Subgroups — C₂×#13116593 #16015900

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88038	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
4	#13116593	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6094	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88126	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88055	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂²×#6090	C₂²	1	1	yes	characteristic
19	C₂×#88043	C₄	1	1	yes	characteristic
20	C₂×#88038	—	2	4	no	—
21	C₂×#88038	C₂²	1	4	yes	—
22	C₂×C₁₆×D₁₃	—	2	4	no	—
23	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
24	#11718350	C₄	1	4	yes	—
25	#11718350	C₂²	1	4	yes	—
26	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
27	#11718093	C₂²	1	4	yes	complemented
28	#11718093	C₂²	1	4	yes	complemented
29	C₄⋊#6090	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#6103	C₈	1	1	yes	characteristic
35	C₂×#6103	C₈	1	1	yes	characteristic
36	C₂×#6101	C₈	1	1	yes	characteristic
37	C₂×#6101	C₈	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
41	C₂×#6094	—	2	4	no	—
42	C₂×#6094	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	D₄	1	4	yes	—
45	C₂×C₈×D₁₃	Q₈	1	4	yes	—
46	C₂×C₈×D₁₃	C₂³	1	2	yes	—
47	#88126	C₈	1	4	yes	—
48	#88126	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₂×C₁₃×C₁₆	—	2	2	no	—
51	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
52	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	#88055	C₂×C₄	1	4	yes	—
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	C₂×#6090	—	2	2	no	—
59	C₂×#6090	—	2	2	no	—
60	C₂×#6090	C₂³	1	2	yes	—
61	C₂×#6090	C₂³	1	2	yes	—
62	#88043	C₂×C₄	1	4	yes	—
63	#88038	—	2	8	no	—
64	C₁₆×D₁₃	—	2	8	no	—
65	C₂×#2151	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
75	C₂×C₄×D₁₃	M₁₆	1	4	yes	—
76	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	#6103	C₂×C₈	1	4	yes	—
86	#6103	C₂×C₈	1	4	yes	—
87	#6101	C₂×C₈	1	4	yes	—
88	#6101	C₂×C₈	1	4	yes	—
89	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	C₄⋊C₄	1	4	yes	—
93	C₂×#1055	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	#6094	—	2	8	no	—
96	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
97	C₈×D₁₃	C₂×D₄	1	4	yes	—
98	C₈×D₁₃	C₂×Q₈	1	4	yes	—
99	C₁₃×C₁₆	—	2	2	no	—
100	C₁₃×C₁₆	—	2	2	no	—
101	#6090	—	2	2	no	—
102	#6090	—	2	2	no	—
103	C₂²×M₃₂	—	13	1	no	—
104	C₂³×C₁₆	—	13	1	no	—
105	C₂×#405	—	13	1	no	—
106	#2151	—	13	8	no	—
107	C₂×C₄⋊C₁₆	—	13	1	no	—
108	C₂×C₄⋊C₁₆	—	13	1	no	—
109	C₂×C₄⋊C₁₆	—	13	1	no	—
110	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₁₃	—	2	2	no	—
113	C₂²×D₁₃	—	2	2	no	—
114	C₂²×D₁₃	C₄⋊C₈	1	4	yes	—
115	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
127	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
129	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
130	C₄×D₁₃	C₄⋊C₈	1	8	yes	—
131	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₄×D₁₃	C₂×M₁₆	1	4	yes	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	#1055	—	2	2	no	—
138	#1055	C₂×C₄⋊C₄	1	4	yes	—
139	C₂²×M₁₆	—	13	1	no	—
140	C₂³×C₈	—	13	1	no	—
141	C₂×#117	—	13	1	no	—
142	C₂×M₃₂	—	13	4	no	—
143	C₂×M₃₂	—	26	4	no	—
144	C₂²×C₁₆	—	13	1	no	—
145	C₂²×C₁₆	—	13	1	no	—
146	C₂²×C₁₆	—	13	4	no	—
147	C₂²×C₁₆	—	26	4	no	—
148	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
150	#405	—	13	4	no	—
151	#405	—	13	4	no	—
152	C₂×#97	—	13	1	no	—
153	C₂×#97	—	13	1	no	—
154	C₂×C₄×C₈	—	13	1	no	—
155	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
156	C₄⋊C₁₆	—	13	4	no	—
157	C₄⋊C₁₆	—	13	4	no	—
158	C₄⋊C₁₆	—	13	4	no	—
159	C₄⋊C₁₆	C₂×D₁₃	1	4	yes	complemented
160	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
161	C₂²×C₁₃	#504	1	2	yes	—
162	C₂²×C₁₃	#504	1	2	yes	—
163	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
164	C₂×D₁₃	—	2	2	no	—
165	C₂×D₁₃	—	2	2	no	—
166	C₂×D₁₃	—	2	8	no	—
167	C₂×D₁₃	C₂×C₄⋊C₈	1	4	yes	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
171	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
173	Dic₁₃	—	2	2	no	—
174	Dic₁₃	—	2	2	no	—
175	Dic₁₃	C₂×C₄⋊C₈	1	4	yes	—
176	C₂³×C₄	—	13	1	no	—
177	C₂×M₁₆	—	13	4	no	—
178	C₂×M₁₆	—	26	4	no	—
