Subgroups — C₂×#13129662 #16778938

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719119	C₂	1	2	yes	maximal
2	#13129662	C₂	1	4	yes	maximal, direct product factor
3	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
6	C₂³×#1055	C₄	1	1	yes	characteristic
7	C₂×C₂²⋊#1057	—	2	2	no	—
8	#11719119	—	2	2	no	—
9	#11719119	—	2	2	no	—
10	#11719119	C₄	1	4	yes	—
11	#11719119	C₂²	1	4	yes	—
12	C₂×C₄⋊#1057	—	2	2	no	—
13	C₂×C₄×#1057	C₄	1	4	yes	—
14	C₂×#88055	—	2	2	no	—
15	C₂×#88055	—	2	2	no	—
16	C₂×#88055	C₄	1	2	yes	—
17	C₂×#88055	C₂²	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
20	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
21	C₂²×#1057	—	2	4	no	—
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	C₈	1	4	yes	complemented
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
33	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
35	C₂²×#1055	—	2	1	no	—
36	C₂²×#1055	—	2	1	no	—
37	C₂²×#1055	—	2	2	no	—
38	C₂²×#1055	C₈	1	4	yes	complemented
39	C₂²×#1055	C₂×C₄	1	2	yes	—
40	C₂²⋊#1057	—	2	4	no	—
41	C₂²⋊#1057	—	2	4	no	—
42	C₄⋊#1057	—	2	4	no	—
43	C₄⋊#1057	—	2	4	no	—
44	C₄×#1057	—	2	4	no	—
45	C₄×#1057	C₂×C₄	1	8	yes	—
46	#88055	—	2	4	no	—
47	#88055	—	2	4	no	—
48	C₂×#1275	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
58	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	4	2	no	—
61	C₂²×Dic₁₃	—	4	2	no	—
62	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
65	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×#1057	—	2	8	no	—
71	C₂×#1057	—	4	8	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
77	C₄×Dic₁₃	C₄²	1	2	yes	—
78	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
79	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
80	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
81	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
82	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
83	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	C₂×C₈	1	4	yes	complemented
90	C₂×#431	—	13	2	no	—
91	C₂²×#97	—	13	1	no	—
92	#1275	—	13	4	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	#95	1	1	yes	characteristic
96	C₂³×C₁₃	#95	1	1	yes	characteristic
97	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
98	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	#97	1	2	yes	—
105	C₂×C₄×C₁₃	#97	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
111	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×Dic₁₃	—	2	4	no	—
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	—	4	2	no	—
121	C₂×Dic₁₃	—	4	1	no	—
122	C₂×Dic₁₃	—	4	1	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	4	1	no	—
125	C₂×Dic₁₃	—	4	1	no	—
126	C₂×Dic₁₃	#95	1	2	yes	—
127	C₂×Dic₁₃	#95	1	2	yes	—
128	C₂×Dic₁₃	C₂×C₂²⋊C₄	1	2	yes	—
129	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	#1057	—	4	8	no	—
131	C₈×C₁₃	—	2	4	no	—
132	#1055	—	2	4	no	—
133	C₂³×C₈	—	13	1	no	—
134	C₂³×C₈	#261	1	1	yes	characteristic
135	C₂²×C₄²	—	13	1	no	—
136	#431	—	13	4	no	—
137	#431	—	13	4	no	—
138	#431	—	26	2	no	—
139	#431	—	26	2	no	—
140	C₂×C₄⋊C₈	—	26	2	no	—
141	C₂×C₂²⋊C₈	—	26	2	no	—
142	C₂×#97	—	13	2	no	—
143	C₂×#97	—	13	2	no	—
144	C₂×#97	—	26	2	no	—
145	C₂×#97	—	26	2	no	—
146	C₂×C₄×C₈	—	13	4	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	#336	1	2	yes	—
154	C₂²×C₁₃	#336	1	2	yes	—
155	C₂²×C₁₃	H(3,ℤ/4)	1	2	yes	—
156	C₂²×C₁₃	#338	1	2	yes	—
157	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₄×C₁₃	#336	1	2	yes	—
161	C₄×C₁₃	#336	1	2	yes	—
162	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
166	Dic₁₃	—	2	2	no	—
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	—	2	2	no	—
169	Dic₁₃	—	2	2	no	—
170	C₂³×C₄	—	13	1	no	—
171	C₂³×C₄	—	13	1	no	—
172	C₂³×C₄	C₂×#261	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	26	2	no	—
179	C₂²×C₈	—	26	1	no	—
180	C₂²×C₈	—	26	1	no	—
181	C₂²×C₈	—	26	4	no	—
182	C₂²×C₈	#1057	1	4	yes	complemented
183	C₂²×C₈	C₂×#261	1	2	yes	—
184	C₂×C₄²	—	13	2	no	—
185	C₂×C₄²	—	13	2	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	13	2	no	—
