Subgroups — C₂×#13130399 #16794938

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×#11719054	C₂	1	2	yes	maximal
2	#13130399	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718256	C₂	1	2	yes	maximal, complemented
4	C₂×#11718255	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88111	—	2	2	no	—
10	C₂×#88110	—	2	2	no	—
11	C₂×C₄⋊#1057	C₂²	1	4	yes	—
12	#11719054	—	2	4	no	—
13	#11719054	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₄	1	1	yes	characteristic
18	C₄×C₈×D₁₃	C₄	1	4	yes	—
19	C₄×C₈×D₁₃	C₂²	1	4	yes	—
20	#11718256	C₂²	1	8	yes	complemented
21	#11718255	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₁₃	Q₈	1	1	yes	characteristic
25	C₂²×#1057	—	2	4	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	Q₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₁₃	—	2	4	no	—
35	C₂×C₈×D₁₃	—	2	4	no	—
36	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
38	C₄²×D₁₃	C₂×C₄	1	2	yes	—
39	C₄²×D₁₃	D₄	1	2	yes	—
40	C₄²×D₁₃	Q₈	1	2	yes	—
41	C₄²×D₁₃	C₂³	1	2	yes	—
42	#88111	—	2	8	no	—
43	#88110	—	2	8	no	—
44	C₄⋊#1057	—	2	4	no	—
45	C₄⋊#1057	C₂³	1	8	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₁₃	C₂³	1	2	yes	—
50	C₈×Dic₁₃	C₂³	1	2	yes	—
51	C₄×#1055	C₂×C₄	1	4	yes	—
52	C₂×#1568	—	13	1	no	maximal, Sylow
53	C₂³×D₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×D₁₃	—	2	4	no	—
61	C₂×C₄×D₁₃	—	2	2	no	—
62	C₂×C₄×D₁₃	—	2	2	no	—
63	C₂×C₄×D₁₃	—	2	4	no	—
64	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
66	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₂×Q₈	1	2	yes	—
70	C₂×#1057	—	2	8	no	—
71	C₂×#1057	—	4	8	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₃	C₂²×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	1	yes	characteristic
81	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
83	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
84	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
85	C₄×Dic₁₃	C₂⁴	1	1	yes	characteristic
86	C₂×#1055	M₁₆	1	4	yes	—
87	C₂×#1055	M₁₆	1	4	yes	—
88	C₂×#1055	C₂²×C₄	1	2	yes	—
89	C₂×#1055	C₂²×C₄	1	2	yes	—
90	C₈×D₁₃	—	2	8	no	—
91	C₈×D₁₃	—	2	8	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×D₁₃	—	2	2	no	—
99	C₂²×D₁₃	—	2	2	no	—
100	C₂²×D₁₃	#108	1	4	yes	—
101	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	#135	1	4	yes	—
116	C₂×Dic₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₄×D₁₃	—	2	4	no	—
119	C₄×D₁₃	—	2	4	no	—
120	C₄×D₁₃	—	2	2	no	—
121	C₄×D₁₃	—	2	2	no	—
122	C₄×D₁₃	—	2	4	no	—
123	C₄×D₁₃	—	2	4	no	—
124	C₄×D₁₃	—	2	4	no	—
125	#1057	—	4	8	no	—
126	C₈×C₁₃	—	2	2	no	—
127	C₈×C₁₃	—	2	2	no	—
128	#1055	C₂×M₁₆	1	4	yes	—
129	#1055	C₂×M₁₆	1	4	yes	—
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	—	13	1	no	—
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	1	no	—
141	C₂×C₄×C₈	—	13	4	no	—
142	C₂×C₄×C₈	—	13	4	no	—
143	C₂×C₄×C₈	#261	1	1	yes	characteristic
144	#335	—	13	8	no	—
145	#334	—	13	8	no	—
146	C₂²×C₁₃	#423	1	2	yes	—
147	C₂²×C₁₃	#427	1	2	yes	—
148	C₂²×C₁₃	C₂×#108	1	2	yes	—
149	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂×D₁₃	—	2	8	no	—
151	C₂×D₁₃	—	2	2	no	—
152	C₂×D₁₃	—	2	2	no	—
153	C₂×D₁₃	C₂×#108	1	4	yes	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	#423	1	2	yes	—
157	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
159	Dic₁₃	—	2	2	no	—
160	Dic₁₃	—	2	1	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	—	13	1	no	—
167	C₂²×C₈	—	13	1	no	—
168	C₂²×C₈	—	13	1	no	—
