Subgroups — C₂×#13113626 #15832005

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718494	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718346	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718498	C₂	1	1	yes	maximal, characteristic
4	#13113626	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718071	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718072	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6094	—	2	1	no	—
10	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88060	—	2	1	no	—
15	C₂×#88060	C₄	1	1	yes	characteristic
16	C₂×#88060	C₄	1	1	yes	characteristic
17	C₂×#88054	—	2	1	no	—
18	C₂×#88054	C₄	1	1	yes	characteristic
19	C₂×#88054	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₁₃	—	2	1	no	—
21	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	#11718498	—	2	2	no	—
25	#11718498	C₂²	1	4	yes	—
26	#11718494	—	2	2	no	—
27	#11718494	C₂²	1	4	yes	—
28	#11718346	C₄	1	4	yes	—
29	#11718346	C₂²	1	4	yes	—
30	C₁₃×#335	C₂²	1	4	yes	complemented
31	#11718072	C₂²	1	4	yes	complemented
32	#11718071	C₂²	1	4	yes	complemented
33	#11718049	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	—	2	1	no	—
38	C₂²×C₈×C₁₃	—	2	1	no	—
39	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
45	C₂²×#1055	—	2	1	no	—
46	C₂²×#1055	—	2	1	no	—
47	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
48	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
49	C₂×#6094	—	2	4	no	—
50	C₂×#6094	—	4	4	no	—
51	C₄²×D₁₃	C₂×C₄	1	2	yes	—
52	C₄²×D₁₃	D₄	1	2	yes	—
53	C₄²×D₁₃	Q₈	1	2	yes	—
54	C₄²×D₁₃	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
58	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	2	yes	—
61	#88060	—	2	4	no	—
62	#88060	C₂×C₄	1	4	yes	—
63	#88060	C₂×C₄	1	4	yes	—
64	#88054	—	2	4	no	—
65	#88054	C₂×C₄	1	4	yes	—
66	#88054	C₂×C₄	1	4	yes	—
67	C₈×Dic₁₃	—	2	4	no	—
68	C₄⋊#1055	—	2	1	no	—
69	C₄⋊#1055	—	2	1	no	—
70	C₄⋊#1055	C₂³	1	2	yes	—
71	C₄⋊#1055	C₂³	1	2	yes	—
72	C₄×#1055	C₂×C₄	1	2	yes	—
73	C₄×#1055	C₂³	1	2	yes	—
74	C₂×#1574	—	13	1	no	maximal, Sylow
75	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₁₃	—	2	2	no	—
83	C₂×C₄×D₁₃	—	2	2	no	—
84	C₂×C₄×D₁₃	—	2	2	no	—
85	C₂×C₄×D₁₃	—	2	2	no	—
86	C₂×C₄×D₁₃	—	2	4	no	—
87	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
89	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
90	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	4	2	no	—
98	C₂×C₈×C₁₃	—	4	2	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
107	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
108	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
109	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
110	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	4	2	no	—
116	C₂×#1055	—	4	2	no	—
117	C₂×#1055	C₂²×C₄	1	2	yes	—
118	C₂×#1055	C₂²×C₄	1	2	yes	—
119	#6094	—	4	8	no	—
120	C₂×#431	—	13	1	no	—
121	C₂×#431	—	13	1	no	—
122	C₂×#423	—	13	1	no	—
123	#1574	—	13	8	no	—
124	C₂×#335	—	13	1	no	—
125	C₂×#335	D₁₃	1	1	yes	characteristic, complemented
126	C₂×#334	—	13	1	no	—
127	C₂×#334	—	13	1	no	—
128	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×D₁₃	—	2	2	no	—
130	C₂²×D₁₃	—	2	2	no	—
131	C₂²×D₁₃	—	2	2	no	—
132	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	#131	1	2	yes	—
