Subgroups — C₂×#13114591 #15860422

Order 3328.

930 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718762	C₂	1	2	yes	maximal
2	#13114591	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718152	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718149	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718123	C₂	1	2	yes	maximal
6	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₁₃×#405	C₂²	1	4	yes	—
14	C₂×C₁₃×#97	C₄	1	2	yes	complemented
15	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	complemented
16	#11718762	C₄	1	8	yes	complemented
17	#11718762	C₂²	1	8	yes	—
18	C₂×#88055	C₄	1	4	yes	complemented
19	C₂×C₈×Dic₁₃	C₄	1	4	yes	complemented
20	#11718152	C₂²	1	4	yes	—
21	#11718149	C₂²	1	4	yes	—
22	#11718123	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₁₃×#117	—	2	1	no	—
37	C₁₃×#117	—	2	1	no	—
38	C₁₃×#117	C₂×C₄	1	2	yes	—
39	C₁₃×#117	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
44	#88180	—	2	2	no	—
45	#88180	—	2	2	no	—
46	#88180	C₂×C₄	1	4	yes	—
47	#88180	C₂³	1	4	yes	—
48	C₂×C₄⋊Dic₁₃	—	2	2	no	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	2	no	—
52	C₂×#6099	—	2	2	no	—
53	C₂×#6099	—	2	2	no	—
54	C₂×#6099	C₂×C₄	1	4	yes	—
55	C₂×#6099	C₂³	1	4	yes	—
56	C₂²×#1055	C₂×C₄	1	4	yes	—
57	C₁₃×#97	C₂×C₄	1	8	yes	complemented
58	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
59	#88055	C₂×C₄	1	16	yes	complemented
60	C₈×Dic₁₃	C₂×C₄	1	16	yes	complemented
61	C₂×#1747	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₄²	1	2	yes	—
80	C₂×C₈×C₁₃	C₄²	1	4	yes	—
81	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₄²	1	4	yes	—
83	C₂×C₈×C₁₃	C₄²	1	2	yes	—
84	C₂×C₈×C₁₃	C₄²	1	2	yes	—
85	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
97	C₂²⋊Dic₁₃	—	2	4	no	—
98	C₂²⋊Dic₁₃	—	2	4	no	—
99	C₄⋊Dic₁₃	—	2	4	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄×Dic₁₃	—	2	4	no	—
102	C₄×Dic₁₃	C₂²×C₄	1	8	yes	—
103	#6099	—	2	4	no	—
104	#6099	—	2	4	no	—
105	#6099	—	2	4	no	—
106	#6099	—	2	4	no	—
107	C₂×#1055	—	2	4	no	—
108	C₂×#1055	—	2	4	no	—
109	C₂×#1055	C₂²×C₄	1	8	yes	—
110	C₂×#405	—	13	2	no	—
111	C₂×#405	D₁₃	1	1	yes	characteristic
112	#1747	—	13	8	no	—
113	C₂×#338	—	13	1	no	—
114	C₂×H(3,ℤ/4)	—	13	1	no	—
115	C₂×H(3,ℤ/4)	—	13	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	#117	1	2	yes	—
118	C₂³×C₁₃	#117	1	1	yes	characteristic
119	C₂³×C₁₃	#117	1	2	yes	—
120	C₂³×C₁₃	#117	1	1	yes	characteristic
121	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₃	—	2	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	2	yes	—
128	C₂×C₄×C₁₃	#117	1	2	yes	—
129	C₂×C₄×C₁₃	#117	1	2	yes	—
130	C₂×C₄×C₁₃	#117	1	2	yes	—
131	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	#117	1	2	yes	—
134	C₂×C₄×C₁₃	#117	1	2	yes	—
135	C₂×C₄×C₁₃	#117	1	2	yes	—
136	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×Dic₁₃	—	2	8	no	—
143	C₂×Dic₁₃	—	4	8	no	—
144	C₈×C₁₃	—	2	4	no	—
145	C₈×C₁₃	C₂×C₄²	1	4	yes	—
146	C₈×C₁₃	C₂×C₄²	1	2	yes	—
147	C₈×C₁₃	C₂×C₄²	1	2	yes	—
148	#1055	—	2	8	no	—
149	C₂²×M₁₆	—	13	2	no	—
150	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₂×#117	—	13	2	no	—
153	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
154	#405	—	13	8	no	—
155	#405	—	13	8	no	—
156	#405	Dic₁₃	1	4	yes	complemented
157	#405	C₂×D₁₃	1	4	yes	—
158	C₂×#97	—	13	4	no	—
159	C₂×#97	Dic₁₃	1	2	yes	complemented
160	C₂×C₄×C₈	—	13	4	no	—
161	C₂×C₄×C₈	Dic₁₃	1	2	yes	complemented
162	#338	—	13	4	no	—
163	H(3,ℤ/4)	—	13	4	no	—
164	H(3,ℤ/4)	—	13	8	no	—
165	C₂²×C₁₃	—	2	4	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	4	2	no	—
168	C₂²×C₁₃	—	4	1	no	—
169	C₂²×C₁₃	—	4	1	no	—
170	C₂²×C₁₃	#376	1	2	yes	—
171	C₂²×C₁₃	#376	1	1	yes	characteristic
172	C₂²×C₁₃	#376	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×#117	1	2	yes	—
174	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	#376	1	1	yes	characteristic
177	C₄×C₁₃	#376	1	2	yes	—
178	C₄×C₁₃	#376	1	1	yes	characteristic
179	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
180	C₄×C₁₃	C₂×#117	1	2	yes	—
181	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
182	Dic₁₃	—	4	8	no	—
183	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	13	4	no	—
187	C₂×M₁₆	—	13	4	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	2	no	—
190	C₂×M₁₆	—	26	2	no	—
191	C₂×M₁₆	—	26	2	no	—
192	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
193	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
194	C₂²×C₈	—	13	4	no	—
195	C₂²×C₈	C₄×D₁₃	1	2	yes	—
196	C₂²×C₈	C₄×D₁₃	1	4	yes	—
197	C₂²×C₈	C₄×D₁₃	1	2	yes	—
198	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
199	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
201	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
203	#117	—	2	1	no	—
204	#117	—	2	1	no	—
205	#117	—	13	4	no	—
206	#117	—	13	4	no	—
207	#117	—	26	2	no	—
