Subgroups — C₂×#11719399 #13128346

Order 1664.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719399	C₂	1	16	yes	maximal, direct product factor
3	C₂×#88152	C₂	1	2	yes	maximal, complemented
4	C₂×#88151	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88148	C₂	1	2	yes	maximal
7	C₂×#88146	C₂	1	2	yes	maximal, complemented
8	C₂×#88146	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#88144	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×D₅₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
17	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
18	C₁₃×#125	C₂²	1	8	yes	complemented
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×#6103	C₂²	1	1	yes	characteristic
25	C₂×#6103	C₂²	1	1	yes	characteristic
26	C₂×#6103	C₂²	1	2	yes	—
27	C₂×#6103	C₂²	1	2	yes	—
28	C₂×#6103	C₂²	1	2	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×C₄⋊Dic₁₃	C₂²	1	2	yes	—
33	C₂×#6101	C₂²	1	2	yes	—
34	C₂×#6101	C₂²	1	2	yes	—
35	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
37	#88152	C₂²	1	16	yes	complemented
38	#88151	C₂²	1	8	yes	complemented
39	#88151	C₂²	1	8	yes	complemented
40	#88148	C₂²	1	16	yes	—
41	#88146	C₂²	1	16	yes	complemented
42	#88146	C₂²	1	8	yes	complemented
43	C₄⋊C₄×D₁₃	C₂²	1	8	yes	complemented
44	#88144	C₂²	1	16	yes	complemented
45	C₄×D₅₂	C₂²	1	8	yes	—
46	C₄×Dic₂₆	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	1	yes	characteristic
48	C₂³×D₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂³	1	2	yes	—
55	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂³	1	2	yes	—
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂×D₅₂	—	2	4	no	—
59	C₂×D₅₂	C₂³	1	4	yes	—
60	C₂×C₄×D₁₃	—	2	8	no	—
61	C₂×C₄×D₁₃	—	2	4	no	—
62	C₂×C₄×D₁₃	—	2	4	no	—
63	C₂×C₄×D₁₃	C₂³	1	4	yes	—
64	C₂×C₄×D₁₃	C₂³	1	4	yes	—
65	C₂×C₄×D₁₃	C₂³	1	8	yes	—
66	C₂×Dic₂₆	—	2	4	no	—
67	C₂×Dic₂₆	C₂³	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
72	C₄²×C₁₃	C₂³	1	4	yes	—
73	#6103	C₂³	1	4	yes	—
74	#6103	C₂³	1	4	yes	—
75	#6103	C₂³	1	8	yes	—
76	#6103	C₂³	1	8	yes	—
77	#6103	C₂³	1	8	yes	—
78	C₄⋊Dic₁₃	C₂³	1	4	yes	—
79	C₄⋊Dic₁₃	C₂³	1	4	yes	—
80	C₄⋊Dic₁₃	C₂³	1	8	yes	—
81	C₄⋊Dic₁₃	C₂³	1	8	yes	—
82	#6101	C₂³	1	8	yes	—
83	#6101	C₂³	1	8	yes	—
84	C₄×Dic₁₃	C₂³	1	8	yes	—
85	C₄×Dic₁₃	C₂³	1	8	yes	—
86	C₂×#552	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₁₃	—	2	2	no	—
89	C₂²×D₁₃	—	2	2	no	—
90	C₂²×D₁₃	—	2	2	no	—
91	C₂²×D₁₃	—	2	2	no	—
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	#41	1	4	yes	—
94	C₂²×D₁₃	C₂⁴	1	2	yes	—
95	C₂²×D₁₃	C₂⁴	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	#41	1	4	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
106	C₂×Dic₁₃	—	2	4	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	2	2	no	—
109	C₂×Dic₁₃	—	2	4	no	—
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	D₅₂	—	2	8	no	—
