Subgroups — C₂×C₁₃×#3185 #15747031

Order 3328.

1954 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#533	C₂	1	2	yes	maximal
2	C₁₃×#3185	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#487	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#470	C₂	1	4	yes	maximal
7	C₂×C₁₃×#441	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
10	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
11	C₁₃×#533	—	2	8	no	—
12	C₁₃×#533	C₂²	1	16	yes	—
13	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#125	—	2	2	no	—
16	C₂×C₁₃×C₂²⋊Q₈	—	2	2	no	—
17	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
19	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
20	C₂×C₁₃×#117	—	2	2	no	—
21	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#498	C₂²	1	8	yes	complemented
23	C₁₃×#494	C₂²	1	8	yes	complemented
24	C₁₃×#487	C₂²	1	16	yes	complemented
25	C₁₃×#470	C₂²	1	32	yes	—
26	C₁₃×#441	C₂²	1	32	yes	complemented
27	C₂×C₁₃×#106	C₂²	1	4	yes	—
28	C₂×C₁₃×#103	C₂²	1	8	yes	—
29	C₄×C₁₃×M₁₆	C₂²	1	8	yes	—
30	C₂×C₁₃×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₁₃	—	2	2	no	—
34	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	D₄	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×Q₁₆	—	2	16	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	16	yes	—
39	C₂×C₁₃×M₁₆	—	2	8	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	8	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₁₃×C₄⋊Q₈	—	2	4	no	—
44	C₁₃×C₄⋊Q₈	—	2	4	no	—
45	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
46	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
47	C₁₃×#125	—	2	16	no	—
48	C₁₃×C₂²⋊Q₈	—	2	16	no	—
49	C₁₃×C₂²⋊Q₈	—	2	8	no	—
50	C₁₃×C₂²⋊Q₈	C₂³	1	16	yes	—
51	C₄×Q₈×C₁₃	—	2	16	no	—
52	C₄×Q₈×C₁₃	—	2	8	no	—
53	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
54	C₁₃×#117	—	2	16	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
60	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
61	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₂×C₄²×C₁₃	—	2	4	no	—
63	C₂×C₄²×C₁₃	D₄	1	2	yes	—
64	C₂×C₄²×C₁₃	D₄	1	4	yes	—
65	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
68	C₁₃×#106	C₂³	1	16	yes	—
69	C₁₃×#103	C₂³	1	32	yes	—
70	C₁₃×#97	C₂³	1	8	yes	—
71	C₄×C₈×C₁₃	C₂³	1	4	yes	—
72	C₄×C₈×C₁₃	C₂³	1	4	yes	—
73	C₂×#3185	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₁₃	—	2	4	no	—
76	C₂×Q₈×C₁₃	—	2	8	no	—
77	C₂×Q₈×C₁₃	—	2	16	no	—
78	C₂×Q₈×C₁₃	—	4	8	no	—
79	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₁₃	—	2	8	no	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	—	2	4	no	—
83	C₂²×C₄×C₁₃	—	2	4	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×Q₁₆	—	2	32	no	—
92	C₁₃×M₁₆	—	2	16	no	—
93	C₂×C₈×C₁₃	—	2	4	no	—
94	C₂×C₈×C₁₃	#41	1	8	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
96	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
97	C₁₃×C₄⋊C₄	—	2	8	no	—
98	C₁₃×C₄⋊C₄	—	2	8	no	—
99	C₁₃×C₄⋊C₄	—	2	8	no	—
100	C₁₃×C₄⋊C₄	—	2	16	no	—
101	C₁₃×C₄⋊C₄	—	2	16	no	—
102	C₁₃×C₄⋊C₄	—	2	4	no	—
103	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
104	C₁₃×C₂²⋊C₄	—	2	16	no	—
105	C₄²×C₁₃	—	2	16	no	—
106	C₄²×C₁₃	C₂×D₄	1	2	yes	—
107	C₄²×C₁₃	C₂×D₄	1	8	yes	—
108	C₄²×C₁₃	C₂×D₄	1	2	yes	—
109	C₄²×C₁₃	C₂⁴	1	2	yes	—
110	C₄²×C₁₃	C₂⁴	1	2	yes	—
111	C₂×#533	C₂×C₁₃	1	2	yes	—
112	#3185	C₂×C₁₃	1	16	yes	direct product factor
113	C₂×#498	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂×#487	C₂×C₁₃	1	2	yes	complemented
116	C₂×#470	C₂×C₁₃	1	4	yes	—
117	C₂×#441	C₂×C₁₃	1	4	yes	complemented
118	C₂×C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	—	2	4	no	—
121	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	16	no	—
124	Q₈×C₁₃	—	4	8	no	—
125	Q₈×C₁₃	—	4	16	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	16	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	8	no	—
132	C₂×C₄×C₁₃	—	2	8	no	—
133	C₂×C₄×C₁₃	—	4	8	no	—
134	C₂×C₄×C₁₃	—	4	8	no	—
135	C₂×C₄×C₁₃	#137	1	8	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
140	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
141	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
142	C₈×C₁₃	—	2	4	no	—
143	C₈×C₁₃	C₂×#41	1	8	yes	—
144	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
145	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
146	#533	—	2	8	no	—
147	#533	C₂²×C₁₃	1	16	yes	—
148	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
150	C₂×#125	—	2	2	no	—
151	C₂×C₂²⋊Q₈	—	2	2	no	—
152	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
153	C₂×C₄×Q₈	—	2	2	no	—
154	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
