Subgroups — C₂×C₁₃×#2214 #15719932

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#510	C₂	1	2	yes	maximal
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2214	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#362	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×Q₁₆	—	2	1	no	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₁₃×#131	—	2	1	no	—
11	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₁₃×#510	—	2	4	no	—
13	C₁₃×#510	C₂²	1	8	yes	—
14	C₂×C₁₃×Q₃₂	—	2	2	no	—
15	C₂×C₁₃×SD₃₂	—	2	2	no	—
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
17	C₁₃×#494	—	2	2	no	—
18	C₁₃×#494	C₂²	1	4	yes	complemented
19	C₁₃×#443	—	2	2	no	—
20	C₁₃×#443	C₂²	1	4	yes	—
21	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#104	—	2	1	no	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#364	C₂²	1	4	yes	complemented
25	C₁₃×#362	C₂²	1	8	yes	complemented
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂²×Q₈×C₁₃	—	2	2	no	—
28	C₁₃×#135	—	2	1	no	—
29	C₁₃×#135	—	2	1	no	—
30	C₁₃×#135	D₄	1	2	yes	—
31	C₁₃×#135	C₂³	1	2	yes	—
32	C₂×C₁₃×Q₁₆	—	2	2	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×Q₁₆	—	2	2	no	—
35	C₂×C₁₃×Q₁₆	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	—	2	2	no	—
37	C₂×C₁₃×Q₁₆	—	2	2	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂×C₁₃×Q₁₆	—	4	1	no	—
40	C₂×C₁₃×Q₁₆	—	4	2	no	—
41	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
42	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
43	C₂×C₁₃×SD₁₆	—	2	1	no	—
44	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic
45	C₁₃×#131	—	2	4	no	—
46	C₁₃×#131	—	4	2	no	—
47	C₂×C₁₃×M₁₆	—	2	1	no	—
48	C₂×C₁₃×M₁₆	—	4	1	no	—
49	C₂²×C₈×C₁₃	—	4	1	no	—
50	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊Q₈	—	2	4	no	—
53	C₁₃×C₄⋊Q₈	—	4	2	no	—
54	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×Q₃₂	—	2	8	no	—
57	C₁₃×SD₃₂	—	2	8	no	—
58	C₁₃×M₃₂	—	2	2	no	—
59	C₁₃×M₃₂	C₂³	1	4	yes	—
60	C₂×C₁₃×C₁₆	—	2	2	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	C₂³	1	2	yes	—
63	C₁₃×#104	—	2	4	no	—
64	C₄×C₈×C₁₃	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	C₂×#2214	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₃×#41	—	2	2	no	—
68	C₁₃×#41	—	2	2	no	—
69	C₁₃×#41	—	4	1	no	—
70	C₁₃×#41	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	2	1	no	—
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	4	2	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂×Q₈×C₁₃	—	4	2	no	—
76	C₂×Q₈×C₁₃	—	4	2	no	—
77	C₂×Q₈×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	2	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	1	no	—
85	C₁₃×Q₁₆	—	2	4	no	—
86	C₁₃×Q₁₆	—	2	4	no	—
87	C₁₃×Q₁₆	—	4	2	no	—
88	C₁₃×Q₁₆	—	4	2	no	—
89	C₁₃×Q₁₆	—	4	4	no	—
90	C₁₃×Q₁₆	—	4	4	no	—
91	C₁₃×Q₁₆	C₂×D₄	1	2	yes	—
92	C₁₃×SD₁₆	—	2	2	no	—
93	C₁₃×SD₁₆	—	2	2	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	C₂×D₄	1	2	yes	—
96	C₁₃×M₁₆	—	2	2	no	—
97	C₁₃×M₁₆	—	4	1	no	—
98	C₁₃×M₁₆	—	4	4	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	—	4	4	no	—
101	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