115	C₄×D₁₃	—	2	16	no	—
116	C₄×D₁₃	—	2	8	no	—
117	C₄×D₁₃	—	2	8	no	—
118	Dic₂₆	—	2	8	no	—
119	#552	—	13	16	no	—
120	C₂×#126	—	13	2	no	—
121	C₂×#125	—	13	2	no	—
122	C₂×#125	D₁₃	1	1	yes	characteristic, complemented
123	C₂×#123	—	13	2	no	—
124	C₂×C₂²⋊Q₈	—	13	1	no	—
125	C₂×C₂²⋊Q₈	—	13	1	no	—
126	C₂×C₄×Q₈	—	13	1	no	—
127	C₂×C₄×D₄	—	13	1	no	—
128	C₂×#117	—	13	1	no	—
129	C₂×#117	—	13	2	no	—
130	C₂²×C₄⋊C₄	—	13	1	no	—
131	C₂²×C₁₃	C₂×#41	1	2	yes	—
132	C₂²×C₁₃	C₂×#41	1	2	yes	—
133	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
134	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₁₃	—	2	8	no	—
136	C₂×D₁₃	—	2	2	no	—
137	C₂×D₁₃	—	2	2	no	—
138	C₂×D₁₃	—	2	2	no	—
139	C₂×D₁₃	—	2	2	no	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	4	4	no	—
142	C₂×D₁₃	C₂×#41	1	4	yes	—
143	C₄×C₁₃	—	2	4	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	C₂×#41	1	4	yes	—
148	Dic₁₃	—	2	4	no	—
149	Dic₁₃	—	2	2	no	—
150	Dic₁₃	—	2	2	no	—
151	Dic₁₃	—	2	4	no	—
152	C₂²×Q₈	—	13	1	no	—
153	C₂²×D₄	—	13	1	no	—
154	C₂³×C₄	—	13	2	no	—
155	C₂³×C₄	—	13	1	no	—
156	C₂³×C₄	—	13	1	no	—
157	#126	—	13	16	no	—
158	#125	—	13	16	no	—
159	#125	C₂×D₁₃	1	8	yes	complemented
160	#123	—	13	16	no	—
161	C₂²⋊Q₈	—	13	8	no	—
162	C₂²⋊Q₈	—	13	8	no	—
163	C₄×Q₈	—	13	8	no	—
164	C₄×D₄	—	13	8	no	—
165	#117	—	13	8	no	—
166	#117	—	13	16	no	—
167	C₂×C₄⋊C₄	—	13	1	no	—
168	C₂×C₄⋊C₄	—	13	2	no	—
169	C₂×C₄⋊C₄	—	13	2	no	—
170	C₂×C₄⋊C₄	—	13	1	no	—
171	C₂×C₄⋊C₄	—	13	2	no	—
172	C₂×C₄⋊C₄	—	13	2	no	—
173	C₂×C₄⋊C₄	—	13	8	no	—
174	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
178	C₂×C₂²⋊C₄	—	13	1	no	—
179	C₂×C₂²⋊C₄	—	13	2	no	—
180	C₂×C₂²⋊C₄	—	13	2	no	—
181	C₂×C₂²⋊C₄	—	13	1	no	—
182	C₂×C₂²⋊C₄	—	13	2	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	13	2	no	—
185	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
186	C₂×C₁₃	#552	1	4	yes	complemented
187	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
188	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
189	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
190	D₁₃	—	2	4	no	—
191	D₁₃	—	4	2	no	—
192	C₂⁴	—	13	1	no	—
193	C₂⁴	—	13	1	no	—
194	C₂×Q₈	—	13	4	no	—
195	C₂×Q₈	—	26	4	no	—
196	C₂×D₄	—	13	4	no	—
197	C₂×D₄	—	26	4	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	2	no	—
200	C₂²×C₄	—	13	2	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	8	no	—
203	C₂²×C₄	—	13	4	no	—
204	C₂²×C₄	—	13	4	no	—
205	C₂²×C₄	—	26	8	no	—
206	C₂²×C₄	—	26	4	no	—
207	C₂²×C₄	—	26	4	no	—
208	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
210	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
212	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	13	4	no	—
214	C₄⋊C₄	—	13	8	no	—
215	C₄⋊C₄	—	13	8	no	—
216	C₄⋊C₄	—	13	4	no	—
217	C₄⋊C₄	—	13	8	no	—
218	C₄⋊C₄	—	13	8	no	—