138	C₂×C₄×C₁₃	#131	1	2	yes	—
139	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
140	C₂×C₄×C₁₃	#137	1	2	yes	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	#108	1	4	yes	—
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₄×D₁₃	—	2	2	no	—
151	C₄×D₁₃	—	2	2	no	—
152	C₄×D₁₃	—	2	2	no	—
153	C₄×D₁₃	—	2	2	no	—
154	C₄×D₁₃	—	2	4	no	—
155	C₄×D₁₃	—	2	4	no	—
156	C₄×D₁₃	—	2	4	no	—
157	C₄×D₁₃	—	2	4	no	—
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	—	4	2	no	—
162	#1055	—	2	2	no	—
163	#1055	—	2	2	no	—
164	#1055	—	4	2	no	—
165	#1055	—	4	2	no	—
166	C₂²×M₁₆	—	26	1	no	—
167	C₂²×C₄²	—	13	1	no	—
168	#431	—	13	4	no	—
169	#431	—	13	4	no	—
170	#431	—	13	4	no	—
171	#431	—	26	2	no	—
172	#423	—	13	4	no	—
173	#423	—	26	2	no	—
174	C₂×C₄⋊C₈	—	13	1	no	—
175	C₂×C₄⋊C₈	—	13	1	no	—
176	C₂×C₄⋊C₈	—	13	1	no	—
177	C₂×C₄⋊C₈	—	13	1	no	—
178	C₂×C₄⋊C₈	—	26	1	no	—
179	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₈	—	13	1	no	—
182	C₂×C₂²⋊C₈	—	13	1	no	—
183	C₂×C₂²⋊C₈	—	26	1	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	—	26	1	no	—
186	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
187	#335	—	13	4	no	—
188	#335	C₂×D₁₃	1	4	yes	complemented
189	#334	—	13	4	no	—
190	#334	—	13	4	no	—
191	C₂²×C₁₃	#424	1	2	yes	—
192	C₂²×C₁₃	#428	1	2	yes	—
193	C₂²×C₁₃	C₂×#108	1	2	yes	—
194	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
195	C₂×D₁₃	—	2	2	no	—
196	C₂×D₁₃	—	2	2	no	—
197	C₂×D₁₃	—	2	2	no	—
198	C₂×D₁₃	—	4	4	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	1	no	—
201	C₄×C₁₃	#424	1	2	yes	—
202	C₄×C₁₃	#428	1	2	yes	—
203	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
204	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
205	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
207	Dic₁₃	—	2	2	no	—
208	Dic₁₃	—	2	2	no	—
209	Dic₁₃	C₂×#108	1	4	yes	—
210	C₂³×C₄	—	13	1	no	—
211	C₂³×C₄	—	13	1	no	—
212	C₂³×C₄	—	13	1	no	—
213	C₂×M₁₆	—	26	4	no	—
214	C₂×M₁₆	—	52	4	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	13	1	no	—
218	C₂²×C₈	—	13	1	no	—
219	C₂²×C₈	—	26	1	no	—
220	C₂²×C₈	—	26	1	no	—
221	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	—	13	1	no	—
224	C₂×C₄²	—	13	1	no	—
225	C₂×C₄²	—	13	1	no	—
226	C₂×C₄²	—	13	2	no	—
227	C₂×C₄²	—	13	2	no	—
228	C₂×C₄²	—	13	2	no	—
229	C₂×C₄²	—	13	2	no	—
230	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	—	13	4	no	—
234	C₄⋊C₈	—	13	4	no	—
235	C₄⋊C₈	—	13	2	no	—
236	C₄⋊C₈	—	13	2	no	—
237	C₄⋊C₈	—	26	4	no	—
238	C₄⋊C₈	—	26	1	no	—
239	C₄⋊C₈	—	26	1	no	—
240	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
241	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
242	C₂²⋊C₈	—	13	4	no	—
243	C₂²⋊C₈	—	13	4	no	—
244	C₂²⋊C₈	—	26	4	no	—
245	C₄×C₈	—	13	2	no	—
246	C₄×C₈	—	13	2	no	—
247	C₄×C₈	—	26	4	no	—
248	C₄×C₈	C₄×D₁₃	1	2	yes	—
249	C₄×C₈	C₂²×D₁₃	1	2	yes	—
250	C₂×C₁₃	#1574	1	4	yes	complemented
251	C₂×C₁₃	C₂×#424	1	1	yes	characteristic
252	C₂×C₁₃	C₂×#428	1	1	yes	characteristic
253	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
254	D₁₃	—	4	2	no	—
255	C₂⁴	—	13	1	no	—
256	C₂²×C₄	—	13	1	no	—
257	C₂²×C₄	—	13	1	no	—
258	C₂²×C₄	—	13	1	no	—
259	C₂²×C₄	—	13	2	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	2	no	—
