Subgroups — C₂×#13115006 #15863373

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88059	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88062	C₂	1	2	yes	maximal
3	C₂×#11718779	C₂	1	2	yes	maximal
4	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#13115006	C₂	1	8	yes	maximal, direct product factor
6	C₂²×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	—	2	4	no	—
13	C₂×C₁₃×#108	C₂²	1	4	yes	—
14	C₂×#88062	—	2	8	no	—
15	C₂×#88062	C₂²	1	8	yes	—
16	#11718779	—	2	4	no	—
17	#11718779	C₂²	1	8	yes	—
18	C₂×#88059	—	2	4	no	—
19	C₂×#88059	C₂²	1	4	yes	—
20	C₂×#88054	—	2	4	no	—
21	C₂×C₈×Dic₁₃	C₄	1	4	yes	—
22	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
23	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
24	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₃	Q₈	1	2	yes	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	—	2	4	no	—
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	D₄	1	4	yes	—
35	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
41	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
42	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
43	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	4	no	—
46	C₂×#6099	—	2	4	no	—
47	C₂×#6099	C₂³	1	4	yes	—
48	C₂²×#1055	—	2	2	no	—
49	C₂²×#1055	C₂×C₄	1	2	yes	—
50	C₂²×#1055	C₂×C₄	1	2	yes	—
51	C₂²×#1055	D₄	1	4	yes	—
52	C₂²×#1055	Q₈	1	4	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#108	—	2	8	no	—
55	#88062	—	2	16	no	—
56	#88059	—	2	8	no	—
57	#88054	—	2	16	no	—
58	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
60	C₂×#1848	—	13	1	no	maximal, Sylow
61	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	8	no	—
73	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
74	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
75	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
76	C₂²×Dic₁₃	C₂×Q₈	1	4	yes	—
77	C₁₃×M₁₆	—	2	4	no	—
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	—	4	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	4	4	no	—
82	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
83	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
87	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
93	#6099	—	2	4	no	—
94	#6099	—	2	4	no	—
95	#6099	—	4	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	4	4	no	—
98	C₂×#1055	C₄⋊C₄	1	8	yes	—
99	C₂×#1055	C₄⋊C₄	1	8	yes	—
100	C₂×#1055	C₂²×C₄	1	4	yes	—
101	C₂×#1055	C₂×D₄	1	4	yes	—
102	C₂×#1055	C₂×Q₈	1	4	yes	—
103	C₂²×#108	—	13	2	no	—
104	C₂²×#108	—	13	1	no	—
105	C₂²×#108	D₁₃	1	1	yes	characteristic
106	C₂×#423	—	13	2	no	—
107	C₂²×C₄×C₈	—	13	1	no	—
108	#1848	—	13	8	no	—
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
114	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#127	1	2	yes	—
122	C₂×C₄×C₁₃	#127	1	2	yes	—
123	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	4	no	—
134	C₂×Dic₁₃	—	2	8	no	—
135	C₂×Dic₁₃	—	2	8	no	—
136	C₂×Dic₁₃	#108	1	8	yes	complemented
137	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₈×C₁₃	—	4	4	no	—
140	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
141	#1055	—	4	4	no	—
142	#1055	C₂×C₄⋊C₄	1	8	yes	—
143	C₂²×M₁₆	—	13	2	no	—
144	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
145	C₂³×C₈	—	13	1	no	—
146	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
147	C₂²×C₄²	—	13	1	no	—
148	C₂×#108	—	2	4	no	—
149	C₂×#108	—	13	8	no	—
150	C₂×#108	—	13	4	no	—
151	C₂×#108	—	26	8	no	—
152	C₂×#108	—	26	4	no	—
153	C₂×#108	C₂×D₁₃	1	4	yes	—
154	#423	—	13	8	no	—
155	#423	—	26	4	no	—
156	C₂×C₄⋊C₈	—	26	4	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	C₂×C₄×C₈	—	13	4	no	—
159	C₂×C₄×C₈	—	13	2	no	—
160	C₂×C₄×C₈	—	13	4	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	4	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	#384	1	2	yes	—
166	C₂²×C₁₃	#384	1	2	yes	—
167	C₂²×C₁₃	C₂×#108	1	4	yes	—
168	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₄×C₁₃	#384	1	2	yes	—
172	C₄×C₁₃	#384	1	2	yes	—
173	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
176	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	Dic₁₃	—	2	4	no	—
178	Dic₁₃	C₂×#108	1	8	yes	complemented
179	C₂³×C₄	—	13	2	no	—
