Subgroups — C₂×#11719379 #13128146

Order 1664.

1896 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88201	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88194	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719379	C₂	1	16	yes	maximal, direct product factor
7	C₂×#88138	C₂	1	4	yes	maximal, complemented
8	C₂×#88136	C₂	1	4	yes	maximal, complemented
9	C₂×#88128	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88126	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×D₁₃	C₂²	1	2	yes	—
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
15	C₂²×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
16	C₂×#6131	—	2	8	no	—
17	C₂×#6131	C₂²	1	8	yes	complemented
18	C₂×D₄×D₁₃	—	2	8	no	—
19	C₂×D₄×D₁₃	C₂²	1	8	yes	complemented
20	C₂²×D₅₂	C₂²	1	1	yes	characteristic
21	C₂²×D₅₂	C₂²	1	2	yes	—
22	C₂²×D₅₂	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₅₂	C₂²	1	4	yes	complemented
24	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₁₃	C₂²	1	2	yes	complemented
26	C₁₃×#124	C₂²	1	8	yes	complemented
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
29	#88201	C₂²	1	8	yes	complemented
30	#88194	C₂²	1	8	yes	complemented
31	C₂×#6103	C₂²	1	2	yes	—
32	C₂×#6103	C₂²	1	4	yes	complemented
33	C₂×#6101	C₂²	1	2	yes	complemented
34	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic, complemented
35	#88138	C₂²	1	32	yes	complemented
36	#88136	C₂²	1	32	yes	complemented
37	#88128	C₂²	1	8	yes	complemented
38	#88126	C₂²	1	8	yes	complemented
39	C₂⁴×C₁₃	C₂³	1	2	yes	—
40	C₂³×D₁₃	—	2	8	no	—
41	C₂³×D₁₃	—	4	8	no	—
42	C₂³×D₁₃	C₂³	1	2	yes	—
43	C₂³×D₁₃	C₂³	1	2	yes	—
44	C₂³×D₁₃	C₂³	1	1	yes	characteristic
45	C₂³×D₁₃	C₂³	1	8	yes	—
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	C₂³	1	4	yes	complemented
48	C₂×D₄×C₁₃	—	2	4	no	—
49	C₂×D₄×C₁₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₃	—	2	8	no	—
54	C₂×C₂²⋊D₁₃	—	4	8	no	—
55	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
56	C₂×C₂²⋊D₁₃	C₂³	1	16	yes	complemented
57	C₂²×Dic₁₃	C₂³	1	2	yes	—
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	#6131	—	2	32	no	—
60	D₄×D₁₃	—	2	32	no	—
61	C₂×D₅₂	—	2	8	no	—
62	C₂×D₅₂	—	2	4	no	—
63	C₂×D₅₂	—	2	4	no	—
64	C₂×D₅₂	—	4	8	no	—
65	C₂×D₅₂	C₂³	1	4	yes	—
66	C₂×D₅₂	C₂³	1	8	yes	—
67	C₂×D₅₂	C₂³	1	4	yes	complemented
68	C₂×D₅₂	C₂³	1	16	yes	complemented
69	C₂×C₄×D₁₃	—	2	8	no	—
70	C₂×C₄×D₁₃	D₄	1	8	yes	complemented
71	C₂×C₄×D₁₃	C₂³	1	4	yes	—
72	C₂×C₄×D₁₃	C₂³	1	8	yes	complemented
73	C₁₃×C₂²⋊C₄	C₂³	1	16	yes	complemented
