Subgroups — C₂×#13130773 #16798269

Order 3328.

1182 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×#88102	C₂	1	2	yes	maximal
2	C₂²×#88101	C₂	1	2	yes	maximal
3	C₂×#11719060	C₂	1	2	yes	maximal
4	#13130773	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊#261	C₂²	1	4	yes	—
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×(C₂×#261)⋊C₄	—	2	4	no	—
11	C₂×#88102	—	2	8	no	—
12	C₂×#88102	C₂²	1	8	yes	—
13	C₂×#88101	—	2	8	no	—
14	C₂×#88101	C₂²	1	8	yes	—
15	#11719060	—	2	4	no	—
16	#11719060	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
19	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
21	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
22	C₄×C₈×D₁₃	C₂²	1	4	yes	—
23	C₂³×#261	—	2	4	no	—
24	C₂²×C₄×D₁₃	D₄	1	2	yes	—
25	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄⋊#261	—	2	8	no	—
27	C₂×C₄⋊#261	—	2	8	no	—
28	C₂×C₄⋊#261	—	2	4	no	—
29	C₂×C₄⋊#261	C₂³	1	8	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	D₄	1	4	yes	—
40	C₂×C₈×D₁₃	D₄	1	4	yes	—
41	C₂×C₈×D₁₃	Q₈	1	4	yes	—
42	C₂×C₈×D₁₃	Q₈	1	4	yes	—
43	C₂×C₈×D₁₃	C₂³	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	D₄	1	2	yes	—
47	C₄²×D₁₃	Q₈	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	2	yes	—
49	(C₂×#261)⋊C₄	—	2	16	no	—
50	#88102	—	2	16	no	—
51	#88101	—	2	16	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
53	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
54	C₈×Dic₁₃	C₂×C₄	1	4	yes	complemented
55	C₄×#1055	C₂×C₄	1	4	yes	complemented
56	C₂×#1846	—	13	1	no	maximal, Sylow
57	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×#261	—	2	8	no	—
59	C₂²×#261	—	4	8	no	—
60	C₂²×#261	—	4	16	no	—
61	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₄×D₁₃	—	2	8	no	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
69	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
76	C₄⋊#261	—	2	8	no	—
77	C₄⋊#261	—	2	8	no	—
78	C₄⋊#261	—	4	8	no	—
79	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
86	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
89	C₂×#1055	C₄⋊C₄	1	4	yes	—
90	C₂×#1055	C₄⋊C₄	1	4	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#1055	C₂²×C₄	1	2	yes	—
93	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
94	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
95	C₈×D₁₃	C₂×D₄	1	4	yes	—
96	C₈×D₁₃	C₂×D₄	1	4	yes	—
97	C₈×D₁₃	C₂×Q₈	1	4	yes	—
98	C₈×D₁₃	C₂×Q₈	1	4	yes	—
99	C₂²×#107	—	13	2	no	—
100	C₂²×#106	—	13	2	no	—
101	C₂²×C₄×C₈	—	13	1	no	—
102	C₂×#382	—	13	2	no	—
103	#1846	—	13	8	no	—
104	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×D₁₃	#127	1	2	yes	—
107	C₂²×D₁₃	C₄⋊Q₈	1	2	yes	—
108	C₂²×D₁₃	#135	1	8	yes	—
109	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×#261	—	4	8	no	—
111	C₂×#261	—	4	16	no	—
112	C₂×#261	—	8	16	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
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	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₄⋊Q₈	1	2	yes	—
123	C₂×Dic₁₃	C₄⋊Q₈	1	2	yes	—
124	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄×D₁₃	—	2	8	no	—
126	C₄×D₁₃	—	2	8	no	—
