Subgroups — C₂×#13113137 #15828544

Order 3328.

802 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×#11718346	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718346	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718346	C₂	1	1	yes	maximal, characteristic
4	#13113137	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718046	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88060	C₄	1	2	yes	—
14	C₂×#88060	C₄	1	2	yes	—
15	C₂×#88060	C₄	1	2	yes	—
16	C₂×#88054	C₄	1	2	yes	—
17	C₂×#88054	C₄	1	2	yes	—
18	C₂×#88054	C₄	1	2	yes	—
19	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	#11718346	C₄	1	4	yes	—
23	#11718346	C₄	1	4	yes	—
24	#11718346	C₄	1	4	yes	—
25	#11718346	C₂²	1	4	yes	—
26	#11718346	C₂²	1	4	yes	—
27	#11718346	C₂²	1	4	yes	—
28	C₁₃×#322	C₂²	1	4	yes	complemented
29	#11718047	C₂²	1	4	yes	complemented
30	#11718046	C₂²	1	4	yes	complemented
31	C₈×#1055	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	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	C₂×C₄	1	1	yes	characteristic
42	C₂²×#1055	C₂×C₄	1	2	yes	—
43	C₂²×#1055	C₂×C₄	1	2	yes	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₄²×D₁₃	C₂×C₄	1	2	yes	—
46	C₄²×D₁₃	C₂×C₄	1	2	yes	—
47	C₄²×D₁₃	C₂×C₄	1	2	yes	—
48	C₄²×D₁₃	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	#88060	C₂×C₄	1	8	yes	—
56	#88060	C₂×C₄	1	8	yes	—
57	#88060	C₂×C₄	1	8	yes	—
58	#88054	C₂×C₄	1	8	yes	—
59	#88054	C₂×C₄	1	8	yes	—
60	#88054	C₂×C₄	1	8	yes	—
61	C₄×#1055	C₂×C₄	1	2	yes	—
62	C₄×#1055	C₂×C₄	1	2	yes	—
63	C₄×#1055	C₂×C₄	1	2	yes	—
64	C₄×#1055	C₂³	1	2	yes	—
65	C₄×#1055	C₂³	1	2	yes	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₂×#1454	—	13	1	no	maximal, Sylow
68	C₂³×D₁₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
74	C₂²×Dic₁₃	C₄²	1	1	yes	characteristic
75	C₂×C₄×D₁₃	—	2	4	no	—
76	C₂×C₄×D₁₃	—	2	4	no	—
77	C₂×C₄×D₁₃	—	2	4	no	—
78	C₂×C₄×D₁₃	C₄²	1	2	yes	—
79	C₂×C₄×D₁₃	C₄²	1	2	yes	—
80	C₂×C₄×D₁₃	C₄²	1	2	yes	—
81	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₄×Dic₁₃	C₄²	1	2	yes	—
95	C₄×Dic₁₃	C₄²	1	2	yes	—
96	C₄×Dic₁₃	C₄²	1	2	yes	—
97	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
98	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
99	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	4	no	—
103	C₂×#1055	C₂²×C₄	1	4	yes	—
104	C₂×#1055	C₂²×C₄	1	4	yes	—
105	C₂×#1055	C₂²×C₄	1	4	yes	—
106	C₂×#431	—	13	1	no	—
107	C₂×#431	—	13	1	no	—
108	C₂×#431	—	13	1	no	—
109	#1454	—	13	8	no	—
110	C₂×#322	—	13	1	no	—
111	C₂×#322	—	13	1	no	—
112	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
113	C₂×C₈²	—	13	1	no	—
114	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂²×D₁₃	—	2	2	no	—
116	C₂²×D₁₃	—	2	2	no	—
117	C₂²×D₁₃	—	2	2	no	—
118	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#131	1	4	yes	—
123	C₂×C₄×C₁₃	#131	1	4	yes	—