74	C₄²×C₁₃	C₂³	1	4	yes	complemented
75	#6103	C₂³	1	8	yes	—
76	#6103	C₂³	1	16	yes	complemented
77	#6101	C₂³	1	8	yes	complemented
78	C₄×Dic₁₃	C₂³	1	4	yes	complemented
79	C₂×#535	—	13	1	no	maximal, Sylow
80	C₂³×C₁₃	—	2	4	no	—
81	C₂³×C₁₃	—	2	8	no	—
82	C₂³×C₁₃	C₂⁴	1	4	yes	—
83	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	4	no	—
85	C₂²×D₁₃	—	2	4	no	—
86	C₂²×D₁₃	—	2	4	no	—
87	C₂²×D₁₃	—	2	8	no	—
88	C₂²×D₁₃	—	2	8	no	—
89	C₂²×D₁₃	—	2	16	no	—
90	C₂²×D₁₃	—	4	16	no	—
91	C₂²×D₁₃	—	4	32	no	—
92	C₂²×D₁₃	C₂×D₄	1	4	yes	—
93	C₂²×D₁₃	C₂⁴	1	4	yes	—
94	C₂²×D₁₃	C₂⁴	1	4	yes	—
95	C₂²×D₁₃	C₂⁴	1	2	yes	—
96	Q₈×C₁₃	—	2	8	no	—
97	D₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
104	C₂²⋊D₁₃	—	2	16	no	—
105	C₂²⋊D₁₃	—	4	16	no	—
106	C₂×Dic₁₃	—	2	4	no	—
107	C₂×Dic₁₃	C₂×D₄	1	4	yes	—
108	C₂×Dic₁₃	C₂⁴	1	4	yes	—
109	C₂×Dic₁₃	C₂⁴	1	2	yes	—
110	D₅₂	—	2	16	no	—
111	D₅₂	—	2	8	no	—
112	D₅₂	—	2	8	no	—
113	D₅₂	—	4	16	no	—
114	C₄×D₁₃	—	2	16	no	—
115	C₄×D₁₃	C₂×D₄	1	16	yes	complemented
116	C₂²×#41	—	13	1	no	—
117	C₂³×D₄	—	13	1	no	—
118	#535	—	13	16	no	—
119	C₂×#127	—	13	1	no	—
120	C₂×#127	—	13	1	no	—
121	C₂×#124	—	13	1	no	—
122	C₂×#124	D₁₃	1	1	yes	characteristic, complemented
123	C₂×#121	—	13	4	no	—
124	C₂×#120	—	13	4	no	—
125	C₂×#117	—	13	1	no	—
126	C₂²×C₁₃	—	2	4	no	—
127	C₂²×C₁₃	—	2	8	no	—
128	C₂²×C₁₃	—	4	8	no	—
129	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
130	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
131	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
132	C₂×D₁₃	—	2	8	no	—
133	C₂×D₁₃	—	2	4	no	—
134	C₂×D₁₃	—	2	4	no	—
135	C₂×D₁₃	—	2	4	no	—
136	C₂×D₁₃	—	2	8	no	—
137	C₂×D₁₃	—	2	8	no	—
138	C₂×D₁₃	—	4	32	no	—
139	C₂×D₁₃	—	4	8	no	—
140	C₂×D₁₃	—	4	8	no	—
141	C₂×D₁₃	C₂²×D₄	1	4	yes	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	C₂²×D₄	1	4	yes	—
145	Dic₁₃	—	2	4	no	—
146	Dic₁₃	C₂²×D₄	1	4	yes	—
147	C₂⁵	—	13	2	no	—
148	C₂×#41	—	13	8	no	—
149	C₂×#41	—	26	8	no	—
150	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
151	C₂²×D₄	—	13	2	no	—
152	C₂²×D₄	—	13	4	no	—
153	C₂²×D₄	—	13	8	no	—
154	C₂²×D₄	—	13	2	no	—
155	C₂²×D₄	—	13	1	no	—
156	C₂²×D₄	—	13	1	no	—
157	C₂²×D₄	—	13	4	no	—
158	C₂²×D₄	—	26	8	no	—
159	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
160	C₂³×C₄	—	13	1	no	—
161	C₂³×C₄	—	13	2	no	—
162	#127	—	13	8	no	—
163	#127	—	13	8	no	—
164	#124	—	13	8	no	—
165	#124	C₂×D₁₃	1	8	yes	complemented
166	#121	—	13	32	no	—
167	#120	—	13	32	no	—
168	#117	—	13	8	no	—
169	C₂×C₄⋊C₄	—	13	2	no	—
