Subgroups — C₂×#13130038 #16792575

Order 3328.

1046 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6118	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13130038	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1057⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#1057	C₂	1	2	yes	maximal, complemented
6	C₂×#11718494	C₂	1	1	yes	maximal, characteristic
7	C₂²×#6118	C₄	1	2	yes	—
8	C₂²×D₁₃⋊C₈	C₄	1	2	yes	—
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88110	C₄	1	2	yes	—
11	C₂×C₄×#1057	C₄	1	2	yes	—
12	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
13	C₂×C₄×#1057	C₂²	1	2	yes	—
14	C₂×C₄×#1057	C₂²	1	1	yes	characteristic
15	C₄×#6118	C₄	1	4	yes	—
16	C₄×#6118	C₂²	1	4	yes	—
17	C₄×D₁₃⋊C₈	C₄	1	4	yes	—
18	C₄×D₁₃⋊C₈	C₂²	1	4	yes	—
19	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₂×#88060	C₄	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
23	#11718494	C₄	1	4	yes	—
24	#11718494	C₂²	1	4	yes	—
25	#1057⋊C₈	C₂²	1	8	yes	complemented
26	C₈×#1057	C₂²	1	8	yes	complemented
27	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#1057	C₂×C₄	1	2	yes	—
31	C₂²×#1057	C₂×C₄	1	4	yes	—
32	C₂²×#1057	C₂×C₄	1	2	yes	—
33	C₂×#6118	C₈	1	16	yes	complemented
34	C₂×#6118	C₂×C₄	1	8	yes	—
35	C₂×D₁₃⋊C₈	—	2	8	no	—
36	C₂×D₁₃⋊C₈	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	2	yes	—
43	C₄²×D₁₃	C₂×C₄	1	2	yes	—
44	C₄²×D₁₃	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	C₂³	1	2	yes	—
47	#88110	C₂×C₄	1	8	yes	—
48	C₄×#1057	C₂×C₄	1	2	yes	—
49	C₄×#1057	C₂×C₄	1	2	yes	—
50	C₄×#1057	C₂×C₄	1	8	yes	—
51	C₄×#1057	C₂×C₄	1	4	yes	—
52	C₄×#1057	C₂³	1	2	yes	—
53	C₄×#1057	C₂³	1	4	yes	—
54	C₄×#1057	C₂³	1	2	yes	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	#88060	C₂×C₄	1	8	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
58	C₈×Dic₁₃	C₂³	1	4	yes	—
59	C₄⋊#1055	C₂×C₄	1	4	yes	—
60	C₂×C₈×M₁₆	—	13	1	no	maximal, Sylow
61	C₂³×D₁₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃	—	2	2	no	—
69	C₂×C₄×D₁₃	—	2	2	no	—
70	C₂×C₄×D₁₃	—	2	4	no	—
71	C₂×C₄×D₁₃	C₄²	1	2	yes	—
72	C₂×C₄×D₁₃	C₄²	1	2	yes	—
73	C₂×C₄×D₁₃	C₄²	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	8	yes	—
75	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
78	C₂×#1057	—	2	4	no	—
79	C₂×#1057	C₂×C₈	1	16	yes	complemented
80	C₂×#1057	M₁₆	1	8	yes	complemented
81	C₂×#1057	C₂²×C₄	1	4	yes	—
82	C₂×#1057	C₂²×C₄	1	8	yes	—
83	C₂×#1057	C₂²×C₄	1	4	yes	—
84	#6118	C₂×C₈	1	32	yes	complemented
85	D₁₃⋊C₈	—	2	16	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₄²	1	2	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄×Dic₁₃	C₄²	1	2	yes	—
91	C₄×Dic₁₃	C₄²	1	2	yes	—
92	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	C₂²×C₄	1	4	yes	—
100	C₂×#431	—	13	1	no	—
101	C₂×C₄×M₁₆	—	13	1	no	—
102	C₂²×C₄×C₈	—	13	1	no	—
103	C₈×M₁₆	—	13	8	no	—
104	C₂×#322	—	13	2	no	—
105	C₂×C₈²	—	13	2	no	—
106	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂²×D₁₃	—	2	2	no	—
108	C₂²×D₁₃	—	2	2	no	—
109	C₂²×D₁₃	C₄×C₈	1	4	yes	—
