—	26	8	no	—
248	C₂×C₈	—	26	8	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	C₂²×#261	1	4	yes	—
251	C₄²	—	13	32	no	—
252	C₄²	—	13	2	no	—
253	C₄²	—	13	2	no	—
254	C₄²	—	13	2	no	—
255	C₄²	—	13	2	no	—
256	C₄²	—	13	2	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	26	8	no	—
259	C₄²	—	26	16	no	—
260	C₄²	—	26	8	no	—
261	C₄²	C₄×#261	1	2	yes	—
262	C₄²	C₂²×#261	1	2	yes	—
263	C₁₃	C₂×#1744	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	13	4	no	—
265	C₂³	—	13	4	no	—
266	C₂³	—	13	4	no	—
267	C₂³	—	13	1	no	—
268	C₂³	—	13	1	no	—
269	C₂³	C₂×C₄×#261	1	1	yes	characteristic
270	C₂×C₄	—	13	4	no	—
271	C₂×C₄	—	13	4	no	—
272	C₂×C₄	—	13	4	no	—
273	C₂×C₄	—	13	1	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	2	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	4	no	—
278	C₂×C₄	—	13	4	no	—
279	C₂×C₄	—	13	8	no	—
280	C₂×C₄	—	13	8	no	—
281	C₂×C₄	—	13	8	no	—
282	C₂×C₄	—	13	8	no	—
283	C₂×C₄	—	13	4	no	—
284	C₂×C₄	—	13	4	no	—
285	C₂×C₄	—	13	16	no	—
286	C₂×C₄	—	26	8	no	—
287	C₂×C₄	—	26	8	no	—
288	C₂×C₄	—	26	32	no	—
289	C₂×C₄	C₈×#261	1	4	yes	—
290	C₂×C₄	#88100	1	4	yes	—
291	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×#261	1	2	yes	—
293	C₂×C₄	C₂×C₄×#261	1	2	yes	—
294	C₂×C₄	#88237	1	4	yes	—
295	C₂×C₄	C₂³×#261	1	1	yes	characteristic
296	C₈	—	2	4	no	—
297	C₈	—	26	8	no	—
298	C₈	—	26	4	no	—
299	C₂²	—	13	16	no	—
300	C₂²	—	13	2	no	—
301	C₂²	—	13	2	no	—
302	C₂²	—	13	2	no	—
303	C₂²	—	13	2	no	—
304	C₂²	—	13	2	no	—
305	C₂²	—	13	2	no	—
306	C₂²	#11719057	1	2	yes	—
307	C₂²	C₂×C₈×#261	1	2	yes	—
308	C₂²	C₂×#88100	1	2	yes	—
309	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
310	C₄	—	13	4	no	—
311	C₄	—	13	4	no	—
312	C₄	—	13	2	no	—
313	C₄	—	13	2	no	—
314	C₄	—	26	16	no	—
315	C₄	#11719057	1	2	yes	—
316	C₄	C₂×C₈×#261	1	4	yes	—
317	C₄	C₂×#88100	1	4	yes	—
318	C₄	C₂×#88237	1	2	yes	—
319	C₂	—	13	4	no	—
320	C₂	—	13	4	no	—
321	C₂	#13130549	1	4	yes	direct product factor
322	C₂	C₂×#11719057	1	1	yes	characteristic
323	C₂	C₂²×C₈×#261	1	1	yes	characteristic
324	C₂	C₂²×#88100	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1744
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13130549	C₂	direct (N × H)
12. C₂×C₄×#1057	C₄	semidirect
13. C₂×C₄×#1057	C₄	semidirect
16. C₄×D₁₃⋊C₈	C₄	semidirect
18. C₂×C₄×C₈×C₁₃	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
20. C₂×C₄×#1055	C₄	semidirect
21. C₄×C₈×D₁₃	C₄	semidirect
31. C₂×C₄×#261	C₈	semidirect
52. C₄×#1057	C₂×C₄	semidirect
53. C₄×#1057	C₂×C₄	semidirect
54. C₄×C₈×C₁₃	C₂×C₄	semidirect
55. C₈×Dic₁₃	C₂×C₄	semidirect
56. C₄×#1055	C₂×C₄	semidirect
85. C₄×#261	C₂×C₈	semidirect
156. C₂×C₄×C₈	#261	semidirect
212. C₄×C₈	C₂×#261	semidirect
213. C₂×C₁₃	#1744	semidirect
217. D₁₃	#1744	semidirect
263. C₁₃	C₂×#1744	semidirect
321. C₂	#13130549	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×C₄×D₁₃⋊C₈	2
C₂×C₄²×#261	2
#13130549	2
C₂×(C₂×#261)⋊C₈	2
C₂×C₄×C₈×D₁₃	2
C₂×#1744	13

Subgroups — C2×#13130549 #16796832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130549 #16796832