179	C₂²×C₈	—	13	1	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	13	4	no	—
182	C₂²×C₈	—	13	4	no	—
183	C₂²×C₈	—	13	2	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	#117	—	13	4	no	—
188	#117	—	13	4	no	—
189	C₂×C₄⋊C₄	—	13	1	no	—
190	C₂×C₄⋊C₄	—	13	1	no	—
191	C₂×C₂²⋊C₄	—	13	1	no	—
192	C₂×C₂²⋊C₄	—	13	1	no	—
193	C₂×C₄²	—	13	1	no	—
194	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
195	M₃₂	—	26	8	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	—	2	2	no	—
198	C₂×C₁₆	—	13	2	no	—
199	C₂×C₁₆	—	13	2	no	—
200	C₂×C₁₆	—	26	8	no	—
201	C₂×C₁₆	—	26	2	no	—
202	C₂×C₁₆	—	26	2	no	—
203	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
204	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
205	#97	—	13	4	no	—
206	#97	—	13	4	no	—
207	C₄×C₈	—	13	2	no	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	C₄×D₁₃	1	2	yes	—
210	C₄×C₈	C₂²×D₁₃	1	2	yes	—
211	C₂×C₁₃	#2151	1	4	yes	complemented
212	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
213	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
215	D₁₃	—	2	4	no	—
216	C₂⁴	—	13	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	4	no	—
219	C₂²×C₄	—	13	2	no	—
220	C₂²×C₄	—	13	4	no	—
221	C₂²×C₄	—	13	2	no	—
222	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	M₁₆	—	26	8	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	13	4	no	—
230	C₂×C₈	—	13	2	no	—
231	C₂×C₈	—	13	8	no	—
232	C₂×C₈	—	13	4	no	—
233	C₂×C₈	—	13	4	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
237	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
238	C₂×C₈	D₄×D₁₃	1	2	yes	—
239	C₂×C₈	Q₈×D₁₃	1	2	yes	—
240	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	13	4	no	—
242	C₄⋊C₄	—	13	4	no	—
243	C₂²⋊C₄	—	13	4	no	—
244	C₂²⋊C₄	—	13	4	no	—
245	C₄²	—	13	2	no	—
246	C₄²	—	13	2	no	—
247	C₄²	C₈×D₁₃	1	2	yes	—
248	C₄²	C₂×C₄×D₁₃	1	2	yes	—
249	C₁₆	—	2	2	no	—
250	C₁₆	—	2	2	no	—
251	C₁₆	—	26	2	no	—
252	C₁₆	—	26	2	no	—
253	C₁₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	2	no	—
255	C₂³	—	13	4	no	—
256	C₂³	—	26	2	no	—
257	C₂³	—	26	2	no	—
258	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	13	2	no	—
262	C₂×C₄	—	13	4	no	—
263	C₂×C₄	—	13	4	no	—
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	13	8	no	—
266	C₂×C₄	—	13	2	no	—
267	C₂×C₄	—	13	2	no	—
268	C₂×C₄	—	26	2	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
274	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
275	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	13	4	no	—
278	C₈	—	26	2	no	—
279	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
280	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
281	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
282	C₂²	—	13	4	no	—
283	C₂²	—	26	2	no	—
284	C₂²	—	26	8	no	—
285	C₂²	—	26	2	no	—
286	C₂²	D₁₃×C₄⋊C₈	1	2	yes	—
287	C₂²	#11718562	1	2	yes	—
288	C₂²	#11718568	1	2	yes	—
289	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	13	4	no	—
293	C₄	—	26	2	no	—
294	C₄	—	26	2	no	—
295	C₄	D₁₃×C₄⋊C₈	1	2	yes	—
296	C₄	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
297	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
298	C₂	—	26	4	no	—
299	C₂	#13116593	1	4	yes	direct product factor
300	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic
301	C₂	C₂×#11718562	1	1	yes	characteristic
302	C₂	C₂×#11718568	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2151
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #13116593	C₂	direct (N × H)
5. C₂×C₄⋊#6090	C₂	semidirect
6. C₂×#11718093	C₂	semidirect
7. C₂×#11718093	C₂	semidirect
8. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
26. C₁₃×C₄⋊C₁₆	C₂²	semidirect
27. #11718093	C₂²	semidirect
28. #11718093	C₂²	semidirect
29. C₄⋊#6090	C₂²	semidirect
110. C₂×C₄⋊C₁₆	D₁₃	semidirect
159. C₄⋊C₁₆	C₂×D₁₃	semidirect
211. C₂×C₁₃	#2151	semidirect
253. C₁₃	C₂×#2151	semidirect
299. C₂	#13116593	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×#88038	2
C₂²×C₁₆×D₁₃	2
C₂×#11718350	2
#13116593	2
C₂×C₄⋊#6090	2
C₂×#11718093	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×#2151	13

Subgroups — C2×#13116593 #16015900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116593 #16015900