Subgroups — C₂×#13114718 #15860928

Order 3328.

1102 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×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718783	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13114718	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718147	C₂	1	2	yes	maximal
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	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂³×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#108	C₄	1	8	yes	complemented
13	C₂×C₁₃×#108	C₂²	1	4	yes	—
14	#11718783	—	2	4	no	—
15	#11718783	C₂²	1	8	yes	—
16	#11718767	C₄	1	4	yes	complemented
17	#11718767	C₂²	1	4	yes	—
18	C₂×#88058	C₄	1	2	yes	complemented
19	C₂×#88057	C₄	1	2	yes	complemented
20	C₂×#88055	—	2	2	no	—
21	C₂×C₈×Dic₁₃	—	2	2	no	—
22	C₂×C₈×Dic₁₃	C₄	1	8	yes	complemented
23	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
24	#11718147	C₂²	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂³×Dic₁₃	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
39	#88180	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	C₂×C₄	1	4	yes	—
42	#88180	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
49	C₂²×#1055	C₂×C₄	1	4	yes	—
50	C₂²×#1055	D₄	1	1	yes	characteristic
51	C₂²×#1055	D₄	1	1	yes	characteristic
52	C₂²×#1055	D₄	1	2	yes	—
53	C₂²×#1055	Q₈	1	4	yes	—
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₁₃×#108	C₂×C₄	1	16	yes	complemented
56	#88058	C₂×C₄	1	8	yes	complemented
57	#88057	C₂×C₄	1	8	yes	complemented
58	#88055	—	2	8	no	—
59	C₈×Dic₁₃	—	2	8	no	—
60	C₈×Dic₁₃	C₂×C₄	1	16	yes	complemented
61	C₂×#1772	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	8	no	—
74	C₂²×Dic₁₃	—	2	4	no	—
75	C₂²×Dic₁₃	—	2	2	no	—
76	C₂²×Dic₁₃	—	2	4	no	—
77	C₁₃×M₁₆	—	2	8	no	—
78	C₁₃×M₁₆	—	2	4	no	—
79	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
80	C₂×C₈×C₁₃	—	2	8	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	C₄²	1	4	yes	—
83	C₂×C₈×C₁₃	C₄²	1	4	yes	—
84	C₂×C₈×C₁₃	C₄²	1	8	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂²⋊Dic₁₃	—	2	4	no	—
90	C₂²⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	—	2	4	no	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
98	C₂×#1055	C₄⋊C₄	1	4	yes	—
99	C₂×#1055	C₄⋊C₄	1	4	yes	—
100	C₂×#1055	C₄⋊C₄	1	8	yes	—
101	C₂×#1055	C₂²×C₄	1	4	yes	—
102	C₂×#1055	C₂×D₄	1	2	yes	—
103	C₂×#1055	C₂×D₄	1	2	yes	—
104	C₂×#1055	C₂×Q₈	1	4	yes	—
105	C₂²×#108	D₁₃	1	1	yes	characteristic
106	C₂×#427	—	13	1	no	—
107	C₂×#405	—	13	2	no	—
108	C₂²×C₄×C₈	—	13	1	no	—
109	#1772	—	13	8	no	—
110	C₂×#339	—	13	2	no	—
111	C₂³×C₁₃	—	2	4	no	—
112	C₂³×C₁₃	#117	1	2	yes	—
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
116	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	#117	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₁₃	—	2	8	no	—
136	C₂×Dic₁₃	—	2	16	no	—
137	C₂×Dic₁₃	—	2	4	no	—
138	C₂×Dic₁₃	—	2	8	no	—
139	C₂×Dic₁₃	—	4	8	no	—
140	C₈×C₁₃	—	2	8	no	—
141	C₈×C₁₃	C₂×C₄²	1	4	yes	—
142	C₈×C₁₃	C₂×C₄²	1	4	yes	—
143	#1055	C₂×C₄⋊C₄	1	4	yes	—
144	#1055	C₂×C₄⋊C₄	1	4	yes	—
145	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
146	C₂³×C₈	—	13	1	no	—
147	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂×#117	—	13	2	no	—
149	C₂²×C₄²	—	13	1	no	—
150	C₂×#108	Dic₁₃	1	8	yes	complemented
151	C₂×#108	C₂×D₁₃	1	4	yes	—
152	#427	—	13	4	no	—
153	#427	—	13	4	no	—
154	C₂×#107	—	13	2	no	—
155	C₂×#106	—	13	2	no	—
156	#405	—	13	8	no	—
157	#405	—	26	4	no	—
158	C₂×#97	—	26	2	no	—
159	C₂×C₄×C₈	—	13	8	no	—
160	C₂×C₄×C₈	—	13	4	no	—
161	C₂×C₄×C₈	—	26	2	no	—
162	#339	—	13	8	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	4	no	—
166	C₂²×C₁₃	—	2	4	no	—
167	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
168	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
169	C₂²×C₁₃	#443	1	4	yes	—
170	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
171	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
173	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
174	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
175	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
176	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
179	Dic₁₃	—	2	8	no	—
180	Dic₁₃	—	4	8	no	—
181	C₂³×C₄	—	26	1	no	—
182	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
186	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	13	4	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	1	no	—
191	C₂²×C₈	—	13	1	no	—
192	C₂²×C₈	—	13	4	no	—
193	C₂²×C₈	—	13	2	no	—
194	C₂²×C₈	C₄×D₁₃	1	4	yes	—
195	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₁₃	1	2	yes	—
198	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
199	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
200	#117	—	13	4	no	—
201	#117	—	13	4	no	—