124	C₂×C₄×C₁₃	#131	1	4	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	—	2	2	no	—
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
132	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
133	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
134	C₄×D₁₃	—	2	4	no	—
135	C₄×D₁₃	—	2	4	no	—
136	C₄×D₁₃	—	2	4	no	—
137	C₄×D₁₃	—	2	4	no	—
138	C₄×D₁₃	—	2	4	no	—
139	C₄×D₁₃	—	2	4	no	—
140	C₈×C₁₃	—	2	4	no	—
141	C₈×C₁₃	—	2	4	no	—
142	C₈×C₁₃	—	2	4	no	—
143	#1055	—	2	4	no	—
144	#1055	—	2	4	no	—
145	#1055	—	2	4	no	—
146	C₂²×C₄²	—	13	1	no	—
147	#431	—	13	4	no	—
148	#431	—	13	4	no	—
149	#431	—	13	4	no	—
150	#431	—	13	4	no	—
151	#431	—	13	4	no	—
152	#431	—	13	4	no	—
153	C₂×C₄⋊C₈	—	13	2	no	—
154	C₂×C₄⋊C₈	—	13	2	no	—
155	C₂×C₄⋊C₈	—	13	2	no	—
156	C₂×C₂²⋊C₈	—	13	2	no	—
157	C₂×C₂²⋊C₈	—	13	2	no	—
158	C₂×C₂²⋊C₈	—	13	2	no	—
159	C₂×C₄×C₈	—	13	1	no	—
160	C₂×C₄×C₈	—	13	1	no	—
161	C₂×C₄×C₈	—	13	1	no	—
162	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
165	#322	—	13	4	no	—
166	#322	—	13	4	no	—
167	#322	C₂×D₁₃	1	4	yes	complemented
168	C₈²	—	13	4	no	—
169	C₂²×C₁₃	#405	1	2	yes	—
170	C₂²×C₁₃	#405	1	2	yes	—
171	C₂²×C₁₃	#405	1	2	yes	—
172	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
173	C₂×D₁₃	—	2	2	no	—
174	C₂×D₁₃	—	2	2	no	—
175	C₂×D₁₃	—	2	2	no	—
176	C₂×D₁₃	—	4	4	no	—
177	C₄×C₁₃	#405	1	2	yes	—
178	C₄×C₁₃	#405	1	2	yes	—
179	C₄×C₁₃	#405	1	2	yes	—
180	C₄×C₁₃	C₂×#131	1	2	yes	—
181	C₄×C₁₃	C₂×#131	1	2	yes	—
182	C₄×C₁₃	C₂×#131	1	2	yes	—
183	Dic₁₃	—	2	2	no	—
184	Dic₁₃	—	2	2	no	—
185	Dic₁₃	—	2	2	no	—
186	C₂³×C₄	—	13	1	no	—
187	C₂³×C₄	—	13	1	no	—
188	C₂³×C₄	—	13	1	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	C₄×D₁₃	1	2	yes	—
193	C₂²×C₈	C₄×D₁₃	1	2	yes	—
194	C₂²×C₈	C₄×D₁₃	1	2	yes	—
195	C₂×C₄²	—	13	1	no	—
196	C₂×C₄²	—	13	1	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	C₂²×D₁₃	1	1	yes	characteristic
203	C₄⋊C₈	—	13	8	no	—
204	C₄⋊C₈	—	13	8	no	—
205	C₄⋊C₈	—	13	8	no	—
206	C₂²⋊C₈	—	13	8	no	—
207	C₂²⋊C₈	—	13	8	no	—
208	C₂²⋊C₈	—	13	8	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	2	no	—
211	C₄×C₈	—	13	2	no	—
212	C₄×C₈	—	13	2	no	—
213	C₄×C₈	—	13	2	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	C₄×D₁₃	1	2	yes	—
216	C₄×C₈	C₄×D₁₃	1	2	yes	—
217	C₄×C₈	C₄×D₁₃	1	2	yes	—
218	C₄×C₈	C₂²×D₁₃	1	2	yes	—
219	C₄×C₈	C₂²×D₁₃	1	2	yes	—
220	C₄×C₈	C₂²×D₁₃	1	2	yes	—
221	C₂×C₁₃	#1454	1	4	yes	complemented
222	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
225	D₁₃	—	4	2	no	—
226	C₂⁴	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	13	1	no	—
229	C₂²×C₄	—	13	1	no	—
230	C₂²×C₄	—	13	2	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	13	2	no	—
233	C₂²×C₄	—	13	2	no	—
234	C₂²×C₄	—	13	2	no	—
235	C₂²×C₄	—	13	2	no	—
236	C₂²×C₄	—	26	4	no	—
237	C₂²×C₄	—	26	4	no	—
238	C₂²×C₄	—	26	4	no	—
