Subgroups — C₂×#11719376 #13128381

Order 1664.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88198	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88190	C₂	1	1	yes	maximal, characteristic
5	C₂×D₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	#11719376	C₂	1	16	yes	maximal, direct product factor
7	C₂×#88133	C₂	1	2	yes	maximal, complemented
8	C₂×#88133	C₂	1	2	yes	maximal, complemented
9	C₂×#88132	C₂	1	2	yes	maximal, complemented
10	C₂×#88131	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×Dic₁₃	C₂²	1	2	yes	—
16	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
17	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
18	C₁₃×#124	C₂²	1	8	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
25	Q₈×Dic₁₃	C₂²	1	8	yes	complemented
26	#88198	C₂²	1	8	yes	complemented
27	#88190	C₂²	1	8	yes	—
28	D₄×Dic₁₃	C₂²	1	8	yes	—
29	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
32	C₂×#6101	C₂²	1	1	yes	characteristic
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	1	yes	characteristic
35	C₂×#6101	C₂²	1	2	yes	—
36	C₂×#6101	C₂²	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
38	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
41	#88133	C₂²	1	16	yes	complemented
42	#88133	C₂²	1	16	yes	complemented
43	#88132	C₂²	1	16	yes	complemented
44	#88131	C₂²	1	16	yes	complemented
45	C₄×Dic₂₆	C₂²	1	8	yes	complemented
46	C₄×Dic₂₆	C₂²	1	8	yes	complemented
47	C₂⁴×C₁₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
50	C₂×D₄×C₁₃	—	2	4	no	—
51	C₂×D₄×C₁₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	8	no	—
57	C₂²×Dic₁₃	C₂³	1	8	yes	—
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂³	1	2	yes	—
62	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₂×Dic₂₆	—	2	4	no	—
65	C₂×Dic₂₆	—	2	4	no	—
66	C₂×Dic₂₆	C₂³	1	4	yes	—
67	C₂×Dic₂₆	C₂³	1	4	yes	—
68	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
69	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
70	C₄²×C₁₃	C₂³	1	4	yes	—
71	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
72	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
73	C₂²⋊Dic₁₃	C₂³	1	8	yes	—
74	C₄⋊Dic₁₃	C₂³	1	4	yes	—
75	C₄⋊Dic₁₃	C₂³	1	4	yes	—
76	C₄⋊Dic₁₃	C₂³	1	8	yes	—
77	#6101	C₂³	1	4	yes	—
78	#6101	C₂³	1	8	yes	—
79	#6101	C₂³	1	4	yes	—
80	#6101	C₂³	1	8	yes	—
81	#6101	C₂³	1	8	yes	—
82	C₄×Dic₁₃	C₂³	1	8	yes	—
83	C₄×Dic₁₃	C₂³	1	4	yes	—
84	C₄×Dic₁₃	C₂³	1	4	yes	—
85	C₄×Dic₁₃	C₂³	1	8	yes	—
86	C₂×#556	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	—	2	4	no	—
89	C₂³×C₁₃	—	2	4	no	—
90	C₂³×C₁₃	C₂⁴	1	4	yes	—
91	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
92	Q₈×C₁₃	—	2	8	no	—
93	D₄×C₁₃	—	2	8	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	#41	1	4	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
99	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂×Dic₁₃	—	2	4	no	—
103	C₂×Dic₁₃	—	2	2	no	—
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	2	2	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	16	no	—
108	C₂×Dic₁₃	#41	1	4	yes	—
109	C₂×Dic₁₃	C₂⁴	1	2	yes	—
110	C₂×Dic₁₃	C₂⁴	1	4	yes	—
111	C₂×Dic₁₃	C₂⁴	1	2	yes	—
112	C₂×Dic₁₃	C₂⁴	1	4	yes	—
113	C₂×Dic₁₃	C₂⁴	1	2	yes	—
114	C₂×Dic₁₃	C₂⁴	1	2	yes	—
115	Dic₂₆	—	2	8	no	—
116	Dic₂₆	—	2	8	no	—
117	#556	—	13	16	no	—
118	C₂×C₄⋊Q₈	—	13	1	no	—
119	C₂×#126	—	13	2	no	—
120	C₂×#126	—	13	2	no	—
121	C₂×#124	—	13	1	no	—
122	C₂×#124	D₁₃	1	1	yes	characteristic
123	C₂×C₂²⋊Q₈	—	13	2	no	—
124	C₂×C₄×Q₈	—	13	1	no	—
125	C₂×C₄×Q₈	—	13	1	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	C₂×C₄×D₄	—	13	1	no	—
128	C₂×#117	—	13	2	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	—	4	8	no	—
133	C₂²×C₁₃	C₂×#41	1	2	yes	—
134	C₂²×C₁₃	C₂×#41	1	2	yes	—
135	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
136	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	4	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	C₂×#41	1	4	yes	—
141	Dic₁₃	—	2	4	no	—
142	Dic₁₃	—	2	2	no	—
143	Dic₁₃	—	2	4	no	—
144	Dic₁₃	—	2	2	no	—
145	Dic₁₃	—	2	2	no	—
146	Dic₁₃	C₂×#41	1	4	yes	—
147	C₂²×Q₈	—	13	1	no	—
148	C₂²×Q₈	—	13	1	no	—
149	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
150	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
151	C₂³×C₄	—	13	2	no	—