202	#117	—	26	2	no	—
203	#117	—	26	2	no	—
204	C₂×C₄⋊C₄	—	13	4	no	—
205	C₂×C₂²⋊C₄	—	26	2	no	—
206	C₂×C₄²	—	13	4	no	—
207	C₂×C₄²	—	13	2	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	—	26	2	no	—
210	C₂×C₄²	—	26	1	no	—
211	#108	C₂×Dic₁₃	1	16	yes	complemented
212	#107	—	13	8	no	—
213	#106	—	13	8	no	—
214	#97	—	26	8	no	—
215	C₄×C₈	—	13	16	no	—
216	C₄×C₈	—	26	8	no	—
217	C₂×C₁₃	—	2	1	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	#1772	1	4	yes	complemented
221	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#443	1	2	yes	—
223	C₂⁴	#6131	1	1	yes	characteristic
224	C₂²×C₄	—	26	8	no	—
225	C₂²×C₄	—	26	4	no	—
226	C₂²×C₄	—	26	2	no	—
227	C₂²×C₄	—	26	4	no	—
228	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
231	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
233	C₂²×C₄	#6129	1	2	yes	—
234	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
235	C₂²×C₄	#6131	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
237	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
238	M₁₆	—	2	8	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	C₂²×Dic₁₃	1	8	yes	—
241	C₂×C₈	—	2	8	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	13	8	no	—
244	C₂×C₈	—	13	4	no	—
245	C₂×C₈	—	13	4	no	—
246	C₂×C₈	—	13	4	no	—
247	C₂×C₈	—	13	2	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	4	no	—
250	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
251	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
252	C₂×C₈	C₄×Dic₁₃	1	8	yes	—
253	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
254	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
255	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
256	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
257	C₄⋊C₄	—	13	8	no	—
258	C₄⋊C₄	—	26	4	no	—
259	C₂²⋊C₄	—	26	4	no	—
260	C₂²⋊C₄	—	26	4	no	—
261	C₄²	—	13	4	no	—
262	C₄²	—	26	4	no	—
263	C₄²	—	26	4	no	—
264	C₄²	—	26	4	no	—
265	C₄²	—	26	2	no	—
266	C₁₃	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	2	4	no	—
268	C₂³	#88146	1	2	yes	—
269	C₂³	#88147	1	1	yes	characteristic
270	C₂³	#88147	1	1	yes	characteristic
271	C₂³	Q₈×Dic₁₃	1	2	yes	—
272	C₂³	C₂×#6131	1	1	yes	characteristic
273	C₂×C₄	—	26	16	no	—
274	C₂×C₄	—	26	8	no	—
275	C₂×C₄	—	26	4	no	—
276	C₂×C₄	—	26	8	no	—
277	C₂×C₄	—	52	8	no	—
278	C₂×C₄	#88141	1	2	yes	—
279	C₂×C₄	#88141	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
281	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
282	C₂×C₄	#88146	1	2	yes	—
283	C₂×C₄	#88147	1	1	yes	characteristic
284	C₂×C₄	#88147	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
287	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
288	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
289	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂×#6129	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
296	C₈	—	2	8	no	—
297	C₈	—	13	4	no	—
298	C₈	—	13	4	no	—
299	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
300	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	#11718556	1	4	yes	—
306	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
307	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
308	C₂²	C₂×#88146	1	1	yes	characteristic
309	C₂²	C₂×#88147	1	1	yes	characteristic
310	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
311	C₄	—	26	8	no	—
312	C₄	—	52	8	no	—
313	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
314	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
315	C₄	C₂×#88141	1	1	yes	characteristic
316	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
317	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
318	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	#13114718	1	4	yes	direct product factor
323	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
324	C₂	C₂×#11718556	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×#108	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂²×#108	D₁₃
hypocenter	C₁₃	C₂×#1772
LCS	G ▷ 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
4. #13114718	C₂	direct (N × H)
12. C₂×C₁₃×#108	C₄	semidirect
16. #11718767	C₄	semidirect
18. C₂×#88058	C₄	semidirect
19. C₂×#88057	C₄	semidirect
22. C₂×C₈×Dic₁₃	C₄	semidirect
55. C₁₃×#108	C₂×C₄	semidirect
56. #88058	C₂×C₄	semidirect
57. #88057	C₂×C₄	semidirect
60. C₈×Dic₁₃	C₂×C₄	semidirect
150. C₂×#108	Dic₁₃	semidirect
211. #108	C₂×Dic₁₃	semidirect
220. C₂×C₁₃	#1772	semidirect
266. C₁₃	C₂×#1772	semidirect
322. C₂	#13114718	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×#11718767	2
C₂×#11718783	2
C₂²×C₈×Dic₁₃	2
#13114718	2
C₂×#11718147	2
C₂²×C₁₃×#108	2
C₂×#1772	13

Subgroups — C2×#13114718 #15860928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114718 #15860928