Subgroups — C₂×C₁₃×#2185 #15719851

Order 3328.

1178 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2185	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₁₃	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	—	2	8	no	—
9	C₂²×C₁₃×Q₁₆	—	2	1	no	—
10	C₂²×C₁₃×Q₁₆	—	2	2	no	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	complemented
13	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#121	—	2	1	no	—
16	C₂×C₁₃×SD₃₂	—	2	8	no	—
17	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	complemented
18	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
19	C₁₃×#466	—	2	2	no	—
20	C₁₃×#466	C₂²	1	4	yes	—
21	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#102	—	2	1	no	—
23	C₁₃×#358	C₂²	1	8	yes	complemented
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₃	—	2	8	no	—
26	C₂²×Q₈×C₁₃	—	2	2	no	—
27	C₂²×Q₈×C₁₃	—	2	4	no	—
28	C₂²×Q₈×C₁₃	—	4	4	no	—
29	C₂²×Q₈×C₁₃	—	4	2	no	—
30	C₂²×Q₈×C₁₃	—	4	1	no	—
31	C₂²×D₄×C₁₃	—	2	1	no	—
32	C₂²×D₄×C₁₃	—	4	1	no	—
33	C₂³×C₄×C₁₃	—	4	1	no	—
34	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₁₃×Q₁₆	—	2	8	no	—
36	C₂×C₁₃×Q₁₆	—	2	16	no	—
37	C₂×C₁₃×Q₁₆	—	2	16	no	—
38	C₂×C₁₃×Q₁₆	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	2	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	4	no	—
41	C₂×C₁₃×Q₁₆	D₄	1	8	yes	complemented
42	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
43	C₂×C₁₃×D₈	—	2	4	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₃	—	2	4	no	—
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	D₄	1	2	yes	—
49	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#121	—	2	4	no	—
52	C₁₃×#121	—	4	2	no	—
53	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
55	C₁₃×SD₃₂	—	2	16	no	—
56	C₂×C₁₃×C₁₆	—	2	4	no	—
57	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
58	C₁₃×#107	—	2	1	no	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#102	—	2	4	no	—
61	C₂×#2185	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	4	1	no	—
63	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
64	C₂×Q₈×C₁₃	—	2	16	no	—
65	C₂×Q₈×C₁₃	—	2	8	no	—
66	C₂×Q₈×C₁₃	—	2	4	no	—
67	C₂×Q₈×C₁₃	—	4	8	no	—
68	C₂×Q₈×C₁₃	—	4	8	no	—
69	C₂×Q₈×C₁₃	—	4	16	no	—
70	C₂×Q₈×C₁₃	—	4	4	no	—
71	C₂×Q₈×C₁₃	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	4	4	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	8	4	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	4	8	no	—
82	C₂²×C₄×C₁₃	—	4	4	no	—
83	C₂²×C₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	D₈	1	2	yes	—
86	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₁₃×Q₁₆	—	2	16	no	—
90	C₁₃×Q₁₆	—	2	8	no	—
91	C₁₃×Q₁₆	—	2	4	no	—
92	C₁₃×Q₁₆	C₂×D₄	1	8	yes	complemented
93	C₁₃×D₈	—	2	4	no	—
94	C₁₃×D₈	—	4	2	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	—	2	4	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	4	no	—
105	C₁₃×C₁₆	—	2	4	no	—
106	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂²×SD₃₂	C₂×C₁₃	1	2	yes	complemented
108	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
109	#2185	C₂×C₁₃	1	8	yes	direct product factor
110	C₂×#358	C₂×C₁₃	1	2	yes	complemented
111	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	4	2	no	—
115	C₂³×C₁₃	—	8	2	no	—
116	C₂³×C₁₃	—	8	2	no	—
117	C₂³×C₁₃	—	8	2	no	—
118	C₂³×C₁₃	SD₃₂	1	4	yes	complemented
119	C₂³×C₁₃	C₂×D₈	1	2	yes	—
120	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	8	no	—
122	Q₈×C₁₃	—	4	16	no	—
123	Q₈×C₁₃	—	4	4	no	—
124	Q₈×C₁₃	—	4	8	no	—
125	D₄×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	8	8	no	—
128	D₄×C₁₃	—	8	2	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	4	16	no	—
134	C₂×C₄×C₁₃	—	4	4	no	—
135	C₂×C₄×C₁₃	—	4	8	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	2	no	—
138	C₂×C₄×C₁₃	#120	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₁₃	—	2	2	no	—
144	C₈×C₁₃	#120	1	2	yes	—
145	C₈×C₁₃	C₂²×D₄	1	2	yes	—
146	C₂³×Q₈	—	2	1	no	—
147	C₂²×Q₁₆	—	2	8	no	—
148	C₂²×Q₁₆	—	2	1	no	—
149	C₂²×Q₁₆	—	2	2	no	—
150	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
151	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	complemented
152	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#121	—	2	1	no	—
155	C₂×SD₃₂	—	2	8	no	—
156	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	complemented
157	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
158	#466	—	2	2	no	—
159	#466	C₂²×C₁₃	1	4	yes	—
160	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#102	—	2	1	no	—
162	#358	C₂²×C₁₃	1	8	yes	complemented
163	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	2	8	no	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	2	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	16	4	no	—
171	C₂²×C₁₃	#447	1	2	yes	—
172	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
173	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	complemented
174	C₂²×C₁₃	#510	1	2	yes	—
175	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	4	8	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	#447	1	2	yes	—