110	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	C₄×C₈	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#131	1	2	yes	—
117	C₂×C₄×C₁₃	#131	1	2	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
121	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
124	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
125	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₄×D₁₃	—	2	2	no	—
128	C₄×D₁₃	—	2	2	no	—
129	C₄×D₁₃	—	2	4	no	—
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	—	2	4	no	—
132	C₄×D₁₃	C₄×C₈	1	8	yes	—
133	C₄×D₁₃	C₂²×C₈	1	8	yes	—
134	#1057	—	2	4	no	—
135	#1057	C₂²×C₈	1	16	yes	complemented
136	#1057	C₂×M₁₆	1	8	yes	complemented
137	C₈×C₁₃	—	2	4	no	—
138	#1055	—	2	4	no	—
139	C₂²×M₁₆	—	13	2	no	—
140	C₂³×C₈	—	13	2	no	—
141	C₂²×C₄²	—	13	1	no	—
142	#431	—	13	4	no	—
143	#431	—	13	4	no	—
144	C₂×C₄⋊C₈	—	13	1	no	—
145	C₂×C₄⋊C₈	#261	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	13	2	no	—
147	C₄×M₁₆	—	13	4	no	—
148	C₄×M₁₆	—	13	4	no	—
149	C₂×#97	—	13	2	no	—
150	C₂×C₄×C₈	—	13	4	no	—
151	C₂×C₄×C₈	—	13	4	no	—
152	C₂×C₄×C₈	—	13	1	no	—
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	2	no	—
155	C₂×C₄×C₈	—	13	2	no	—
156	C₂×C₄×C₈	—	13	2	no	—
157	#322	—	13	8	no	—
158	C₈²	—	13	8	no	—
159	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
160	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
161	C₂²×C₁₃	#405	1	2	yes	—
162	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
163	C₂×D₁₃	—	2	2	no	—
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	—	2	8	no	—
166	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
169	C₄×C₁₃	#405	1	2	yes	—
170	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
172	Dic₁₃	—	2	2	no	—
173	Dic₁₃	C₂×C₄×C₈	1	2	yes	—
174	Dic₁₃	C₄×M₁₆	1	2	yes	—
175	Dic₁₃	C₂³×C₈	1	2	yes	—
176	Dic₁₃	C₂²×M₁₆	1	2	yes	—
177	C₂³×C₄	—	13	1	no	—
178	C₂³×C₄	—	13	1	no	—
179	C₂³×C₄	—	13	1	no	—
180	C₂×M₁₆	—	13	8	no	—
181	C₂×M₁₆	—	13	16	no	—
182	C₂²×C₈	—	13	2	no	—
183	C₂²×C₈	—	13	4	no	—
184	C₂²×C₈	—	13	8	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	13	2	no	—
187	C₂²×C₈	—	26	8	no	—
188	C₂²×C₈	C₂×#261	1	2	yes	—
189	C₂×C₄²	—	13	1	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	13	1	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	—	13	2	no	—
196	C₂×C₄²	C₂×#261	1	1	yes	characteristic
197	C₄⋊C₈	—	13	4	no	—
198	C₄⋊C₈	C₂×#261	1	4	yes	—
199	C₂²⋊C₈	—	13	8	no	—
200	#97	—	13	8	no	—
201	C₄×C₈	—	13	2	no	—
202	C₄×C₈	—	13	8	no	—
203	C₄×C₈	—	13	2	no	—
204	C₄×C₈	—	13	4	no	—
205	C₄×C₈	—	13	2	no	—
206	C₄×C₈	—	13	4	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	—	13	4	no	—
209	C₄×C₈	—	13	4	no	—
210	C₂×C₁₃	C₈×M₁₆	1	4	yes	complemented
211	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
212	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
214	D₁₃	—	2	4	no	—
215	C₂⁴	—	13	1	no	—
216	C₂²×C₄	—	13	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	8	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	1	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	26	4	no	—
227	C₂²×C₄	—	26	2	no	—
228	C₂²×C₄	—	26	2	no	—