189	C₂×C₄²	—	13	2	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	13	1	no	—
192	C₄⋊C₈	—	26	4	no	—
193	C₄⋊C₈	—	26	4	no	—
194	C₂²⋊C₈	—	26	4	no	—
195	C₂²⋊C₈	—	26	4	no	—
196	#97	—	26	4	no	—
197	#97	—	26	4	no	—
198	C₄×C₈	—	13	8	no	—
199	C₄×C₈	—	26	4	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	#1275	1	4	yes	complemented
204	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
205	C₂×C₁₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#338	1	1	yes	characteristic
207	C₂⁴	C₄⋊#261	1	1	yes	characteristic
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	13	2	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	52	2	no	—
217	C₂²×C₄	—	52	2	no	—
218	C₂²×C₄	D₁₃⋊C₈	1	2	yes	—
219	C₂²×C₄	#6118	1	2	yes	—
220	C₂²×C₄	C₄×#261	1	1	yes	characteristic
221	C₂²×C₄	C₄×#261	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
223	C₂²×C₄	C₂×#1057	1	2	yes	—
224	C₂²×C₄	#6122	1	2	yes	—
225	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
227	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	—	26	4	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	8	no	—
238	C₂×C₈	—	52	8	no	—
239	C₂×C₈	C₂×#1057	1	4	yes	complemented
240	C₄²	—	13	2	no	—
241	C₄²	—	13	2	no	—
242	C₄²	—	13	2	no	—
243	C₄²	—	13	2	no	—
244	C₄²	—	13	2	no	—
245	C₄²	—	13	2	no	—
246	C₄²	—	13	2	no	—
247	C₄²	—	13	2	no	—
248	C₁₃	C₂×#1275	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₄⋊#1057	1	2	yes	—
252	C₂³	#88112	1	2	yes	—
253	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
254	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
255	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	4	no	—
262	C₂×C₄	—	52	1	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	1	no	—
265	C₂×C₄	—	52	2	no	—
266	C₂×C₄	—	52	1	no	—
267	C₂×C₄	—	52	1	no	—
268	C₂×C₄	C₄×#1057	1	2	yes	—
269	C₂×C₄	C₄×#1057	1	2	yes	—
270	C₂×C₄	C₄⋊#1057	1	2	yes	—
271	C₂×C₄	#88110	1	2	yes	—
272	C₂×C₄	#88110	1	2	yes	—
273	C₂×C₄	#88111	1	2	yes	—
274	C₂×C₄	#88111	1	2	yes	—
275	C₂×C₄	#88112	1	2	yes	—
276	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂²⋊#1057	1	2	yes	—
279	C₂×C₄	C₂²⋊#1057	1	2	yes	—
280	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂×#6118	1	1	yes	characteristic
282	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
283	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
284	C₂×C₄	C₂×#6122	1	1	yes	characteristic
285	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	26	4	no	—
288	C₈	—	52	8	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	#11718267	1	2	yes	—
296	C₂²	#11718269	1	2	yes	—
297	C₂²	#11718290	1	2	yes	—
298	C₂²	#11718290	1	2	yes	—
299	C₂²	C₂×C₄⋊#1057	1	1	yes	characteristic
300	C₂²	C₂×#88112	1	1	yes	characteristic
301	C₂²	C₂×(C₂×#261)⋊C₄	1	1	yes	characteristic
302	C₄	—	26	2	no	—
303	C₄	—	26	2	no	—
304	C₄	—	26	2	no	—
305	C₄	—	26	2	no	—
306	C₄	#11718290	1	2	yes	—
307	C₄	#11718290	1	2	yes	—
308	C₄	C₂×C₄×#1057	1	1	yes	characteristic
309	C₄	C₂×#88110	1	1	yes	characteristic
310	C₄	C₂×#88111	1	1	yes	characteristic
311	C₄	C₂×C₂²⋊#1057	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	#13129662	1	4	yes	direct product factor
316	C₂	C₂×#11718290	1	1	yes	characteristic
317	C₂	C₂×#11718267	1	1	yes	characteristic
318	C₂	C₂×#11718269	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×#261
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	#95
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄×Dic₁₃	C₂³
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	C₂×#1275
LCS	G ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #13129662	C₂	direct (N × H)
25. C₂²×C₈×C₁₃	C₈	semidirect
38. C₂²×#1055	C₈	semidirect
76. C₂×C₈×C₁₃	C₂×C₈	semidirect
89. C₂×#1055	C₂×C₈	semidirect
182. C₂²×C₈	#1057	semidirect
203. C₂×C₁₃	#1275	semidirect
239. C₂×C₈	C₂×#1057	semidirect
248. C₁₃	C₂×#1275	semidirect
315. C₂	#13129662	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×#11719119	2
#13129662	2
C₂²×#88055	2
C₂×#1275	13

Subgroups — C2×#13129662 #16778938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13129662 #16778938