170	C₂×C₂²⋊C₄	—	13	2	no	—
171	C₂×C₂²⋊C₄	—	13	4	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₃	1	4	yes	complemented
173	C₂×C₄²	—	13	1	no	—
174	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
175	C₂×C₁₃	—	4	4	no	—
176	C₂×C₁₃	#535	1	4	yes	complemented
177	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
178	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
179	D₁₃	—	2	4	no	—
180	D₁₃	—	4	4	no	—
181	D₁₃	—	4	4	no	—
182	C₂⁴	—	13	1	no	—
183	C₂⁴	—	13	2	no	—
184	C₂⁴	—	13	2	no	—
185	C₂⁴	—	13	8	no	—
186	C₂⁴	—	26	8	no	—
187	C₂⁴	—	52	8	no	—
188	C₂⁴	C₂²×D₁₃	1	2	yes	—
189	#41	—	26	32	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	C₂²×D₁₃	1	4	yes	complemented
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	13	8	no	—
194	C₂×D₄	—	13	16	no	—
195	C₂×D₄	—	13	8	no	—
196	C₂×D₄	—	13	4	no	—
197	C₂×D₄	—	13	4	no	—
198	C₂×D₄	—	13	16	no	—
199	C₂×D₄	—	26	8	no	—
200	C₂×D₄	—	26	32	no	—
201	C₂×D₄	—	26	8	no	—
202	C₂×D₄	—	26	4	no	—
203	C₂×D₄	—	26	4	no	—
204	C₂×D₄	—	52	8	no	—
205	C₂×D₄	—	52	8	no	—
206	C₂×D₄	C₂²×D₁₃	1	4	yes	—
207	C₂²×C₄	—	13	1	no	—
208	C₂²×C₄	—	13	8	no	—
209	C₂²×C₄	—	13	4	no	—
210	C₂²×C₄	—	13	2	no	—
211	C₂²×C₄	—	13	8	no	—
212	C₂²×C₄	—	26	8	no	—
213	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
214	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
215	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	13	8	no	—
217	C₂²⋊C₄	—	13	8	no	—
218	C₂²⋊C₄	—	13	16	no	—
219	C₂²⋊C₄	C₂²×D₁₃	1	16	yes	complemented
220	C₄²	—	13	4	no	—
221	C₄²	C₂²×D₁₃	1	4	yes	complemented
222	C₁₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	13	4	no	—
226	C₂³	—	13	2	no	—
227	C₂³	—	13	4	no	—
228	C₂³	—	13	4	no	—
229	C₂³	—	26	16	no	—
230	C₂³	—	26	4	no	—
231	C₂³	—	26	4	no	—
232	C₂³	—	26	4	no	—
233	C₂³	—	26	8	no	—
234	C₂³	—	26	8	no	—
235	C₂³	—	52	16	no	—
236	C₂³	—	52	32	no	—
237	C₂³	C₂³×D₁₃	1	4	yes	—
238	C₂³	C₂³×D₁₃	1	1	yes	characteristic
239	Q₈	—	2	8	no	—
240	D₄	—	2	8	no	—
241	D₄	—	26	16	no	—
242	D₄	—	26	16	no	—
243	D₄	—	26	8	no	—
244	D₄	—	26	8	no	—
245	D₄	—	52	16	no	—
246	D₄	—	52	16	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	13	4	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	16	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	26	16	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	D₄×D₁₃	1	4	yes	—
256	C₂×C₄	C₂³×D₁₃	1	4	yes	—
257	C₂×C₄	C₂³×D₁₃	1	4	yes	—
258	C₂×C₄	C₂³×D₁₃	1	2	yes	—
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	13	4	no	—
263	C₂²	—	26	8	no	—
264	C₂²	—	26	4	no	—
