Subgroups — C₂×#13130398 #16794937

Order 3328.

766 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×#11719130	C₂	1	2	yes	maximal
2	#13130398	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718254	C₂	1	2	yes	maximal, complemented
4	C₂×#11718255	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₂³×#1055	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊#1057	—	2	2	no	—
10	#11719130	—	2	4	no	—
11	#11719130	C₂²	1	8	yes	—
12	C₂×#88110	—	2	2	no	—
13	C₂×C₄⋊#1057	C₂²	1	4	yes	—
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₄	1	4	yes	—
16	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
20	#11718255	C₂²	1	8	yes	complemented
21	#11718254	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₁₃	D₄	1	1	yes	characteristic
24	C₂³×Dic₁₃	Q₈	1	1	yes	characteristic
25	C₂²×#1057	—	2	4	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
34	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
35	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
38	C₂²×#1055	—	2	4	no	—
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₂²×#1055	C₂×C₄	1	4	yes	—
41	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
42	C₂²⋊#1057	—	2	8	no	—
43	#88110	—	2	8	no	—
44	C₄⋊#1057	—	2	4	no	—
45	C₄⋊#1057	C₂³	1	8	yes	—
46	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
50	C₈×Dic₁₃	C₂³	1	2	yes	—
51	C₈×Dic₁₃	C₂³	1	2	yes	—
52	C₂×#1568	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	—	2	4	no	—
61	C₂²×Dic₁₃	—	2	4	no	—
62	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
63	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
66	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
67	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×#1057	—	2	8	no	—
71	C₂×#1057	—	4	8	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	8	no	—
74	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
79	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
80	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
81	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
84	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
85	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	—	2	8	no	—
89	C₂×#1055	M₁₆	1	4	yes	—
90	C₂×#1055	C₂²×C₄	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	2	yes	—
92	C₂×#432	—	13	2	no	—
93	C₂²×C₄×C₈	—	13	1	no	—
94	#1568	—	13	8	no	—
95	C₂×#335	—	13	2	no	—
96	C₂×#334	—	13	2	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	#108	1	4	yes	—
100	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	—	2	4	no	—
113	C₂×Dic₁₃	—	2	2	no	—
114	C₂×Dic₁₃	—	2	4	no	—
115	C₂×Dic₁₃	—	2	2	no	—
116	C₂×Dic₁₃	—	2	4	no	—
117	C₂×Dic₁₃	—	2	4	no	—
118	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	#135	1	4	yes	—
123	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
125	#1057	—	4	8	no	—
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
128	#1055	—	2	2	no	—
129	#1055	C₂×M₁₆	1	4	yes	—
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	#261	1	1	yes	characteristic
132	C₂²×C₄²	—	13	1	no	—
133	#432	—	13	8	no	—
134	#432	—	26	4	no	—
135	C₂×C₄⋊C₈	—	13	4	no	—
136	C₂×C₂²⋊C₈	—	26	2	no	—
137	C₂×#97	—	26	2	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	C₂×C₄×C₈	—	13	4	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	C₂×C₄×C₈	—	13	4	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	#335	—	13	8	no	—
145	#334	—	13	8	no	—
146	C₂²×C₁₃	—	2	8	no	—
147	C₂²×C₁₃	—	2	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	#423	1	2	yes	—
150	C₂²×C₁₃	#427	1	2	yes	—
151	C₂²×C₁₃	C₂×#108	1	4	yes	—
152	C₂²×C₁₃	C₂×#108	1	2	yes	—
153	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	#423	1	2	yes	—
156	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
157	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
158	Dic₁₃	—	2	2	no	—
159	Dic₁₃	—	2	1	no	—
160	Dic₁₃	—	2	2	no	—
161	Dic₁₃	—	2	1	no	—
