Subgroups — C₂×C₁₃×#2237 #15719981

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2237	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₂²⋊Q₈	—	2	1	no	—
12	C₂×C₁₃×#117	—	2	1	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#467	—	2	2	no	—
16	C₁₃×#467	C₂²	1	4	yes	—
17	C₁₃×#427	—	2	2	no	—
18	C₁₃×#427	C₂²	1	4	yes	—
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#106	—	2	1	no	—
23	C₂×C₁₃×#103	—	2	1	no	—
24	C₁₃×#367	C₂²	1	4	yes	complemented
25	C₁₃×#366	C₂²	1	4	yes	complemented
26	C₁₃×#358	C₂²	1	4	yes	complemented
27	C₁₃×#358	C₂²	1	4	yes	complemented
28	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₂²⋊Q₈	—	2	4	no	—
42	C₁₃×C₂²⋊Q₈	—	4	2	no	—
43	C₁₃×#117	—	2	4	no	—
44	C₁₃×#117	—	4	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
51	C₂×C₄²×C₁₃	—	4	1	no	—
52	C₂×C₁₃×C₁₆	—	2	2	no	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#107	C₂³	1	2	yes	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#106	—	2	4	no	—
63	C₁₃×#103	—	2	4	no	—
64	C₂×#2237	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	—	4	1	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂×Q₈×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	D₈	1	2	yes	—
77	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×Q₁₆	—	4	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	4	no	—
99	C₁₃×C₂²⋊C₄	—	4	4	no	—
100	C₁₃×C₂²⋊C₄	—	4	4	no	—
101	C₄²×C₁₃	—	4	4	no	—
102	C₁₃×C₁₆	—	2	2	no	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₂×#467	C₂×C₁₃	1	1	yes	characteristic
105	C₂×#427	C₂×C₁₃	1	1	yes	characteristic
106	#2237	C₂×C₁₃	1	8	yes	direct product factor
107	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#358	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂×#358	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	C₂×D₈	1	2	yes	—
116	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₁₃	—	4	2	no	—
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	8	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	—	8	2	no	—
130	C₂×C₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	—	8	2	no	—
132	C₂×C₄×C₁₃	#123	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#137	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	#123	1	2	yes	—
139	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
140	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
142	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₂²⋊Q₈	—	2	1	no	—
144	C₂×#117	—	2	1	no	—
145	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
146	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
147	#467	—	2	2	no	—
148	#467	C₂²×C₁₃	1	4	yes	—
149	#427	—	2	2	no	—
150	#427	C₂²×C₁₃	1	4	yes	—
151	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#106	—	2	1	no	—
155	C₂×#103	—	2	1	no	—
156	#367	C₂²×C₁₃	1	4	yes	complemented
157	#366	C₂²×C₁₃	1	4	yes	complemented
158	#358	C₂²×C₁₃	1	4	yes	complemented
159	#358	C₂²×C₁₃	1	4	yes	complemented
160	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	2	2	no	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	#480	1	2	yes	—
166	C₂²×C₁₃	#508	1	2	yes	—
167	C₂²×C₁₃	#510	1	2	yes	—
168	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	8	2	no	—
171	C₄×C₁₃	—	8	2	no	—
172	C₄×C₁₃	—	8	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#480	1	2	yes	—
175	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
176	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
177	C₂²×Q₈	—	2	1	no	—
178	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
179	C₂×Q₁₆	—	2	1	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	D₄×C₁₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²⋊Q₈	—	2	4	no	—
190	C₂²⋊Q₈	—	4	2	no	—
191	#117	—	2	4	no	—
192	#117	—	4	2	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₂²⋊C₄	—	4	1	no	—
199	C₂×C₄²	—	4	1	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
203	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
204	#107	—	2	1	no	—
205	#107	—	2	1	no	—
206	#107	—	2	1	no	—
207	#107	C₂³×C₁₃	1	2	yes	—
208	#107	C₂³×C₁₃	1	2	yes	—
209	#107	C₂³×C₁₃	1	2	yes	—
210	#106	—	2	4	no	—
211	#103	—	2	4	no	—
212	C₂×C₁₃	—	4	2	no	—
213	C₂×C₁₃	#2237	1	4	yes	direct product factor
214	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
215	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
216	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
217	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
229	C₂²×C₄	C₁₃×D₈	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	4	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
240	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₁₃×#41	1	2	yes	—
242	C₂×C₈	C₁₃×#41	1	2	yes	—
243	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	4	4	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₁₃	C₂×#2237	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₁₃×D₈	1	2	yes	—
261	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	2	no	—
275	C₂×C₄	—	8	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	C₁₃×#123	1	2	yes	—
278	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
279	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
280	C₂×C₄	C₁₃×#137	1	2	yes	—
281	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	C₁₃×#123	1	2	yes	—
284	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
285	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₁₃×#480	1	2	yes	—
291	C₂²	C₁₃×#508	1	2	yes	—
292	C₂²	C₁₃×#510	1	2	yes	—
293	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	8	2	no	—
298	C₄	—	8	2	no	—
299	C₄	C₁₃×#480	1	2	yes	—
300	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
301	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	C₁₃×#2237	1	4	yes	direct product factor
304	C₂	C₂×C₁₃×#480	1	1	yes	characteristic
305	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
306	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2237	C₂	direct (N × H)
4. C₂×C₁₃×#367	C₂	semidirect
5. C₂×C₁₃×#366	C₂	semidirect
6. C₂×C₁₃×#358	C₂	semidirect
7. C₂×C₁₃×#358	C₂	semidirect
24. C₁₃×#367	C₂²	semidirect
25. C₁₃×#366	C₂²	semidirect
26. C₁₃×#358	C₂²	semidirect
27. C₁₃×#358	C₂²	semidirect
64. C₂×#2237	C₁₃	direct (N × H)
106. #2237	C₂×C₁₃	direct (N × H)
107. C₂×#367	C₂×C₁₃	semidirect
108. C₂×#366	C₂×C₁₃	semidirect
109. C₂×#358	C₂×C₁₃	semidirect
110. C₂×#358	C₂×C₁₃	semidirect
156. #367	C₂²×C₁₃	semidirect
157. #366	C₂²×C₁₃	semidirect
158. #358	C₂²×C₁₃	semidirect
159. #358	C₂²×C₁₃	semidirect
213. C₂×C₁₃	#2237	direct (N × H)
256. C₁₃	C₂×#2237	direct (N × H)
303. C₂	C₁₃×#2237	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×#467	2
C₂×C₁₃×#427	2
C₁₃×#2237	2
C₂×C₁₃×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×#358	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2237	13

Subgroups — C2×C13×#2237 #15719981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2237 #15719981