239	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	13	4	no	—
246	C₂×C₈	—	13	4	no	—
247	C₂×C₈	—	13	4	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	4	no	—
251	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
252	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
253	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
254	C₄²	—	13	2	no	—
255	C₄²	—	13	2	no	—
256	C₄²	—	13	2	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	13	2	no	—
259	C₄²	—	13	2	no	—
260	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₄²	C₂³×D₁₃	1	1	yes	characteristic
264	C₁₃	C₂×#1454	1	1	yes	characteristic, complemented, Sylow
265	C₂³	—	13	2	no	—
266	C₂³	—	26	2	no	—
267	C₂³	—	26	2	no	—
268	C₂³	—	26	2	no	—
269	C₂³	C₄²×D₁₃	1	1	yes	characteristic
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	13	2	no	—
273	C₂×C₄	—	26	2	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	—	26	4	no	—
277	C₂×C₄	—	26	4	no	—
278	C₂×C₄	—	26	4	no	—
279	C₂×C₄	—	26	4	no	—
280	C₂×C₄	—	26	4	no	—
281	C₂×C₄	—	26	4	no	—
282	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
283	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
285	C₂×C₄	#88155	1	4	yes	—
286	C₂×C₄	#88161	1	4	yes	—
287	C₂×C₄	#88161	1	4	yes	—
288	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₈	—	26	4	no	—
295	C₈	—	26	4	no	—
296	C₈	—	26	4	no	—
297	C₂²	—	26	2	no	—
298	C₂²	—	26	2	no	—
299	C₂²	—	26	2	no	—
300	C₂²	—	52	4	no	—
301	C₂²	#11718350	1	2	yes	—
302	C₂²	#11718369	1	2	yes	—
303	C₂²	#11718369	1	2	yes	—
304	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
305	C₄	—	26	2	no	—
306	C₄	—	26	2	no	—
307	C₄	—	26	2	no	—
308	C₄	#11718350	1	2	yes	—
309	C₄	#11718369	1	2	yes	—
310	C₄	#11718369	1	2	yes	—
311	C₄	C₂×#88155	1	2	yes	—
312	C₄	C₂×#88161	1	2	yes	—
313	C₄	C₂×#88161	1	2	yes	—
314	C₂	—	52	2	no	—
315	C₂	#13113137	1	4	yes	direct product factor
316	C₂	C₂×#11718350	1	1	yes	characteristic
317	C₂	C₂×#11718369	1	1	yes	characteristic
318	C₂	C₂×#11718369	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂³×C₁₃	C₂×C₄²
prime radical	C₂³×D₁₃	C₄²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	C₂×#1454
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #13113137	C₂	direct (N × H)
5. C₂×#11718046	C₂	semidirect
6. C₂×#11718047	C₂	semidirect
7. C₂×C₈×#1055	C₂	semidirect
8. C₂×C₁₃×#322	C₂	semidirect
28. C₁₃×#322	C₂²	semidirect
29. #11718047	C₂²	semidirect
30. #11718046	C₂²	semidirect
31. C₈×#1055	C₂²	semidirect
112. C₂×#322	D₁₃	semidirect
167. #322	C₂×D₁₃	semidirect
221. C₂×C₁₃	#1454	semidirect
264. C₁₃	C₂×#1454	semidirect
315. C₂	#13113137	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×#11718346	2
#13113137	2
C₂×#11718046	2
C₂×#11718047	2
C₂×C₈×#1055	2
C₂×C₁₃×#322	2
C₂×#1454	13

Subgroups — C2×#13113137 #15828544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13113137 #15828544