208	#117	—	26	2	no	—
209	#117	C₂×Dic₁₃	1	2	yes	—
210	#117	C₂²×D₁₃	1	2	yes	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	26	2	no	—
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₂²⋊C₄	—	26	2	no	—
215	C₂×C₄²	—	13	4	no	—
216	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
217	#97	—	13	16	no	—
218	#97	C₂×Dic₁₃	1	8	yes	complemented
219	C₄×C₈	—	13	16	no	—
220	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
221	C₂×C₁₃	—	2	2	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	—	2	2	no	—
224	C₂×C₁₃	—	2	1	no	—
225	C₂×C₁₃	#1747	1	2	yes	complemented
226	C₂×C₁₃	C₂×#376	1	1	yes	characteristic
227	C₂⁴	#6127	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	26	4	no	—
230	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
232	C₂²×C₄	#6127	1	1	yes	characteristic
233	C₂²×C₄	#6127	1	1	yes	characteristic
234	C₂²×C₄	#6127	1	1	yes	characteristic
235	C₂²×C₄	#6127	1	2	yes	—
236	C₂²×C₄	#6127	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	26	4	no	—
243	M₁₆	—	26	4	no	—
244	M₁₆	—	26	4	no	—
245	M₁₆	—	26	4	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	13	8	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	4	no	—
251	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
252	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
253	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
254	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
255	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
256	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
257	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
259	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
260	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
261	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
262	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	26	4	no	—
266	C₄⋊C₄	—	26	4	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	26	4	no	—
270	C₂²⋊C₄	—	26	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	13	8	no	—
273	C₄²	—	26	4	no	—
274	C₄²	C₂²×Dic₁₃	1	4	yes	—
275	C₁₃	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	4	2	no	—
277	C₂³	#88126	1	2	yes	—
278	C₂³	#88126	1	2	yes	—
279	C₂³	#88180	1	1	yes	characteristic
280	C₂³	#88180	1	1	yes	characteristic
281	C₂³	C₂×#6127	1	1	yes	characteristic
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	26	8	no	—
285	C₂×C₄	—	52	8	no	—
286	C₂×C₄	#88126	1	2	yes	—
287	C₂×C₄	#88126	1	2	yes	—
288	C₂×C₄	#88126	1	2	yes	—
289	C₂×C₄	#88126	1	2	yes	—
290	C₂×C₄	#88126	1	2	yes	—
291	C₂×C₄	#88126	1	2	yes	—
292	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
293	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
295	C₂×C₄	#88180	1	1	yes	characteristic
296	C₂×C₄	#88180	1	1	yes	characteristic
297	C₂×C₄	#88180	1	2	yes	—
298	C₂×C₄	#88180	1	1	yes	characteristic
299	C₂×C₄	#88180	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
301	C₂×C₄	C₂×#6127	1	1	yes	characteristic
302	C₂×C₄	C₂×#6127	1	2	yes	—
303	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	26	8	no	—
306	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
307	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
308	C₈	C₂×C₄×Dic₁₃	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	#11718593	1	2	yes	—
315	C₂²	#11718593	1	1	yes	characteristic
316	C₂²	#11718593	1	1	yes	characteristic
317	C₂²	C₂×#88126	1	2	yes	—
318	C₂²	C₂×#88180	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	52	8	no	—
321	C₄	#11718593	1	1	yes	characteristic
322	C₄	#11718593	1	2	yes	—
323	C₄	#11718593	1	1	yes	characteristic
324	C₄	C₂×#88126	1	2	yes	—
325	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
326	C₄	C₂×#88180	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	#13114591	1	2	yes	direct product factor
332	C₂	C₂×#11718593	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	#376
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	C₂×#1747
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13114591	C₂	direct (N × H)
12. C₁₃×#405	C₄	semidirect
14. C₂×C₁₃×#97	C₄	semidirect
15. C₂×C₄×C₈×C₁₃	C₄	semidirect
16. #11718762	C₄	semidirect
18. C₂×#88055	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
57. C₁₃×#97	C₂×C₄	semidirect
58. C₄×C₈×C₁₃	C₂×C₄	semidirect
59. #88055	C₂×C₄	semidirect
60. C₈×Dic₁₃	C₂×C₄	semidirect
156. #405	Dic₁₃	semidirect
159. C₂×#97	Dic₁₃	semidirect
161. C₂×C₄×C₈	Dic₁₃	semidirect
218. #97	C₂×Dic₁₃	semidirect
220. C₄×C₈	C₂×Dic₁₃	semidirect
225. C₂×C₁₃	#1747	semidirect
275. C₁₃	C₂×#1747	semidirect
331. C₂	#13114591	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×#11718762	2
#13114591	2
C₂×#11718152	2
C₂×#11718149	2
C₂×#11718123	2
C₂×C₁₃×#405	2
C₂×#1747	13

Subgroups — C2×#13114591 #15860422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114591 #15860422