Subgroups — C₂×#11719285 #13128105

Order 1664.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88201	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88200	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719285	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88152	C₂	1	2	yes	maximal, complemented
6	C₂×#88149	C₂	1	1	yes	maximal, characteristic
7	C₂×#88148	C₂	1	2	yes	maximal, complemented
8	C₂×#88144	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₅₂	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88126	C₂	1	2	yes	maximal, complemented
12	C₂×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×D₅₂	C₂²	1	2	yes	—
15	C₂²×D₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
20	C₄×Q₈×C₁₃	C₂²	1	8	yes	complemented
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×C₄²×C₁₃	C₂²	1	2	yes	—
25	#88201	C₂²	1	8	yes	complemented
26	#88200	C₂²	1	8	yes	complemented
27	C₂×#6103	C₂²	1	2	yes	—
28	C₂×#6103	C₂²	1	2	yes	—
29	C₂×#6103	C₂²	1	2	yes	—
30	C₂×#6103	C₂²	1	2	yes	—
31	C₂×#6103	C₂²	1	2	yes	—
32	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
33	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
34	C₂×#6101	C₂²	1	2	yes	—
35	C₂×#6101	C₂²	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
39	#88152	C₂²	1	16	yes	complemented
40	#88149	C₂²	1	8	yes	—
41	#88148	C₂²	1	16	yes	complemented
42	#88144	C₂²	1	8	yes	complemented
43	C₄×D₅₂	C₂²	1	16	yes	complemented
44	C₄×D₅₂	C₂²	1	8	yes	complemented
45	#88126	C₂²	1	16	yes	complemented
46	C₄²×D₁₃	C₂²	1	8	yes	complemented
47	C₂³×D₁₃	C₂³	1	2	yes	—
48	C₂³×D₁₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	—	2	4	no	—
50	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
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	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂³	1	2	yes	—
60	C₂×D₅₂	—	2	8	no	—
61	C₂×D₅₂	—	2	4	no	—
62	C₂×D₅₂	C₂³	1	8	yes	—
63	C₂×D₅₂	C₂³	1	4	yes	—
64	C₂×C₄×D₁₃	—	2	4	no	—
65	C₂×C₄×D₁₃	—	2	4	no	—
66	C₂×C₄×D₁₃	—	2	4	no	—
67	C₂×C₄×D₁₃	—	2	8	no	—
68	C₂×C₄×D₁₃	C₂³	1	4	yes	—
69	C₂×C₄×D₁₃	C₂³	1	4	yes	—
70	C₂×C₄×D₁₃	C₂³	1	4	yes	—
71	C₂×C₄×D₁₃	C₂³	1	8	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
74	C₄²×C₁₃	C₂³	1	4	yes	—
75	C₄²×C₁₃	C₂³	1	8	yes	—
76	#6103	C₂³	1	8	yes	—
77	#6103	C₂³	1	8	yes	—
78	#6103	C₂³	1	8	yes	—
79	#6103	C₂³	1	8	yes	—
80	#6103	C₂³	1	8	yes	—
81	C₄⋊Dic₁₃	C₂³	1	8	yes	—
82	C₄⋊Dic₁₃	C₂³	1	4	yes	—
83	#6101	C₂³	1	8	yes	—
84	#6101	C₂³	1	8	yes	—
85	C₄×Dic₁₃	C₂³	1	4	yes	—
86	C₄×Dic₁₃	C₂³	1	4	yes	—
87	C₄×Dic₁₃	C₂³	1	4	yes	—
88	C₂×#529	—	13	1	no	maximal, Sylow
89	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂²×D₁₃	—	2	2	no	—
91	C₂²×D₁₃	—	2	2	no	—
92	C₂²×D₁₃	—	2	4	no	—
93	C₂²×D₁₃	—	2	4	no	—
94	C₂²×D₁₃	—	2	4	no	—
95	C₂²×D₁₃	#41	1	4	yes	—
96	C₂²×D₁₃	C₂⁴	1	4	yes	—
97	C₂²×D₁₃	C₂⁴	1	2	yes	—
98	Q₈×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	#41	1	4	yes	—
103	C₂×C₄×C₁₃	#41	1	4	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
109	C₂×Dic₁₃	—	2	2	no	—
110	C₂×Dic₁₃	—	2	2	no	—