102	C₂×C₈×C₁₃	D₈	1	1	yes	characteristic
103	C₂×C₈×C₁₃	D₈	1	2	yes	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
109	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	4	8	no	—
112	C₄²×C₁₃	—	2	1	no	—
113	C₄²×C₁₃	C₂×D₄	1	2	yes	—
114	C₁₃×C₁₆	—	2	4	no	—
115	C₂×#510	C₂×C₁₃	1	2	yes	—
116	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
118	#2214	C₂×C₁₃	1	8	yes	direct product factor
119	C₂×#364	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂×#362	C₂×C₁₃	1	2	yes	complemented
121	C₂³×C₁₃	—	4	1	no	—
122	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	2	no	—
124	Q₈×C₁₃	—	4	1	no	—
125	Q₈×C₁₃	—	4	4	no	—
126	Q₈×C₁₃	—	4	4	no	—
127	Q₈×C₁₃	—	8	2	no	—
128	Q₈×C₁₃	—	8	2	no	—
129	D₄×C₁₃	—	2	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	1	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	8	2	no	—
140	C₂×C₄×C₁₃	—	8	2	no	—
141	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₁₃	—	4	2	no	—
148	C₈×C₁₃	#121	1	1	yes	characteristic
149	C₈×C₁₃	#121	1	1	yes	characteristic
150	C₈×C₁₃	C₂×D₈	1	2	yes	—
151	C₈×C₁₃	C₂×D₈	1	2	yes	—
152	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×Q₁₆	—	2	1	no	—
156	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
157	C₂×#131	—	2	1	no	—
158	C₂×C₄⋊Q₈	—	2	1	no	—
159	#510	—	2	4	no	—
160	#510	C₂²×C₁₃	1	8	yes	—
161	C₂×Q₃₂	—	2	2	no	—
162	C₂×SD₃₂	—	2	2	no	—
163	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
164	#494	—	2	2	no	—
165	#494	C₂²×C₁₃	1	4	yes	complemented
166	#443	—	2	2	no	—
167	#443	C₂²×C₁₃	1	4	yes	—
168	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×#104	—	2	1	no	—
170	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
171	#364	C₂²×C₁₃	1	4	yes	complemented
172	#362	C₂²×C₁₃	1	8	yes	complemented
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	4	2	no	—
175	C₂²×C₁₃	—	8	2	no	—
176	C₂²×C₁₃	#459	1	1	yes	characteristic
177	C₂²×C₁₃	#459	1	1	yes	characteristic
178	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₄×C₁₃	—	2	4	no	—
180	C₄×C₁₃	—	4	2	no	—
181	C₄×C₁₃	—	8	4	no	—
182	C₄×C₁₃	#459	1	1	yes	characteristic
183	C₄×C₁₃	#459	1	1	yes	characteristic
184	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
185	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₂×#41	—	2	1	no	—
187	C₂²×Q₈	—	2	2	no	—
188	#135	—	2	1	no	—
189	#135	—	2	1	no	—
190	#135	D₄×C₁₃	1	2	yes	—
191	#135	C₂³×C₁₃	1	2	yes	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	—	2	2	no	—
194	C₂×Q₁₆	—	2	4	no	—
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×Q₁₆	—	2	2	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	4	1	no	—
199	C₂×Q₁₆	—	4	1	no	—
200	C₂×Q₁₆	—	4	2	no	—
201	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic
202	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
203	C₂×SD₁₆	—	2	1	no	—
204	C₂×D₈	D₄×C₁₃	1	1	yes	characteristic
205	#131	—	2	4	no	—
206	#131	—	4	2	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	4	1	no	—
209	C₂²×C₈	—	4	1	no	—
210	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	4	no	—
213	C₄⋊Q₈	—	4	2	no	—