262	C₂²×C₄	—	13	2	no	—
263	C₂²×C₄	—	13	2	no	—
264	C₂²×C₄	—	13	2	no	—
265	C₂²×C₄	—	26	4	no	—
266	C₂²×C₄	—	26	2	no	—
267	C₂²×C₄	—	26	2	no	—
268	C₂²×C₄	—	26	2	no	—
269	C₂²×C₄	—	26	2	no	—
270	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
272	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
273	M₁₆	—	52	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	—	4	2	no	—
280	C₂×C₈	—	13	2	no	—
281	C₂×C₈	—	13	2	no	—
282	C₂×C₈	—	26	2	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	—	26	2	no	—
286	C₂×C₈	—	52	2	no	—
287	C₂×C₈	—	52	2	no	—
288	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
290	C₄²	—	13	2	no	—
291	C₄²	—	13	2	no	—
292	C₄²	—	13	2	no	—
293	C₄²	—	13	2	no	—
294	C₄²	—	13	2	no	—
295	C₄²	—	13	2	no	—
296	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
297	C₄²	D₄×D₁₃	1	1	yes	characteristic
298	C₄²	Q₈×D₁₃	1	1	yes	characteristic
299	C₄²	C₂³×D₁₃	1	1	yes	characteristic
300	C₁₃	C₂×#1574	1	1	yes	characteristic, complemented, Sylow
301	C₂³	—	13	2	no	—
302	C₂³	—	26	2	no	—
303	C₂³	—	26	2	no	—
304	C₂³	—	26	2	no	—
305	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	13	2	no	—
308	C₂×C₄	—	13	4	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	13	2	no	—
311	C₂×C₄	—	26	4	no	—
312	C₂×C₄	—	26	4	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	2	no	—
315	C₂×C₄	—	26	2	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	—	26	2	no	—
319	C₂×C₄	—	26	4	no	—
320	C₂×C₄	—	26	4	no	—
321	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
324	C₂×C₄	#88155	1	2	yes	—
325	C₂×C₄	#88161	1	2	yes	—
326	C₂×C₄	#88165	1	2	yes	—
327	C₂×C₄	#88172	1	2	yes	—
328	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	—	26	2	no	—
336	C₈	—	26	2	no	—
337	C₈	—	52	2	no	—
338	C₈	—	52	2	no	—
339	C₂²	—	26	2	no	—
340	C₂²	—	26	2	no	—
341	C₂²	—	26	2	no	—
342	C₂²	—	52	4	no	—
343	C₂²	#11718500	1	2	yes	—
344	C₂²	#11718542	1	2	yes	—
345	C₂²	D₁₃×#108	1	2	yes	—
346	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	13	4	no	—
350	C₄	—	26	2	no	—
351	C₄	—	26	2	no	—
352	C₄	#11718500	1	2	yes	—
353	C₄	#11718542	1	2	yes	—
354	C₄	C₂×#88155	1	1	yes	characteristic
355	C₄	C₂×#88161	1	1	yes	characteristic
356	C₄	C₂×#88165	1	1	yes	characteristic
357	C₄	C₂×#88172	1	1	yes	characteristic
358	C₂	—	52	2	no	—
359	C₂	#13113626	1	4	yes	direct product factor
360	C₂	C₂×#11718500	1	1	yes	characteristic
361	C₂	C₂×#11718542	1	1	yes	characteristic
362	C₂	C₂×D₁₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	C₂×#1574
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #13113626	C₂	direct (N × H)
5. C₂×#11718071	C₂	semidirect
6. C₂×#11718049	C₂	semidirect
7. C₂×#11718072	C₂	semidirect
8. C₂×C₁₃×#335	C₂	semidirect
30. C₁₃×#335	C₂²	semidirect
31. #11718072	C₂²	semidirect
32. #11718071	C₂²	semidirect
33. #11718049	C₂²	semidirect
125. C₂×#335	D₁₃	semidirect
188. #335	C₂×D₁₃	semidirect
250. C₂×C₁₃	#1574	semidirect
300. C₁₃	C₂×#1574	semidirect
359. C₂	#13113626	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×#11718494	2
C₂×#11718346	2
C₂×#11718498	2
#13113626	2
C₂×#11718071	2
C₂×#11718049	2
C₂×#11718072	2
C₂×C₁₃×#335	2
C₂×#1574	13

Subgroups — C2×#13113626 #15832005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113626 #15832005