265	C₂²	—	26	4	no	—
266	C₂²	—	26	4	no	—
267	C₂²	—	26	8	no	—
268	C₂²	—	26	8	no	—
269	C₂²	—	52	32	no	—
270	C₂²	—	52	8	no	—
271	C₂²	—	52	8	no	—
272	C₂²	C₂×D₄×D₁₃	1	2	yes	—
273	C₂²	#88256	1	4	yes	—
274	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	4	no	—
277	C₄	—	13	4	no	—
278	C₄	—	26	4	no	—
279	C₄	C₂×D₄×D₁₃	1	4	yes	—
280	C₂	—	4	4	no	—
281	C₂	—	26	4	no	—
282	C₂	—	52	4	no	—
283	C₂	—	52	4	no	—
284	C₂	#11719379	1	4	yes	direct product factor
285	C₂	C₂²×D₄×D₁₃	1	1	yes	characteristic
286	C₂	C₂×#88256	1	2	yes	—

	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
hypercenter	C₂×#124	D₁₃
hypocenter	C₁₃	C₂×#535
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
1. C₂²×#6131	C₂	semidirect
2. C₂²×D₄×D₁₃	C₂	semidirect
3. C₂×C₁₃×#124	C₂	semidirect
4. C₂×#88201	C₂	semidirect
5. C₂×#88194	C₂	semidirect
6. #11719379	C₂	direct (N × H)
7. C₂×#88138	C₂	semidirect
8. C₂×#88136	C₂	semidirect
9. C₂×#88128	C₂	semidirect
10. C₂×#88126	C₂	semidirect
12. C₂²×Q₈×C₁₃	C₂²	semidirect
15. C₂²×C₂²⋊D₁₃	C₂²	semidirect
17. C₂×#6131	C₂²	semidirect
19. C₂×D₄×D₁₃	C₂²	semidirect
22. C₂²×D₅₂	C₂²	semidirect
23. C₂²×D₅₂	C₂²	semidirect
25. C₂²×C₄×D₁₃	C₂²	semidirect
26. C₁₃×#124	C₂²	semidirect
27. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₄²×C₁₃	C₂²	semidirect
29. #88201	C₂²	semidirect
30. #88194	C₂²	semidirect
32. C₂×#6103	C₂²	semidirect
33. C₂×#6101	C₂²	semidirect
34. C₂×C₄×Dic₁₃	C₂²	semidirect
35. #88138	C₂²	semidirect
36. #88136	C₂²	semidirect
37. #88128	C₂²	semidirect
38. #88126	C₂²	semidirect
47. C₂×Q₈×C₁₃	C₂³	semidirect
56. C₂×C₂²⋊D₁₃	C₂³	semidirect
67. C₂×D₅₂	C₂³	semidirect
68. C₂×D₅₂	C₂³	semidirect
70. C₂×C₄×D₁₃	D₄	semidirect
72. C₂×C₄×D₁₃	C₂³	semidirect
73. C₁₃×C₂²⋊C₄	C₂³	semidirect
74. C₄²×C₁₃	C₂³	semidirect
76. #6103	C₂³	semidirect
77. #6101	C₂³	semidirect
78. C₄×Dic₁₃	C₂³	semidirect
115. C₄×D₁₃	C₂×D₄	semidirect
122. C₂×#124	D₁₃	semidirect
150. C₂²×Q₈	C₂×D₁₃	semidirect
165. #124	C₂×D₁₃	semidirect
172. C₂×C₂²⋊C₄	C₂×D₁₃	semidirect
174. C₂×C₄²	C₂×D₁₃	semidirect
176. C₂×C₁₃	#535	semidirect
191. C₂×Q₈	C₂²×D₁₃	semidirect
219. C₂²⋊C₄	C₂²×D₁₃	semidirect
221. C₄²	C₂²×D₁₃	semidirect
222. C₁₃	C₂×#535	semidirect
284. C₂	#11719379	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#6131	2
C₂²×D₄×D₁₃	2
C₂×C₁₃×#124	2
C₂×#88201	2
C₂×#88194	2
#11719379	2
C₂×#88138	2
C₂×#88136	2
C₂×#88128	2
C₂×#88126	2
C₂×#535	13

Subgroups — C2×#11719379 #13128146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719379 #13128146