214	C₂×C₄⋊C₄	—	4	2	no	—
215	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
216	Q₃₂	—	2	8	no	—
217	SD₃₂	—	2	8	no	—
218	M₃₂	—	2	2	no	—
219	M₃₂	C₂³×C₁₃	1	4	yes	—
220	C₂×C₁₆	—	2	2	no	—
221	#108	—	2	1	no	—
222	#108	C₂³×C₁₃	1	2	yes	—
223	#104	—	2	4	no	—
224	C₄×C₈	D₄×C₁₃	1	2	yes	—
225	C₄×C₈	C₂³×C₁₃	1	2	yes	—
226	C₂×C₁₃	—	2	1	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	8	2	no	—
229	C₂×C₁₃	#2214	1	2	yes	direct product factor
230	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
231	#41	—	2	2	no	—
232	#41	—	2	2	no	—
233	#41	—	4	1	no	—
234	#41	—	4	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	Q₁₆	—	2	1	no	—
249	Q₁₆	—	2	4	no	—
250	Q₁₆	—	2	4	no	—
251	Q₁₆	—	4	2	no	—
252	Q₁₆	—	4	2	no	—
253	Q₁₆	—	4	4	no	—
254	Q₁₆	—	4	4	no	—
255	Q₁₆	C₂×D₄×C₁₃	1	2	yes	—
256	SD₁₆	—	2	2	no	—
257	SD₁₆	—	2	2	no	—
258	D₈	—	2	1	no	—
259	D₈	C₂×D₄×C₁₃	1	2	yes	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	4	no	—
265	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
266	C₂×C₈	C₁₃×D₈	1	1	yes	characteristic
267	C₂×C₈	C₁₃×D₈	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
272	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
273	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
274	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
275	C₄⋊C₄	—	4	8	no	—
276	C₄²	—	2	1	no	—
277	C₄²	C₂×D₄×C₁₃	1	2	yes	—
278	C₁₆	—	2	4	no	—
279	C₁₃	C₂×#2214	1	1	yes	characteristic, direct product factor, Sylow
280	C₂³	—	4	1	no	—
281	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
301	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	C₁₃×#121	1	1	yes	characteristic
308	C₈	C₁₃×#121	1	1	yes	characteristic
309	C₈	C₂×C₁₃×D₈	1	2	yes	—
310	C₈	C₂×C₁₃×D₈	1	2	yes	—
311	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
312	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	C₁₃×#459	1	1	yes	characteristic
317	C₂²	C₁₃×#459	1	1	yes	characteristic
318	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	4	no	—
322	C₄	C₁₃×#459	1	1	yes	characteristic
323	C₄	C₁₃×#459	1	1	yes	characteristic
324	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
325	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	2	no	—
329	C₂	C₁₃×#2214	1	2	yes	direct product factor
330	C₂	C₂×C₁₃×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#2214	C₂	direct (N × H)
5. C₂×C₁₃×#364	C₂	semidirect
6. C₂×C₁₃×#362	C₂	semidirect
18. C₁₃×#494	C₂²	semidirect
24. C₁₃×#364	C₂²	semidirect
25. C₁₃×#362	C₂²	semidirect
66. C₂×#2214	C₁₃	direct (N × H)
116. C₂×#494	C₂×C₁₃	semidirect
118. #2214	C₂×C₁₃	direct (N × H)
119. C₂×#364	C₂×C₁₃	semidirect
120. C₂×#362	C₂×C₁₃	semidirect
165. #494	C₂²×C₁₃	semidirect
171. #364	C₂²×C₁₃	semidirect
172. #362	C₂²×C₁₃	semidirect
229. C₂×C₁₃	#2214	direct (N × H)
279. C₁₃	C₂×#2214	direct (N × H)
329. C₂	C₁₃×#2214	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×#510	2
C₂×C₁₃×#494	2
C₂×C₁₃×#443	2
C₁₃×#2214	2
C₂×C₁₃×#364	2
C₂×C₁₃×#362	2
C₂×#2214	13

Subgroups — C2×C13×#2214 #15719932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2214 #15719932