229	C₂²×C₄	C₄×#261	1	1	yes	characteristic
230	C₂²×C₄	C₄×#261	1	1	yes	characteristic
231	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
232	M₁₆	—	13	32	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	13	4	no	—
235	C₂×C₈	—	13	8	no	—
236	C₂×C₈	—	13	8	no	—
237	C₂×C₈	—	13	16	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	4	no	—
240	C₂×C₈	—	26	16	no	—
241	C₂×C₈	—	26	4	no	—
242	C₂×C₈	—	26	4	no	—
243	C₂×C₈	C₂²×#261	1	4	yes	—
244	C₄²	—	13	1	no	—
245	C₄²	—	13	1	no	—
246	C₄²	—	13	2	no	—
247	C₄²	—	13	2	no	—
248	C₄²	—	13	2	no	—
249	C₄²	—	13	2	no	—
250	C₄²	—	13	1	no	—
251	C₄²	—	13	1	no	—
252	C₄²	C₄×#261	1	2	yes	—
253	C₄²	C₂²×#261	1	2	yes	—
254	C₁₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	13	4	no	—
256	C₂³	—	13	2	no	—
257	C₂³	—	26	2	no	—
258	C₂³	—	26	2	no	—
259	C₂³	C₂×C₄×#261	1	1	yes	characteristic
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	13	4	no	—
262	C₂×C₄	—	13	4	no	—
263	C₂×C₄	—	13	1	no	—
264	C₂×C₄	—	13	1	no	—
265	C₂×C₄	—	13	1	no	—
266	C₂×C₄	—	13	1	no	—
267	C₂×C₄	—	13	8	no	—
268	C₂×C₄	—	13	8	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	26	4	no	—
271	C₂×C₄	—	26	4	no	—
272	C₂×C₄	—	26	2	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	C₈×#261	1	4	yes	—
277	C₂×C₄	C₂×C₄×#261	1	2	yes	—
278	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×#261	1	2	yes	—
280	C₂×C₄	#88238	1	2	yes	—
281	C₂×C₄	#88240	1	2	yes	—
282	C₂×C₄	C₂³×#261	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	13	8	no	—
285	C₈	—	13	16	no	—
286	C₈	—	26	4	no	—
287	C₈	—	26	4	no	—
288	C₂²	—	13	4	no	—
289	C₂²	—	26	8	no	—
290	C₂²	—	26	2	no	—
291	C₂²	—	26	2	no	—
292	C₂²	#11719072	1	2	yes	—
293	C₂²	C₂×C₈×#261	1	2	yes	—
294	C₂²	M₁₆×#261	1	2	yes	—
295	C₂²	C₂²×C₄×#261	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	13	2	no	—
298	C₄	—	13	2	no	—
299	C₄	—	13	2	no	—
300	C₄	—	13	2	no	—
301	C₄	—	26	2	no	—
302	C₄	#11719072	1	2	yes	—
303	C₄	C₂×C₈×#261	1	4	yes	—
304	C₄	C₂×#88238	1	1	yes	characteristic
305	C₄	C₂×#88240	1	1	yes	characteristic
306	C₂	—	26	4	no	—
307	C₂	#13130038	1	4	yes	direct product factor
308	C₂	C₂²×C₈×#261	1	1	yes	characteristic
309	C₂	C₂×M₁₆×#261	1	1	yes	characteristic
310	C₂	C₂×#11719072	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×C₄⋊C₈	C₄
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	#261
hypocenter	C₁₃	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #13130038	C₂	direct (N × H)
4. C₂×#1057⋊C₈	C₂	semidirect
5. C₂×C₈×#1057	C₂	semidirect
25. #1057⋊C₈	C₂²	semidirect
26. C₈×#1057	C₂²	semidirect
33. C₂×#6118	C₈	semidirect
79. C₂×#1057	C₂×C₈	semidirect
80. C₂×#1057	M₁₆	semidirect
84. #6118	C₂×C₈	semidirect
135. #1057	C₂²×C₈	semidirect
136. #1057	C₂×M₁₆	semidirect
210. C₂×C₁₃	C₈×M₁₆	semidirect
254. C₁₃	C₂×C₈×M₁₆	semidirect
307. C₂	#13130038	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×C₄×#6118	2
C₂×C₄×D₁₃⋊C₈	2
#13130038	2
C₂×#1057⋊C₈	2
C₂×C₈×#1057	2
C₂×#11718494	2
C₂×C₈×M₁₆	13

Subgroups — C2×#13130038 #16792575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130038 #16792575