152	C₄⋊Q₈	—	13	8	no	—
153	#126	—	13	16	no	—
154	#126	—	13	16	no	—
155	#124	—	13	8	no	—
156	#124	C₂×D₁₃	1	8	yes	—
157	C₂²⋊Q₈	—	13	16	no	—
158	C₄×Q₈	—	13	8	no	—
159	C₄×Q₈	—	13	8	no	—
160	C₄×Q₈	—	13	8	no	—
161	C₄×D₄	—	13	8	no	—
162	#117	—	13	16	no	—
163	C₂×C₄⋊C₄	—	13	1	no	—
164	C₂×C₄⋊C₄	—	13	2	no	—
165	C₂×C₄⋊C₄	—	13	1	no	—
166	C₂×C₄⋊C₄	—	13	2	no	—
167	C₂×C₄⋊C₄	—	13	2	no	—
168	C₂×C₄⋊C₄	—	13	1	no	—
169	C₂×C₄⋊C₄	—	13	1	no	—
170	C₂×C₄⋊C₄	—	13	2	no	—
171	C₂×C₂²⋊C₄	—	13	2	no	—
172	C₂×C₂²⋊C₄	—	13	2	no	—
173	C₂×C₂²⋊C₄	—	13	2	no	—
174	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
176	C₂×C₄²	—	13	2	no	—
177	C₂×C₄²	—	13	2	no	—
178	C₂×C₄²	—	13	1	no	—
179	C₂×C₄²	—	13	1	no	—
180	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
181	C₂×C₁₃	—	4	4	no	—
182	C₂×C₁₃	#556	1	4	yes	complemented
183	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
184	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
185	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
186	C₂⁴	C₂²×D₁₃	1	2	yes	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	13	4	no	—
189	C₂×Q₈	—	13	4	no	—
190	C₂×Q₈	—	26	4	no	—
191	C₂×Q₈	—	26	4	no	—
192	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×D₁₃	1	4	yes	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	8	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	2	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	2	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	26	8	no	—
203	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
205	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
207	C₄⋊C₄	—	13	4	no	—
208	C₄⋊C₄	—	13	8	no	—
209	C₄⋊C₄	—	13	4	no	—
210	C₄⋊C₄	—	13	8	no	—
211	C₄⋊C₄	—	13	8	no	—
212	C₄⋊C₄	—	13	4	no	—
213	C₄⋊C₄	—	13	4	no	—
214	C₄⋊C₄	—	13	8	no	—
215	C₂²⋊C₄	—	13	8	no	—
216	C₂²⋊C₄	—	13	8	no	—
217	C₂²⋊C₄	—	13	8	no	—
218	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
219	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
220	C₄²	—	13	8	no	—
221	C₄²	—	13	8	no	—
222	C₄²	—	13	4	no	—
223	C₄²	—	13	4	no	—
224	C₄²	C₂²×D₁₃	1	4	yes	—
225	C₁₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	4	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	2	4	no	—
229	C₂³	C₂³×D₁₃	1	4	yes	—
230	C₂³	C₂³×D₁₃	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	26	8	no	—
233	Q₈	—	26	8	no	—
234	D₄	—	2	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	2	no	—
240	C₂×C₄	—	13	2	no	—
241	C₂×C₄	—	13	4	no	—
242	C₂×C₄	—	13	2	no	—
243	C₂×C₄	—	13	4	no	—
244	C₂×C₄	—	13	2	no	—
245	C₂×C₄	—	26	16	no	—
246	C₂×C₄	—	26	4	no	—
247	C₂×C₄	—	26	4	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	#6129	1	4	yes	—
252	C₂×C₄	C₂³×D₁₃	1	2	yes	—
253	C₂×C₄	C₂³×D₁₃	1	4	yes	—
254	C₂×C₄	C₂³×D₁₃	1	2	yes	—
255	C₂×C₄	C₂³×D₁₃	1	2	yes	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	4	8	no	—
260	C₂²	C₂×#6129	1	2	yes	—
261	C₂²	#41×D₁₃	1	2	yes	—
262	C₂²	#88257	1	2	yes	—
263	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	13	4	no	—
268	C₄	—	26	4	no	—
269	C₄	—	26	4	no	—
270	C₄	—	26	2	no	—
271	C₄	—	26	2	no	—
272	C₄	—	26	2	no	—
273	C₄	C₂×#6129	1	4	yes	—
274	C₂	—	4	4	no	—
275	C₂	#11719376	1	4	yes	direct product factor
276	C₂	C₂²×#6129	1	1	yes	characteristic
277	C₂	C₂×#41×D₁₃	1	1	yes	characteristic
278	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	C₂×#556
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₁₃	C₂	semidirect
3. C₂×#88198	C₂	semidirect
6. #11719376	C₂	direct (N × H)
7. C₂×#88133	C₂	semidirect
8. C₂×#88133	C₂	semidirect
9. C₂×#88132	C₂	semidirect
10. C₂×#88131	C₂	semidirect
11. C₂×C₄×Dic₂₆	C₂	semidirect
12. C₂×C₄×Dic₂₆	C₂	semidirect
25. Q₈×Dic₁₃	C₂²	semidirect
26. #88198	C₂²	semidirect
41. #88133	C₂²	semidirect
42. #88133	C₂²	semidirect
43. #88132	C₂²	semidirect
44. #88131	C₂²	semidirect
45. C₄×Dic₂₆	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
182. C₂×C₁₃	#556	semidirect
225. C₁₃	C₂×#556	semidirect
275. C₂	#11719376	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×C₁₃×#124	2
C₂×Q₈×Dic₁₃	2
C₂×#88198	2
C₂×#88190	2
C₂×D₄×Dic₁₃	2
#11719376	2
C₂×#88133	2
C₂×#88132	2
C₂×#88131	2
C₂×C₄×Dic₂₆	2
C₂×#556	13

Subgroups — C2×#11719376 #13128381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719376 #13128381