127	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₄×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₄×D₁₃	#127	1	2	yes	—
130	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
131	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
132	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
133	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
134	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₄×D₁₃	C₂²×Q₈	1	1	yes	characteristic
136	C₄×D₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
138	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
139	#1055	C₂×C₄⋊C₄	1	4	yes	—
140	#1055	C₂×C₄⋊C₄	1	4	yes	—
141	C₂³×C₈	—	13	1	no	—
142	C₂³×C₈	—	13	1	no	—
143	C₂²×C₄⋊C₄	—	13	4	no	—
144	C₂²×C₄²	—	13	1	no	—
145	C₂×#107	—	13	8	no	—
146	C₂×#107	—	26	8	no	—
147	C₂×#106	—	13	8	no	—
148	C₂×#106	—	26	8	no	—
149	C₂×C₄×C₈	—	13	1	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₂×C₄×C₈	—	13	1	no	—
152	C₂×C₄×C₈	—	13	4	no	—
153	C₂×C₄×C₈	—	13	4	no	—
154	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
155	#382	—	13	8	no	—
156	#382	—	26	4	no	—
157	C₂×#95	—	26	4	no	—
158	C₂²×C₁₃	#384	1	2	yes	—
159	C₂²×C₁₃	#427	1	4	yes	—
160	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂×D₁₃	#384	1	2	yes	—
162	C₂×D₁₃	#427	1	4	yes	—
163	C₂×D₁₃	#495	1	8	yes	—
164	C₂×D₁₃	#499	1	8	yes	—
165	C₂×D₁₃	C₂×#127	1	1	yes	characteristic
166	C₂×D₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂×D₁₃	C₂×#135	1	4	yes	—
168	#261	—	8	8	no	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	#384	1	2	yes	—
171	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	Dic₁₃	—	2	4	no	—
174	Dic₁₃	#384	1	2	yes	—
175	Dic₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
176	Dic₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
177	C₂³×C₄	—	13	1	no	—
178	C₂³×C₄	—	13	2	no	—
179	C₂³×C₄	—	26	4	no	—
180	C₂²×C₈	—	13	1	no	—
181	C₂²×C₈	—	13	1	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	4	no	—
184	C₂²×C₈	—	13	4	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	C₂×#261	1	1	yes	characteristic
191	C₂²×C₈	C₂×#261	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	13	8	no	—
193	C₂×C₄⋊C₄	—	26	8	no	—
194	C₂×C₄⋊C₄	—	26	8	no	—
195	C₂×C₄⋊C₄	—	26	4	no	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	13	1	no	—
198	C₂×C₄²	—	13	2	no	—
199	C₂×C₄²	—	13	2	no	—
200	C₂×C₄²	—	13	2	no	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	C₂×#261	1	1	yes	characteristic
203	#107	—	26	16	no	—
204	#106	—	26	16	no	—
205	C₄×C₈	—	13	4	no	—
206	C₄×C₈	—	13	4	no	—
207	C₄×C₈	—	13	4	no	—
208	C₄×C₈	C₂×#261	1	4	yes	complemented
209	#95	—	26	16	no	—
210	C₂×C₁₃	#1846	1	4	yes	complemented
211	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
212	C₂×C₁₃	C₂×#427	1	2	yes	—
213	D₁₃	#1846	1	4	yes	complemented
214	D₁₃	C₂×#495	1	2	yes	—
215	D₁₃	C₂×#499	1	2	yes	—
216	C₂⁴	—	13	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	1	no	—
219	C₂²×C₄	—	13	1	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	1	no	—
226	C₂²×C₄	—	13	2	no	—
227	C₂²×C₄	—	13	4	no	—
