Subgroups — C₂×#13116640 #16015991

Order 3328.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
2	#13116640	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
6	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₄	1	1	yes	characteristic
11	C₂²×#6099	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	#11718762	C₄	1	4	yes	—
15	#11718762	C₂²	1	4	yes	—
16	C₂×#88055	C₄	1	1	yes	characteristic
17	C₂×#88055	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	C₂²×#6090	C₂²	1	1	yes	characteristic
22	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
23	C₂²⋊#6090	C₂²	1	4	yes	—
24	#11718094	C₂²	1	4	yes	complemented
25	#11718093	C₂²	1	4	yes	complemented
26	#11718093	C₂²	1	4	yes	complemented
27	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃	C₈	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₃	C₈	1	1	yes	characteristic
37	#88180	C₈	1	4	yes	—
38	#88180	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
41	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#6099	—	2	4	no	—
44	C₂×#6099	C₂×C₄	1	4	yes	—
45	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₂×C₁₃×C₁₆	—	2	2	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	#88055	C₂×C₄	1	4	yes	—
52	#88055	C₂×C₄	1	4	yes	—
53	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	C₈×Dic₁₃	C₂³	1	2	yes	—
57	C₂×#6090	—	2	2	no	—
58	C₂×#6090	—	2	2	no	—
59	C₂×#6090	C₂³	1	2	yes	—
60	C₂×#6090	C₂³	1	2	yes	—
61	C₂×#2164	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₃	C₂×C₈	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×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
83	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
85	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
86	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
90	#6099	—	2	8	no	—
91	C₂×#1055	—	2	2	no	—
92	C₂×#1055	M₁₆	1	4	yes	—
93	C₂×#1055	C₂²×C₄	1	2	yes	—
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	C₁₃×C₁₆	—	2	2	no	—
96	C₁₃×C₁₆	—	2	2	no	—
97	#6090	—	2	2	no	—
98	#6090	—	2	2	no	—
99	C₂×#405	—	13	1	no	—
100	#2164	—	13	8	no	—
101	C₂×C₄⋊C₁₆	—	13	1	no	—
102	C₂×C₄⋊C₁₆	—	13	1	no	—
103	C₂×C₂²⋊C₁₆	—	13	1	no	—
104	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
105	C₂×#346	—	13	1	no	—
106	C₂×C₄×C₁₆	—	13	1	no	—
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
111	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	#117	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	#117	1	2	yes	—
131	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
133	#1055	—	2	2	no	—
134	#1055	C₂×M₁₆	1	4	yes	—
135	C₂²×M₁₆	—	13	1	no	—
136	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
137	C₂×#117	—	13	1	no	—
138	C₂²×C₁₆	—	13	1	no	—
139	C₂²×C₁₆	—	13	1	no	—
140	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
142	#405	—	13	4	no	—
143	#405	—	13	4	no	—
144	C₂×#97	—	13	1	no	—
145	C₂×#97	—	13	1	no	—
146	C₂×C₄×C₈	—	13	1	no	—
147	C₂×C₄×C₈	—	13	1	no	—
148	C₄⋊C₁₆	—	13	4	no	—
149	C₄⋊C₁₆	—	13	4	no	—
150	C₂²⋊C₁₆	—	13	4	no	—
151	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
152	#346	—	13	4	no	—
153	C₄×C₁₆	—	13	4	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	4	4	no	—
158	C₂²×C₁₃	#431	1	2	yes	—
159	C₂²×C₁₃	#504	1	2	yes	—
160	C₂²×C₁₃	#504	1	2	yes	—
161	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	#431	1	2	yes	—
164	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₁₃	C₂²×M₁₆	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	C₄×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	13	4	no	—
172	C₂×M₁₆	—	26	4	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₄×D₁₃	1	2	yes	—
179	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	#117	—	13	4	no	—
182	#117	—	13	4	no	—
183	C₂×C₄⋊C₄	—	13	1	no	—
184	C₂×C₄⋊C₄	—	13	1	no	—
185	C₂×C₂²⋊C₄	—	13	1	no	—
186	C₂×C₂²⋊C₄	—	13	1	no	—
187	C₂×C₄²	—	13	1	no	—
188	C₂×C₄²	—	13	1	no	—
189	C₂×C₁₆	—	2	2	no	—
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	13	2	no	—
192	C₂×C₁₆	—	13	2	no	—
193	C₂×C₁₆	—	26	2	no	—
194	C₂×C₁₆	—	26	2	no	—
195	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
196	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
197	#97	—	13	4	no	—
198	#97	—	13	4	no	—
199	C₄×C₈	—	13	2	no	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	—	13	2	no	—
202	C₄×C₈	—	13	2	no	—
203	C₂×C₁₃	—	4	2	no	—
204	C₂×C₁₃	#2164	1	4	yes	complemented
205	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
208	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	13	1	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	1	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₈×D₁₃	1	2	yes	—
217	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	M₁₆	—	26	8	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	13	4	no	—
225	C₂×C₈	—	13	2	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂×C₈	#6129	1	2	yes	—
231	C₂×C₈	#6129	1	2	yes	—
232	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
233	C₄⋊C₄	—	13	4	no	—
234	C₄⋊C₄	—	13	4	no	—
235	C₂²⋊C₄	—	13	4	no	—
236	C₂²⋊C₄	—	13	4	no	—
237	C₄²	—	13	2	no	—
238	C₄²	—	13	2	no	—
239	C₄²	—	13	2	no	—
240	C₄²	—	13	2	no	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	26	2	no	—
244	C₁₆	—	26	2	no	—
245	C₁₃	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	—	2	2	no	—
249	C₂³	C₂×C₈×D₁₃	1	2	yes	—
250	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	2	no	—
257	C₂×C₄	—	13	2	no	—
258	C₂×C₄	—	13	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	2	no	—
263	C₂×C₄	#88131	1	2	yes	—
264	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
265	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
267	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	13	4	no	—
270	C₈	—	26	2	no	—
271	C₈	#88131	1	2	yes	—
272	C₈	C₂×#6129	1	1	yes	characteristic
273	C₈	C₂×#6129	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	#11718379	1	2	yes	—
279	C₂²	#11718562	1	2	yes	—
280	C₂²	#11718568	1	2	yes	—
281	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	26	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	26	2	no	—
286	C₄	—	26	2	no	—
287	C₄	#11718379	1	2	yes	—
288	C₄	C₂×#88131	1	1	yes	characteristic
289	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
290	C₂	—	4	2	no	—
291	C₂	#13116640	1	4	yes	direct product factor
292	C₂	C₂×#11718379	1	1	yes	characteristic
293	C₂	C₂×#11718562	1	1	yes	characteristic
294	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×C₂²⋊C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2164
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116640	C₂	direct (N × H)
3. C₂×#11718093	C₂	semidirect
4. C₂×#11718093	C₂	semidirect
6. C₂×#11718094	C₂	semidirect
7. C₂×C₁₆×Dic₁₃	C₂	semidirect
24. #11718094	C₂²	semidirect
25. #11718093	C₂²	semidirect
26. #11718093	C₂²	semidirect
27. C₁₆×Dic₁₃	C₂²	semidirect
204. C₂×C₁₃	#2164	semidirect
245. C₁₃	C₂×#2164	semidirect
291. C₂	#13116640	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×#11718762	2
#13116640	2
C₂×#11718093	2
C₂×C₂²⋊#6090	2
C₂×#11718094	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2164	13

Subgroups — C2×#13116640 #16015991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116640 #16015991