111	C₂×Dic₁₃	—	2	4	no	—
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	C₂⁴	1	2	yes	—
114	C₂×Dic₁₃	C₂⁴	1	2	yes	—
115	C₂×Dic₁₃	C₂⁴	1	2	yes	—
116	C₂×Dic₁₃	C₂⁴	1	4	yes	—
117	D₅₂	—	2	16	no	—
118	D₅₂	—	2	8	no	—
119	C₄×D₁₃	—	2	8	no	—
120	C₄×D₁₃	—	2	8	no	—
121	C₄×D₁₃	—	2	8	no	—
122	C₄×D₁₃	—	2	16	no	—
123	#529	—	13	16	no	—
124	C₂×#126	—	13	2	no	—
125	C₂×#125	—	13	1	no	—
126	C₂×#124	—	13	1	no	—
127	C₂×#123	—	13	2	no	—
128	C₂×C₂²⋊Q₈	—	13	1	no	—
129	C₂×#121	—	13	1	no	—
130	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
131	C₂×C₄×D₄	—	13	2	no	—
132	C₂×C₄×D₄	—	13	1	no	—
133	C₂×#117	—	13	2	no	—
134	C₂²×C₄²	—	13	1	no	—
135	C₂²×C₁₃	C₂×#41	1	2	yes	—
136	C₂²×C₁₃	C₂×#41	1	2	yes	—
137	C₂²×C₁₃	C₂×#41	1	2	yes	—
138	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
139	C₂×D₁₃	—	2	8	no	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	2	no	—
142	C₂×D₁₃	—	2	4	no	—
143	C₂×D₁₃	—	2	4	no	—
144	C₂×D₁₃	—	2	4	no	—
145	C₂×D₁₃	—	4	8	no	—
146	C₂×D₁₃	C₂×#41	1	4	yes	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	C₂×#41	1	4	yes	—
151	C₄×C₁₃	C₂×#41	1	4	yes	—
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	2	no	—
154	Dic₁₃	—	2	4	no	—
155	Dic₁₃	—	2	2	no	—
156	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂²×D₄	—	13	2	no	—
158	C₂²×D₄	—	13	1	no	—
159	C₂³×C₄	—	13	1	no	—
160	C₂³×C₄	—	13	1	no	—
161	C₂³×C₄	—	13	2	no	—
162	C₂³×C₄	—	13	1	no	—
163	#126	—	13	16	no	—
164	#125	—	13	8	no	—
165	#124	—	13	8	no	—
166	#123	—	13	16	no	—
167	C₂²⋊Q₈	—	13	8	no	—
168	#121	—	13	8	no	—
169	C₄×Q₈	C₂×D₁₃	1	8	yes	complemented
170	C₄×D₄	—	13	8	no	—
171	C₄×D₄	—	13	16	no	—
172	#117	—	13	16	no	—
173	C₂×C₄⋊C₄	—	13	2	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₄⋊C₄	—	13	1	no	—
176	C₂×C₄⋊C₄	—	13	2	no	—
177	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
179	C₂×C₂²⋊C₄	—	13	2	no	—
180	C₂×C₂²⋊C₄	—	13	2	no	—
181	C₂×C₂²⋊C₄	—	13	2	no	—
182	C₂×C₂²⋊C₄	—	13	2	no	—
183	C₂×C₂²⋊C₄	—	13	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	1	no	—
187	C₂×C₄²	—	13	8	no	—
188	C₂×C₄²	C₂×D₁₃	1	2	yes	—
189	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
190	C₂×C₁₃	#529	1	4	yes	complemented
191	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
192	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
193	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
194	D₁₃	—	2	4	no	—
195	D₁₃	—	4	4	no	—
196	C₂⁴	—	13	1	no	—
197	C₂⁴	—	13	2	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
200	C₂×D₄	—	13	8	no	—
201	C₂×D₄	—	13	4	no	—
202	C₂×D₄	—	26	8	no	—
203	C₂×D₄	—	26	4	no	—
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	13	2	no	—
208	C₂²×C₄	—	13	4	no	—
209	C₂²×C₄	—	13	4	no	—
210	C₂²×C₄	—	13	4	no	—
211	C₂²×C₄	—	13	8	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	—	26	4	no	—
214	C₂²×C₄	—	26	4	no	—
215	C₂²×C₄	—	26	8	no	—
216	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
218	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	13	8	no	—
222	C₄⋊C₄	—	13	8	no	—
223	C₄⋊C₄	—	13	4	no	—