169	C₂²×C₈	—	13	4	no	—
170	C₂²×C₈	—	13	4	no	—
171	C₂²×C₈	—	26	4	no	—
172	C₂²×C₈	—	26	4	no	—
173	C₂²×C₈	—	26	4	no	—
174	C₂²×C₈	C₂×#261	1	1	yes	characteristic
175	C₂²×C₈	C₂×#261	1	1	yes	characteristic
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	—	13	1	no	—
178	C₂×C₄²	—	13	1	no	—
179	C₂×C₄²	—	13	2	no	—
180	C₂×C₄²	—	13	2	no	—
181	C₂×C₄²	—	13	2	no	—
182	C₂×C₄²	—	13	2	no	—
183	C₂×C₄²	C₂×#261	1	1	yes	characteristic
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	2	no	—
192	C₄×C₈	—	13	4	no	—
193	C₄×C₈	C₂×#261	1	4	yes	—
194	C₂×C₁₃	#1568	1	4	yes	complemented
195	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
197	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
198	D₁₃	—	2	4	no	—
199	C₂⁴	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	1	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	2	no	—
208	C₂²×C₄	—	13	2	no	—
209	C₂²×C₄	—	26	4	no	—
210	C₂²×C₄	—	26	4	no	—
211	C₂²×C₄	—	26	2	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
215	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	13	4	no	—
219	C₂×C₈	—	13	4	no	—
220	C₂×C₈	—	13	2	no	—
221	C₂×C₈	—	13	2	no	—
222	C₂×C₈	—	26	8	no	—
223	C₂×C₈	—	26	8	no	—
224	C₂×C₈	—	26	8	no	—
225	C₂×C₈	—	52	8	no	—
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	1	no	—
233	C₄²	—	13	2	no	—
234	C₄²	—	13	1	no	—
235	C₄²	—	13	1	no	—
236	C₄²	C₄⋊#261	1	2	yes	—
237	C₄²	C₂²×#261	1	2	yes	—
238	C₁₃	C₂×#1568	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	13	4	no	—
240	C₂³	—	13	2	no	—
241	C₂³	—	26	2	no	—
242	C₂³	—	26	2	no	—
243	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	13	4	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	26	4	no	—
253	C₂×C₄	—	26	4	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	2	no	—
261	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
262	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
264	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
265	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
266	C₂×C₄	C₂³×#261	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	13	4	no	—
270	C₈	—	13	4	no	—
271	C₈	—	52	8	no	—
272	C₂²	—	13	4	no	—
273	C₂²	—	26	8	no	—
274	C₂²	—	26	2	no	—
275	C₂²	—	26	2	no	—
276	C₂²	#11719053	1	2	yes	—
277	C₂²	#11719092	1	2	yes	—
278	C₂²	D₁₃⋊#108	1	2	yes	—
279	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	13	2	no	—
283	C₄	—	13	2	no	—
284	C₄	—	26	2	no	—
285	C₄	—	26	1	no	—
286	C₄	—	26	1	no	—
287	C₄	#11719053	1	2	yes	—
288	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
289	C₄	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
290	C₂	—	26	4	no	—
291	C₂	#13130399	1	4	yes	direct product factor
292	C₂	C₂×#11719053	1	1	yes	characteristic
293	C₂	C₂×#11719092	1	1	yes	characteristic
294	C₂	C₂×D₁₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₄⋊C₄
cosocle	C₄×Dic₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1568
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13130399	C₂	direct (N × H)
3. C₂×#11718256	C₂	semidirect
4. C₂×#11718255	C₂	semidirect
20. #11718256	C₂²	semidirect
21. #11718255	C₂²	semidirect
194. C₂×C₁₃	#1568	semidirect
238. C₁₃	C₂×#1568	semidirect
291. C₂	#13130399	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×#11719054	2
#13130399	2
C₂×#11718256	2
C₂×#11718255	2
C₂×C₄×C₈×D₁₃	2
C₂×#1568	13

Subgroups — C2×#13130399 #16794938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130399 #16794938