155	C₂×#117	—	2	2	no	—
156	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
157	#498	C₂²×C₁₃	1	8	yes	complemented
158	#494	C₂²×C₁₃	1	8	yes	complemented
159	#487	C₂²×C₁₃	1	16	yes	complemented
160	#470	C₂²×C₁₃	1	32	yes	—
161	#441	C₂²×C₁₃	1	32	yes	complemented
162	C₂×#106	C₂²×C₁₃	1	4	yes	—
163	C₂×#103	C₂²×C₁₃	1	8	yes	—
164	C₄×M₁₆	C₂²×C₁₃	1	8	yes	—
165	C₂×#97	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₂²×C₁₃	—	2	2	no	—
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	#532	1	2	yes	—
172	C₂²×C₁₃	C₂×#137	1	4	yes	—
173	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	4	no	—
176	C₄×C₁₃	—	4	8	no	—
177	C₄×C₁₃	—	4	8	no	—
178	C₄×C₁₃	#532	1	2	yes	—
179	C₄×C₁₃	C₂×#137	1	8	yes	—
180	C₄×C₁₃	C₂²×#41	1	2	yes	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
183	C₂³×C₄	D₄×C₁₃	1	2	yes	—
184	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	16	no	—
186	C₂×Q₁₆	C₂³×C₁₃	1	16	yes	—
187	C₂×M₁₆	—	2	8	no	—
188	C₂×M₁₆	C₂³×C₁₃	1	8	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	4	no	—
193	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
194	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
195	#125	—	2	16	no	—
196	C₂²⋊Q₈	—	2	16	no	—
197	C₂²⋊Q₈	—	2	8	no	—
198	C₂²⋊Q₈	C₂³×C₁₃	1	16	yes	—
199	C₄×Q₈	—	2	16	no	—
200	C₄×Q₈	—	2	8	no	—
201	C₄×Q₈	C₂³×C₁₃	1	16	yes	—
202	#117	—	2	16	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	D₄×C₁₃	1	2	yes	—
212	C₂×C₄²	D₄×C₁₃	1	4	yes	—
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
216	#106	C₂³×C₁₃	1	16	yes	—
217	#103	C₂³×C₁₃	1	32	yes	—
218	#97	C₂³×C₁₃	1	8	yes	—
219	C₄×C₈	C₂³×C₁₃	1	4	yes	—
220	C₄×C₈	C₂³×C₁₃	1	4	yes	—
221	C₂×C₁₃	—	4	2	no	—
222	C₂×C₁₃	#3185	1	4	yes	direct product factor
223	C₂×C₁₃	C₂×#532	1	1	yes	characteristic
224	C₂×C₁₃	C₂²×#137	1	2	yes	—
225	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂×Q₈	—	2	4	no	—
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	16	no	—
229	C₂×Q₈	—	4	8	no	—
230	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
239	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
242	Q₁₆	—	2	32	no	—
243	M₁₆	—	2	16	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₁₃×#41	1	8	yes	—
246	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
247	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	16	no	—
252	C₄⋊C₄	—	2	16	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	C₂⁴×C₁₃	1	8	yes	—
255	C₂²⋊C₄	—	2	16	no	—
256	C₄²	—	2	16	no	—
257	C₄²	C₂×D₄×C₁₃	1	2	yes	—
258	C₄²	C₂×D₄×C₁₃	1	8	yes	—
259	C₄²	C₂×D₄×C₁₃	1	2	yes	—
260	C₄²	C₂⁴×C₁₃	1	2	yes	—
261	C₄²	C₂⁴×C₁₃	1	2	yes	—
262	C₁₃	C₂×#3185	1	1	yes	characteristic, direct product factor, Sylow
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
266	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	Q₈	—	2	16	no	—
268	Q₈	—	4	8	no	—
269	Q₈	—	4	16	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	C₁₃×#137	1	8	yes	—
280	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
281	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
284	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
285	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	C₂×C₁₃×#41	1	8	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	C₁₃×#532	1	2	yes	—
292	C₂²	C₂×C₁₃×#137	1	4	yes	—
293	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	8	no	—
298	C₄	C₁₃×#532	1	2	yes	—
299	C₄	C₂×C₁₃×#137	1	8	yes	—
300	C₄	C₂²×C₁₃×#41	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₁₃×#3185	1	4	yes	direct product factor
303	C₂	C₂×C₁₃×#532	1	1	yes	characteristic
304	C₂	C₂²×C₁₃×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#3185	C₂	direct (N × H)
3. C₂×C₁₃×#498	C₂	semidirect
4. C₂×C₁₃×#494	C₂	semidirect
5. C₂×C₁₃×#487	C₂	semidirect
7. C₂×C₁₃×#441	C₂	semidirect
22. C₁₃×#498	C₂²	semidirect
23. C₁₃×#494	C₂²	semidirect
24. C₁₃×#487	C₂²	semidirect
26. C₁₃×#441	C₂²	semidirect
73. C₂×#3185	C₁₃	direct (N × H)
112. #3185	C₂×C₁₃	direct (N × H)
113. C₂×#498	C₂×C₁₃	semidirect
114. C₂×#494	C₂×C₁₃	semidirect
115. C₂×#487	C₂×C₁₃	semidirect
117. C₂×#441	C₂×C₁₃	semidirect
157. #498	C₂²×C₁₃	semidirect
158. #494	C₂²×C₁₃	semidirect
159. #487	C₂²×C₁₃	semidirect
161. #441	C₂²×C₁₃	semidirect
222. C₂×C₁₃	#3185	direct (N × H)
262. C₁₃	C₂×#3185	direct (N × H)
302. C₂	C₁₃×#3185	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×#533	2
C₁₃×#3185	2
C₂×C₁₃×#498	2
C₂×C₁₃×#494	2
C₂×C₁₃×#487	2
C₂×C₁₃×#470	2
C₂×C₁₃×#441	2
C₂×C₄×C₁₃×M₁₆	2
C₂×#3185	13

Subgroups — C2×C13×#3185 #15747031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3185 #15747031