180	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	13	4	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	4	no	—
187	C₂×M₁₆	—	26	4	no	—
188	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	4	no	—
194	C₂²×C₈	—	13	2	no	—
195	C₂²×C₈	—	26	2	no	—
196	C₂²×C₈	Dic₂₆	1	4	yes	—
197	C₂²×C₈	C₄×D₁₃	1	2	yes	—
198	C₂²×C₈	C₄×D₁₃	1	2	yes	—
199	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
200	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
201	C₂×C₄²	—	13	1	no	—
202	C₂×C₄²	—	13	2	no	—
203	C₂×C₄²	—	13	1	no	—
204	C₂×C₄²	—	13	2	no	—
205	C₂×C₄²	—	13	2	no	—
206	C₂×C₄²	—	13	2	no	—
207	C₂×C₄²	—	13	2	no	—
208	#108	—	2	8	no	—
209	#108	—	26	16	no	—
210	#108	—	26	8	no	—
211	C₄⋊C₈	—	26	16	no	—
212	C₄×C₈	—	13	8	no	—
213	C₄×C₈	—	13	8	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	#1848	1	4	yes	complemented
218	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
219	C₂×C₁₃	C₂²×#108	1	2	yes	—
220	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	4	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	4	no	—
225	C₂²×C₄	—	26	8	no	—
226	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₁₃	1	2	yes	—
231	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
232	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
233	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
235	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	4	4	no	—
239	M₁₆	—	26	4	no	—
240	M₁₆	—	26	4	no	—
241	M₁₆	—	52	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	13	8	no	—
246	C₂×C₈	—	13	4	no	—
247	C₂×C₈	—	13	4	no	—
248	C₂×C₈	—	13	4	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	52	4	no	—
251	C₂×C₈	#6101	1	8	yes	—
252	C₂×C₈	#6101	1	8	yes	—
253	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
254	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
255	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
256	C₄²	—	13	2	no	—
257	C₄²	—	13	4	no	—
258	C₄²	—	13	2	no	—
259	C₄²	—	13	2	no	—
260	C₄²	—	13	2	no	—
261	C₄²	—	13	4	no	—
262	C₁₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	2	4	no	—
264	C₂³	#88142	1	2	yes	—
265	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
266	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
267	C₂³	#88198	1	2	yes	—
268	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
269	C₂×C₄	—	13	4	no	—
270	C₂×C₄	—	13	8	no	—
271	C₂×C₄	—	13	4	no	—
272	C₂×C₄	—	26	8	no	—
273	C₂×C₄	—	26	8	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	#88142	1	2	yes	—
276	C₂×C₄	#88142	1	2	yes	—
277	C₂×C₄	#88142	1	2	yes	—
278	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
279	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
281	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#6101	1	2	yes	—
283	C₂×C₄	C₂×#6101	1	2	yes	—
284	C₂×C₄	#88194	1	2	yes	—
285	C₂×C₄	#88194	1	2	yes	—
286	C₂×C₄	#88198	1	2	yes	—
287	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₈	—	13	8	no	—
295	C₈	—	52	4	no	—
296	C₈	C₂×#6101	1	8	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	D₁₃×#108	1	4	yes	—
302	C₂²	#11718637	1	2	yes	—
303	C₂²	#11718637	1	2	yes	—
304	C₂²	C₂×#88142	1	1	yes	characteristic
305	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
306	C₂²	C₂×#88198	1	1	yes	characteristic
307	C₄	—	13	8	no	—
308	C₄	—	26	4	no	—
309	C₄	#11718637	1	2	yes	—
310	C₄	#11718637	1	2	yes	—
311	C₄	C₂×#88142	1	1	yes	characteristic
312	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
313	C₄	C₂²×#6101	1	1	yes	characteristic
314	C₄	C₂×#88194	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	#13115006	1	4	yes	direct product factor
319	C₂	C₂×#11718637	1	1	yes	characteristic
320	C₂	C₂×D₁₃×#108	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#1848
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #13115006	C₂	direct (N × H)
136. C₂×Dic₁₃	#108	semidirect
178. Dic₁₃	C₂×#108	semidirect
217. C₂×C₁₃	#1848	semidirect
262. C₁₃	C₂×#1848	semidirect
318. C₂	#13115006	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88059	2
C₂²×#88062	2
C₂×#11718779	2
C₂²×C₈×Dic₁₃	2
#13115006	2
C₂²×C₁₃×#108	2
C₂×#1848	13

Subgroups — C2×#13115006 #15863373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13115006 #15863373