Subgroups — C₂×C₁₃×#2234 #15719980

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	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×#357	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#121	—	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×#466	—	2	2	no	—
16	C₁₃×#466	C₂²	1	4	yes	—
17	C₁₃×#427	—	2	2	no	—
18	C₁₃×#427	C₂²	1	4	yes	complemented
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×#102	—	2	1	no	—
24	C₁₃×#367	C₂²	1	4	yes	complemented
25	C₁₃×#366	C₂²	1	4	yes	complemented
26	C₁₃×#357	C₂²	1	4	yes	complemented
27	C₁₃×#357	C₂²	1	4	yes	complemented
28	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	C₂²×D₄×C₁₃	—	2	1	no	—
30	C₂²×D₄×C₁₃	—	4	1	no	—
31	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₁₃×D₈	—	2	1	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	—	2	2	no	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#121	—	2	4	no	—
43	C₁₃×#121	—	4	2	no	—
44	C₁₃×#117	—	2	4	no	—
45	C₁₃×#117	—	4	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	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×#102	—	2	4	no	—
64	C₂×#2234	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂⁴×C₁₃	—	4	1	no	—
66	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	1	no	—
78	C₂²×C₄×C₁₃	D₈	1	2	yes	—
79	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×D₈	—	4	2	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₂×C₈×C₁₃	#41	1	2	yes	—
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₂²⋊C₄	—	4	4	no	—
101	C₁₃×C₂²⋊C₄	—	4	4	no	—
102	C₄²×C₁₃	—	4	4	no	—
103	C₁₃×C₁₆	—	2	2	no	—
104	C₁₃×C₁₆	—	2	2	no	—
105	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
106	C₂×#427	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#2234	C₂×C₁₃	1	8	yes	direct product factor
108	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂×#357	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#357	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	2	no	—
115	C₂³×C₁₃	—	2	2	no	—
116	C₂³×C₁₃	—	4	2	no	—
117	C₂³×C₁₃	—	8	2	no	—
118	C₂³×C₁₃	—	8	2	no	—
119	C₂³×C₁₃	—	8	2	no	—
120	C₂³×C₁₃	C₂×D₈	1	2	yes	—
121	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
122	D₄×C₁₃	—	4	2	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	8	8	no	—
125	D₄×C₁₃	—	8	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	8	2	no	—
134	C₂×C₄×C₁₃	—	8	2	no	—
135	C₂×C₄×C₁₃	—	8	2	no	—
136	C₂×C₄×C₁₃	#123	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#137	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	#123	1	2	yes	—
143	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×#121	—	2	1	no	—
148	C₂×#117	—	2	1	no	—
149	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	#466	—	2	2	no	—
152	#466	C₂²×C₁₃	1	4	yes	—
153	#427	—	2	2	no	—
154	#427	C₂²×C₁₃	1	4	yes	complemented
155	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#106	—	2	1	no	—
159	C₂×#102	—	2	1	no	—
160	#367	C₂²×C₁₃	1	4	yes	complemented
161	#366	C₂²×C₁₃	1	4	yes	complemented
162	#357	C₂²×C₁₃	1	4	yes	complemented
163	#357	C₂²×C₁₃	1	4	yes	complemented
164	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	8	2	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	16	4	no	—
173	C₂²×C₁₃	#480	1	2	yes	—
174	C₂²×C₁₃	#508	1	2	yes	—
175	C₂²×C₁₃	#509	1	2	yes	—
176	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#480	1	2	yes	—
182	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
183	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
184	C₂²×D₄	—	2	1	no	—
185	C₂²×D₄	—	4	1	no	—
186	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	C₂³×C₁₃	1	2	yes	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₁₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	#121	—	2	4	no	—
198	#121	—	4	2	no	—
199	#117	—	2	4	no	—
200	#117	—	4	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	—	4	1	no	—
205	C₂×C₂²⋊C₄	—	4	1	no	—
206	C₂×C₄²	—	4	1	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
210	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
211	#107	—	2	1	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	C₂³×C₁₃	1	2	yes	—
215	#107	C₂³×C₁₃	1	2	yes	—
216	#107	C₂³×C₁₃	1	2	yes	—
217	#106	—	2	4	no	—
218	#102	—	2	4	no	—
219	C₂×C₁₃	—	4	2	no	—
220	C₂×C₁₃	—	16	2	no	—
221	C₂×C₁₃	#2234	1	4	yes	direct product factor
222	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
225	C₂⁴	—	4	1	no	—
226	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	8	4	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
239	C₂²×C₄	C₁₃×D₈	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	D₈	—	2	2	no	—
243	D₈	—	2	2	no	—
244	D₈	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₁₃×#41	1	2	yes	—
252	C₂×C₈	C₁₃×#41	1	2	yes	—
253	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₄²	—	4	4	no	—
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₁₃	C₂×#2234	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	C₂×C₁₃×D₈	1	2	yes	—
274	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	C₁₃×#123	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
291	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
292	C₂×C₄	C₁₃×#137	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	C₁₃×#123	1	2	yes	—
296	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
297	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	C₁₃×#480	1	2	yes	—
307	C₂²	C₁₃×#508	1	2	yes	—
308	C₂²	C₁₃×#509	1	2	yes	—
309	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₁₃×#480	1	2	yes	—
315	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
316	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	C₁₃×#2234	1	4	yes	direct product factor
320	C₂	C₂×C₁₃×#480	1	1	yes	characteristic
321	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
322	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#466	C₂
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
2. C₂×C₁₃×#427	C₂	semidirect
3. C₁₃×#2234	C₂	direct (N × H)
4. C₂×C₁₃×#367	C₂	semidirect
5. C₂×C₁₃×#366	C₂	semidirect
6. C₂×C₁₃×#357	C₂	semidirect
7. C₂×C₁₃×#357	C₂	semidirect
8. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
18. C₁₃×#427	C₂²	semidirect
24. C₁₃×#367	C₂²	semidirect
25. C₁₃×#366	C₂²	semidirect
26. C₁₃×#357	C₂²	semidirect
27. C₁₃×#357	C₂²	semidirect
28. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
64. C₂×#2234	C₁₃	direct (N × H)
106. C₂×#427	C₂×C₁₃	semidirect
107. #2234	C₂×C₁₃	direct (N × H)
108. C₂×#367	C₂×C₁₃	semidirect
109. C₂×#366	C₂×C₁₃	semidirect
110. C₂×#357	C₂×C₁₃	semidirect
111. C₂×#357	C₂×C₁₃	semidirect
112. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
154. #427	C₂²×C₁₃	semidirect
160. #367	C₂²×C₁₃	semidirect
161. #366	C₂²×C₁₃	semidirect
162. #357	C₂²×C₁₃	semidirect
163. #357	C₂²×C₁₃	semidirect
164. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
221. C₂×C₁₃	#2234	direct (N × H)
265. C₁₃	C₂×#2234	direct (N × H)
319. C₂	C₁₃×#2234	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×#466	2
C₂×C₁₃×#427	2
C₁₃×#2234	2
C₂×C₁₃×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2234	13

Subgroups — C2×C13×#2234 #15719980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2234 #15719980