224	C₄⋊C₄	—	13	8	no	—
225	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
226	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
227	C₂²⋊C₄	—	13	8	no	—
228	C₂²⋊C₄	—	13	8	no	—
229	C₂²⋊C₄	—	13	8	no	—
230	C₂²⋊C₄	—	13	8	no	—
231	C₂²⋊C₄	—	13	8	no	—
232	C₄²	—	13	4	no	—
233	C₄²	—	13	4	no	—
234	C₄²	—	13	4	no	—
235	C₄²	C₂²×D₁₃	1	4	yes	—
236	C₄²	C₂²×D₁₃	1	8	yes	—
237	C₁₃	C₂×#529	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	13	4	no	—
239	C₂³	—	13	2	no	—
240	C₂³	—	13	4	no	—
241	C₂³	—	26	2	no	—
242	C₂³	—	26	2	no	—
243	C₂³	—	26	4	no	—
244	C₂³	—	26	4	no	—
245	C₂³	—	26	4	no	—
246	C₂³	C₂³×D₁₃	1	1	yes	characteristic
247	Q₈	—	2	8	no	—
248	D₄	—	26	16	no	—
249	D₄	—	26	8	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	13	2	no	—
255	C₂×C₄	—	13	2	no	—
256	C₂×C₄	—	13	4	no	—
257	C₂×C₄	—	26	8	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	26	2	no	—
260	C₂×C₄	—	26	4	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	26	8	no	—
263	C₂×C₄	—	26	8	no	—
264	C₂×C₄	—	26	16	no	—
265	C₂×C₄	#6127	1	4	yes	—
266	C₂×C₄	#6131	1	4	yes	—
267	C₂×C₄	C₂³×D₁₃	1	2	yes	—
268	C₂×C₄	C₂³×D₁₃	1	4	yes	—
269	C₂×C₄	C₂³×D₁₃	1	4	yes	—
270	C₂×C₄	C₂³×D₁₃	1	2	yes	—
271	C₂×C₄	C₂³×D₁₃	1	2	yes	—
272	C₂²	—	13	4	no	—
273	C₂²	—	26	8	no	—
274	C₂²	—	26	2	no	—
275	C₂²	—	26	2	no	—
276	C₂²	—	26	4	no	—
277	C₂²	—	26	4	no	—
278	C₂²	—	26	4	no	—
279	C₂²	—	52	8	no	—
280	C₂²	C₂×#6127	1	2	yes	—
281	C₂²	C₂×#6131	1	2	yes	—
282	C₂²	#41×D₁₃	1	2	yes	—
283	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	26	2	no	—
288	C₄	—	26	2	no	—
289	C₄	—	26	4	no	—
290	C₄	—	26	2	no	—
291	C₄	C₂×#6127	1	4	yes	—
292	C₄	C₂×#6131	1	4	yes	—
293	C₂	—	26	4	no	—
294	C₂	—	52	4	no	—
295	C₂	#11719285	1	4	yes	direct product factor
296	C₂	C₂²×#6127	1	1	yes	characteristic
297	C₂	C₂²×#6131	1	1	yes	characteristic
298	C₂	C₂×#41×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂×#88149	C₂
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#529
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
2. C₂×#88201	C₂	semidirect
3. C₂×#88200	C₂	semidirect
4. #11719285	C₂	direct (N × H)
5. C₂×#88152	C₂	semidirect
7. C₂×#88148	C₂	semidirect
8. C₂×#88144	C₂	semidirect
9. C₂×C₄×D₅₂	C₂	semidirect
10. C₂×C₄×D₅₂	C₂	semidirect
11. C₂×#88126	C₂	semidirect
12. C₂×C₄²×D₁₃	C₂	semidirect
20. C₄×Q₈×C₁₃	C₂²	semidirect
25. #88201	C₂²	semidirect
26. #88200	C₂²	semidirect
39. #88152	C₂²	semidirect
41. #88148	C₂²	semidirect
42. #88144	C₂²	semidirect
43. C₄×D₅₂	C₂²	semidirect
44. C₄×D₅₂	C₂²	semidirect
45. #88126	C₂²	semidirect
46. C₄²×D₁₃	C₂²	semidirect
130. C₂×C₄×Q₈	D₁₃	semidirect
169. C₄×Q₈	C₂×D₁₃	semidirect
190. C₂×C₁₃	#529	semidirect
237. C₁₃	C₂×#529	semidirect
295. C₂	#11719285	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×C₄×Q₈×C₁₃	2
C₂×#88201	2
C₂×#88200	2
#11719285	2
C₂×#88152	2
C₂×#88149	2
C₂×#88148	2
C₂×#88144	2
C₂×C₄×D₅₂	2
C₂×#88126	2
C₂×C₄²×D₁₃	2
C₂×#529	13

Subgroups — C2×#11719285 #13128105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719285 #13128105