162	Dic₁₃	#427	1	2	yes	—
163	Dic₁₃	C₂×#135	1	2	yes	—
164	C₂³×C₄	—	13	1	no	—
165	C₂³×C₄	—	13	1	no	—
166	C₂³×C₄	C₂×#261	1	1	yes	characteristic
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	13	1	no	—
169	C₂²×C₈	—	13	4	no	—
170	C₂²×C₈	—	13	1	no	—
171	C₂²×C₈	—	26	4	no	—
172	C₂²×C₈	—	26	4	no	—
173	C₂²×C₈	C₂×#261	1	1	yes	characteristic
174	C₂²×C₈	C₂×#261	1	4	yes	—
175	C₂²×C₈	C₂×#261	1	1	yes	characteristic
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	2	no	—
178	C₂×C₄²	—	13	2	no	—
179	C₂×C₄²	—	13	1	no	—
180	C₂×C₄²	—	13	2	no	—
181	C₂×C₄²	—	13	1	no	—
182	C₂×C₄²	—	13	1	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₄⋊C₈	—	13	8	no	—
185	C₄⋊C₈	—	26	4	no	—
186	C₂²⋊C₈	—	26	8	no	—
187	#97	—	26	8	no	—
188	C₄×C₈	—	13	2	no	—
189	C₄×C₈	—	13	2	no	—
190	C₄×C₈	—	13	2	no	—
191	C₄×C₈	—	13	4	no	—
192	C₄×C₈	—	13	4	no	—
193	C₄×C₈	—	13	2	no	—
194	C₂×C₁₃	—	2	4	no	—
195	C₂×C₁₃	#1568	1	4	yes	complemented
196	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
198	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
199	C₂⁴	C₄⋊#261	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	2	no	—
204	C₂²×C₄	—	13	2	no	—
205	C₂²×C₄	—	13	2	no	—
206	C₂²×C₄	—	13	2	no	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	13	1	no	—
210	C₂²×C₄	—	26	4	no	—
211	C₂²×C₄	—	26	4	no	—
212	C₂²×C₄	C₄⋊#261	1	2	yes	—
213	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
214	C₂²×C₄	C₂²×#261	1	2	yes	—
215	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	13	2	no	—
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	—	26	8	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	—	26	8	no	—
224	C₂×C₈	—	52	8	no	—
225	C₂×C₈	#6122	1	4	yes	—
226	C₂×C₈	C₂²×#261	1	2	yes	—
227	C₂×C₈	C₂²×#261	1	2	yes	—
228	C₄²	—	13	2	no	—
229	C₄²	—	13	1	no	—
230	C₄²	—	13	2	no	—
231	C₄²	—	13	2	no	—
232	C₄²	—	13	2	no	—
233	C₄²	—	13	1	no	—
234	C₄²	—	13	2	no	—
235	C₄²	—	13	1	no	—
236	C₄²	—	13	1	no	—
237	C₄²	—	13	2	no	—
238	C₁₃	C₂×#1568	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	#88104	1	4	yes	—
242	C₂³	C₂×C₄⋊#261	1	2	yes	—
243	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	13	4	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	2	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	13	1	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	4	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
265	C₂×C₄	C₂×#6122	1	2	yes	—
266	C₂×C₄	C₂³×#261	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	13	4	no	—
269	C₈	—	26	2	no	—
270	C₈	—	52	8	no	—
271	C₈	C₂×#6122	1	4	yes	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	#11719092	1	2	yes	—
276	C₂²	C₂×#88104	1	4	yes	—
277	C₂²	C₂×#88104	1	2	yes	—
278	C₂²	#11719129	1	2	yes	—
279	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	13	2	no	—
282	C₄	—	13	2	no	—
283	C₄	—	26	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	26	1	no	—
286	C₄	—	26	1	no	—
287	C₄	#11719129	1	2	yes	—
288	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
289	C₄	C₂²×#6122	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	#13130398	1	4	yes	direct product factor
292	C₂	C₂×#11719129	1	1	yes	characteristic
293	C₂	C₂×#11719092	1	1	yes	characteristic
294	C₂	C₂²×#88104	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂³×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂³×C₈	#261
hypocenter	C₁₃	C₂×#1568
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #13130398	C₂	direct (N × H)
3. C₂×#11718254	C₂	semidirect
4. C₂×#11718255	C₂	semidirect
20. #11718255	C₂²	semidirect
21. #11718254	C₂²	semidirect
195. C₂×C₁₃	#1568	semidirect
238. C₁₃	C₂×#1568	semidirect
291. C₂	#13130398	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×#11719130	2
#13130398	2
C₂×#11718254	2
C₂×#11718255	2
C₂²×C₈×Dic₁₃	2
C₂×#1568	13

Subgroups — C2×#13130398 #16794937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130398 #16794937