228	C₂²×C₄	—	13	4	no	—
229	C₂²×C₄	—	26	8	no	—
230	C₂²×C₄	—	26	8	no	—
231	C₂²×C₄	—	52	8	no	—
232	C₂²×C₄	—	52	16	no	—
233	C₂²×C₄	C₄⋊#261	1	2	yes	—
234	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
235	C₂×C₈	—	13	4	no	—
236	C₂×C₈	—	13	4	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	13	4	no	—
241	C₂×C₈	—	13	8	no	—
242	C₂×C₈	—	13	8	no	—
243	C₂×C₈	—	13	4	no	—
244	C₂×C₈	—	13	4	no	—
245	C₂×C₈	C₄⋊#261	1	4	yes	—
246	C₂×C₈	C₄⋊#261	1	4	yes	—
247	C₂×C₈	C₂²×#261	1	2	yes	—
248	C₂×C₈	C₂²×#261	1	2	yes	—
249	C₄⋊C₄	—	26	8	no	—
250	C₄⋊C₄	—	26	8	no	—
251	C₄⋊C₄	—	52	8	no	—
252	C₄²	—	13	4	no	—
253	C₄²	—	13	2	no	—
254	C₄²	—	13	4	no	—
255	C₄²	—	13	2	no	—
256	C₄²	C₄⋊#261	1	2	yes	—
257	C₄²	C₂²×#261	1	2	yes	—
258	C₁₃	C₂×#1846	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	13	2	no	—
260	C₂³	—	13	2	no	—
261	C₂³	—	13	8	no	—
262	C₂³	—	13	1	no	—
263	C₂³	—	13	1	no	—
264	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	13	1	no	—
269	C₂×C₄	—	13	1	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	1	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	13	2	no	—
277	C₂×C₄	—	13	2	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	4	no	—
281	C₂×C₄	—	26	8	no	—
282	C₂×C₄	—	26	8	no	—
283	C₂×C₄	—	26	4	no	—
284	C₂×C₄	—	52	8	no	—
285	C₂×C₄	—	52	16	no	—
286	C₂×C₄	—	104	16	no	—
287	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
288	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
291	C₂×C₄	C₂³×#261	1	1	yes	characteristic
292	C₈	—	13	4	no	—
293	C₈	—	13	4	no	—
294	C₈	C₂×C₄⋊#261	1	4	yes	—
295	C₈	C₂×C₄⋊#261	1	4	yes	—
296	C₂²	—	13	1	no	—
297	C₂²	—	13	1	no	—
298	C₂²	—	13	2	no	—
299	C₂²	—	13	8	no	—
300	C₂²	—	13	8	no	—
301	C₂²	—	13	4	no	—
302	C₂²	—	13	4	no	—
303	C₂²	#11719059	1	2	yes	—
304	C₂²	#11719092	1	4	yes	—
305	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	13	1	no	—
308	C₄	—	13	1	no	—
309	C₄	—	13	2	no	—
310	C₄	—	26	4	no	—
311	C₄	—	104	8	no	—
312	C₄	#11719059	1	2	yes	—
313	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
314	C₄	C₂²×C₄⋊#261	1	1	yes	characteristic
315	C₂	—	13	2	no	—
316	C₂	—	13	4	no	—
317	C₂	—	13	2	no	—
318	C₂	#13130773	1	4	yes	direct product factor
319	C₂	C₂×#11719059	1	1	yes	characteristic
320	C₂	C₂×#11719092	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1846
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #13130773	C₂	direct (N × H)
17. C₂×C₄×C₈×C₁₃	C₄	semidirect
18. C₂×C₈×Dic₁₃	C₄	semidirect
19. C₂×C₈×Dic₁₃	C₄	semidirect
20. C₂×C₄×#1055	C₄	semidirect
21. C₄×C₈×D₁₃	C₄	semidirect
52. C₄×C₈×C₁₃	C₂×C₄	semidirect
53. C₈×Dic₁₃	C₂×C₄	semidirect
54. C₈×Dic₁₃	C₂×C₄	semidirect
55. C₄×#1055	C₂×C₄	semidirect
154. C₂×C₄×C₈	#261	semidirect
208. C₄×C₈	C₂×#261	semidirect
210. C₂×C₁₃	#1846	semidirect
213. D₁₃	#1846	semidirect
258. C₁₃	C₂×#1846	semidirect
318. C₂	#13130773	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×#88102	2
C₂²×#88101	2
C₂×#11719060	2
#13130773	2
C₂×C₄×C₈×D₁₃	2
C₂×#1846	13

Subgroups — C2×#13130773 #16798269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130773 #16798269