180	C₄×C₁₃	C₂×#120	1	1	yes	characteristic
181	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂²×Q₈	—	2	8	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	2	4	no	—
185	C₂²×Q₈	—	4	4	no	—
186	C₂²×Q₈	—	4	1	no	—
187	C₂²×Q₈	—	4	2	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	—	4	1	no	—
190	C₂³×C₄	—	4	1	no	—
191	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
192	C₂×Q₁₆	—	2	8	no	—
193	C₂×Q₁₆	—	2	16	no	—
194	C₂×Q₁₆	—	2	16	no	—
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×Q₁₆	—	2	4	no	—
198	C₂×Q₁₆	D₄×C₁₃	1	8	yes	complemented
199	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	C₂³×C₁₃	1	2	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₁₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
208	#121	—	2	4	no	—
209	#121	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	SD₃₂	—	2	16	no	—
213	C₂×C₁₆	—	2	4	no	—
214	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
215	#107	—	2	1	no	—
216	#107	C₂³×C₁₃	1	2	yes	—
217	#102	—	2	4	no	—
218	C₂×C₁₃	—	2	4	no	—
219	C₂×C₁₃	—	16	2	no	—
220	C₂×C₁₃	#2185	1	4	yes	direct product factor
221	C₂×C₁₃	C₂×#447	1	1	yes	characteristic
222	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
224	C₂⁴	—	4	1	no	—
225	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
226	C₂×Q₈	—	2	16	no	—
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	4	8	no	—
230	C₂×Q₈	—	4	16	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	4	4	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	4	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	8	4	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	8	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
248	C₂²×C₄	C₁₃×D₈	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	Q₁₆	—	2	16	no	—
252	Q₁₆	—	2	8	no	—
253	Q₁₆	—	2	4	no	—
254	Q₁₆	C₂×D₄×C₁₃	1	8	yes	complemented
255	D₈	—	2	4	no	—
256	D₈	—	4	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
263	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₁₆	—	2	4	no	—
268	C₁₃	C₂×#2185	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	—	8	2	no	—
274	C₂³	—	8	2	no	—
275	C₂³	C₁₃×SD₃₂	1	4	yes	complemented
276	C₂³	C₂×C₁₃×D₈	1	2	yes	—
277	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	4	16	no	—
280	Q₈	—	4	8	no	—
281	Q₈	—	4	4	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	8	no	—
285	D₄	—	8	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	16	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	C₁₃×#120	1	2	yes	—
296	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
297	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
298	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
299	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
300	C₈	—	2	2	no	—
301	C₈	C₁₃×#120	1	2	yes	—
302	C₈	C₂²×D₄×C₁₃	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	4	no	—
310	C₂²	C₁₃×#447	1	2	yes	—
311	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
312	C₂²	C₂×C₁₃×SD₃₂	1	4	yes	complemented
313	C₂²	C₁₃×#510	1	2	yes	—
314	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
315	C₄	—	2	2	no	—
316	C₄	—	4	8	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#447	1	2	yes	—
319	C₄	C₂×C₁₃×#120	1	1	yes	characteristic
320	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₂	—	2	4	no	—
322	C₂	—	16	2	no	—
323	C₂	C₁₃×#2185	1	4	yes	direct product factor
324	C₂	C₂×C₁₃×#447	1	1	yes	characteristic
325	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
326	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	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
1. C₂³×C₁₃×Q₁₆	C₂	semidirect
2. C₂²×C₁₃×SD₃₂	C₂	semidirect
4. C₁₃×#2185	C₂	direct (N × H)
5. C₂×C₁₃×#358	C₂	semidirect
6. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
12. C₂²×C₁₃×Q₁₆	C₂²	semidirect
17. C₂×C₁₃×SD₃₂	C₂²	semidirect
23. C₁₃×#358	C₂²	semidirect
24. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
41. C₂×C₁₃×Q₁₆	D₄	semidirect
61. C₂×#2185	C₁₃	direct (N × H)
92. C₁₃×Q₁₆	C₂×D₄	semidirect
106. C₂³×Q₁₆	C₂×C₁₃	semidirect
107. C₂²×SD₃₂	C₂×C₁₃	semidirect
109. #2185	C₂×C₁₃	direct (N × H)
110. C₂×#358	C₂×C₁₃	semidirect
111. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
118. C₂³×C₁₃	SD₃₂	semidirect
151. C₂²×Q₁₆	C₂²×C₁₃	semidirect
156. C₂×SD₃₂	C₂²×C₁₃	semidirect
162. #358	C₂²×C₁₃	semidirect
163. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
173. C₂²×C₁₃	C₂×SD₃₂	semidirect
198. C₂×Q₁₆	D₄×C₁₃	semidirect
220. C₂×C₁₃	#2185	direct (N × H)
254. Q₁₆	C₂×D₄×C₁₃	semidirect
268. C₁₃	C₂×#2185	direct (N × H)
275. C₂³	C₁₃×SD₃₂	semidirect
312. C₂²	C₂×C₁₃×SD₃₂	semidirect
323. C₂	C₁₃×#2185	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#466	2
C₁₃×#2185	2
C₂×C₁₃×#358	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2185	13

Subgroups — C2×C13×#2185 #15719851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2185 #15719851