219	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
220	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
221	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
222	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
223	C₂²⋊C₄	—	13	4	no	—
224	C₂²⋊C₄	—	13	8	no	—
225	C₂²⋊C₄	—	13	8	no	—
226	C₂²⋊C₄	—	13	4	no	—
227	C₂²⋊C₄	—	13	8	no	—
228	C₄²	—	13	8	no	—
229	C₄²	—	13	8	no	—
230	C₄²	C₂²×D₁₃	1	4	yes	—
231	C₁₃	C₂×#552	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	13	4	no	—
233	C₂³	—	13	2	no	—
234	C₂³	—	13	2	no	—
235	C₂³	—	26	2	no	—
236	C₂³	—	26	2	no	—
237	C₂³	—	26	2	no	—
238	C₂³	—	26	2	no	—
239	C₂³	—	26	2	no	—
240	C₂³	C₂³×D₁₃	1	1	yes	characteristic
241	Q₈	—	26	8	no	—
242	D₄	—	26	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	13	4	no	—
249	C₂×C₄	—	13	4	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	26	16	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	8	no	—
257	C₂×C₄	—	26	8	no	—
258	C₂×C₄	#6131	1	4	yes	—
259	C₂×C₄	C₂³×D₁₃	1	2	yes	—
260	C₂×C₄	C₂³×D₁₃	1	4	yes	—
261	C₂×C₄	C₂³×D₁₃	1	2	yes	—
262	C₂×C₄	C₂³×D₁₃	1	4	yes	—
263	C₂×C₄	C₂³×D₁₃	1	2	yes	—
264	C₂²	—	13	4	no	—
265	C₂²	—	26	2	no	—
266	C₂²	—	26	2	no	—
267	C₂²	—	26	8	no	—
268	C₂²	—	26	2	no	—
269	C₂²	—	26	2	no	—
270	C₂²	—	26	2	no	—
271	C₂²	—	52	4	no	—
272	C₂²	C₂×#6131	1	2	yes	—
273	C₂²	#41×D₁₃	1	2	yes	—
274	C₂²	#88257	1	2	yes	—
275	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	26	2	no	—
281	C₄	—	26	2	no	—
282	C₄	—	26	4	no	—
283	C₄	—	26	4	no	—
284	C₄	C₂×#6131	1	4	yes	—
285	C₂	—	26	4	no	—
286	C₂	—	52	2	no	—
287	C₂	#11719399	1	4	yes	direct product factor
288	C₂	C₂²×#6131	1	1	yes	characteristic
289	C₂	C₂×#41×D₁₃	1	1	yes	characteristic
290	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×#125	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	C₂×#552
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#125	C₂	semidirect
2. #11719399	C₂	direct (N × H)
3. C₂×#88152	C₂	semidirect
4. C₂×#88151	C₂	semidirect
5. C₂×#88151	C₂	semidirect
7. C₂×#88146	C₂	semidirect
8. C₂×#88146	C₂	semidirect
9. C₂×C₄⋊C₄×D₁₃	C₂	semidirect
10. C₂×#88144	C₂	semidirect
12. C₂×C₄×Dic₂₆	C₂	semidirect
18. C₁₃×#125	C₂²	semidirect
37. #88152	C₂²	semidirect
38. #88151	C₂²	semidirect
39. #88151	C₂²	semidirect
41. #88146	C₂²	semidirect
42. #88146	C₂²	semidirect
43. C₄⋊C₄×D₁₃	C₂²	semidirect
44. #88144	C₂²	semidirect
46. C₄×Dic₂₆	C₂²	semidirect
122. C₂×#125	D₁₃	semidirect
159. #125	C₂×D₁₃	semidirect
186. C₂×C₁₃	#552	semidirect
231. C₁₃	C₂×#552	semidirect
287. C₂	#11719399	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×#125	2
#11719399	2
C₂×#88152	2
C₂×#88151	2
C₂×#88148	2
C₂×#88146	2
C₂×C₄⋊C₄×D₁₃	2
C₂×#88144	2
C₂×C₄×D₅₂	2
C₂×C₄×Dic₂₆	2
C₂×#552	13

Subgroups — C2×#11719399 #13128346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719399 #13128346