Subgroups — C₁₃×#29206 #15715545

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2058	C₂	1	4	yes	maximal
4	C₁₃×#1990	C₂	1	2	yes	maximal
5	C₁₃×#1948	C₂	1	4	yes	maximal
6	C₁₃×#1844	C₂	1	2	yes	maximal
7	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₁₃×#106	C₂²	1	8	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#391	—	2	1	no	—
17	C₁₃×#389	—	2	1	no	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#384	—	2	1	no	—
20	C₁₃×#384	—	2	1	no	—
21	C₁₃×#384	C₂²	1	2	yes	—
22	C₁₃×#384	C₂²	1	1	yes	characteristic
23	C₁₃×#384	C₂²	1	2	yes	—
24	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄³×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	8	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂²×Q₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	4	4	no	—
34	C₁₃×C₄⋊Q₈	—	4	4	no	—
35	C₁₃×C₄⋊Q₈	—	4	4	no	—
36	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	complemented
37	C₄×Q₈×C₁₃	—	4	4	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#106	—	4	8	no	—
59	C₁₃×C₄⋊C₈	Q₈	1	8	yes	complemented
60	C₄×C₈×C₁₃	—	2	4	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	#29206	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	—	4	4	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	4	2	no	—
68	C₂×Q₈×C₁₃	—	4	4	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	—	2	4	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	2	8	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	4	no	—
93	C₁₃×C₄⋊C₄	—	4	4	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	4	4	no	—
100	C₄²×C₁₃	SD₁₆	1	4	yes	—
101	C₄²×C₁₃	SD₁₆	1	4	yes	—
102	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
104	C₄²×C₁₃	#41	1	2	yes	—
105	#2868	C₂×C₁₃	1	1	yes	characteristic
106	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
107	#2058	C₂×C₁₃	1	4	yes	—
108	#1990	C₂×C₁₃	1	2	yes	—
109	#1948	C₂×C₁₃	1	4	yes	—
110	#1844	C₂×C₁₃	1	2	yes	—
111	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₃	—	4	8	no	—
114	Q₈×C₁₃	—	8	8	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
141	C₂×C₄×C₁₃	#137	1	2	yes	—
142	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
146	C₈×C₁₃	—	2	8	no	—
147	C₂×C₄⋊Q₈	—	2	1	no	—
148	C₂×C₄⋊Q₈	—	2	1	no	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
151	C₂×C₄×Q₈	—	2	1	no	—
152	C₂×#106	C₂²×C₁₃	1	8	yes	—
153	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
155	#391	—	2	1	no	—
156	#389	—	2	1	no	—
157	#389	C₂²×C₁₃	1	2	yes	—
158	#384	—	2	1	no	—
159	#384	—	2	1	no	—
160	#384	C₂²×C₁₃	1	1	yes	characteristic
161	#384	C₂²×C₁₃	1	2	yes	—
162	#384	C₂²×C₁₃	1	2	yes	—
163	C₄×C₄⋊C₄	—	2	1	no	—
164	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₄³	C₂²×C₁₃	1	1	yes	characteristic
166	#340	C₂²×C₁₃	1	8	yes	—
167	#336	C₂²×C₁₃	1	2	yes	—
168	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
170	C₂²×C₁₃	#535	1	1	yes	characteristic
171	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
172	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	4	no	—
181	C₄×C₁₃	#475	1	4	yes	—
182	C₄×C₁₃	#476	1	4	yes	—
183	C₄×C₁₃	#498	1	4	yes	—
184	C₄×C₁₃	#554	1	2	yes	—
185	C₄×C₁₃	Q₈²	1	2	yes	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×Q₈	—	2	2	no	—
188	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	4	4	no	—
191	C₄⋊Q₈	—	4	4	no	—
192	C₄⋊Q₈	—	4	4	no	—
193	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	complemented
194	C₄×Q₈	—	4	4	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	#106	—	4	8	no	—
216	C₄⋊C₈	Q₈×C₁₃	1	8	yes	complemented
217	C₄×C₈	—	2	4	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	C₂×C₁₃	#2567	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#475	1	1	yes	characteristic
223	C₂×C₁₃	C₂×#498	1	1	yes	characteristic
224	C₂×C₁₃	#3165	1	1	yes	characteristic
225	C₂×C₁₃	#3206	1	1	yes	characteristic
226	C₂×C₁₃	#3208	1	1	yes	characteristic
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×Q₈	—	4	4	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂×Q₈×C₁₃	1	8	yes	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	4	no	—
263	C₄²	C₁₃×SD₁₆	1	4	yes	—
264	C₄²	C₁₃×SD₁₆	1	4	yes	—
265	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
266	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
267	C₄²	C₁₃×#41	1	2	yes	—
268	C₁₃	#29206	1	1	yes	characteristic, direct product factor, Sylow
269	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	Q₈	—	4	8	no	—
271	Q₈	—	8	8	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
288	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
289	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
290	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
292	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
293	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
294	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
295	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
296	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
297	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
298	C₂×C₄	C₁₃×#137	1	2	yes	—
299	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
300	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
302	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
303	C₈	—	2	8	no	—
304	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
306	C₂²	C₁₃×#535	1	1	yes	characteristic
307	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
308	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
309	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	4	no	—
317	C₄	C₁₃×#475	1	4	yes	—
318	C₄	C₁₃×#476	1	4	yes	—
319	C₄	C₁₃×#498	1	4	yes	—
320	C₄	C₁₃×#554	1	2	yes	—
321	C₄	Q₈²×C₁₃	1	2	yes	—
322	C₂	C₁₃×#2567	1	1	yes	characteristic
323	C₂	C₂×C₁₃×#476	1	1	yes	characteristic
324	C₂	C₂×C₁₃×#475	1	1	yes	characteristic
325	C₂	C₂×C₁₃×#498	1	1	yes	characteristic
326	C₂	C₁₃×#3165	1	1	yes	characteristic
327	C₂	C₁₃×#3206	1	1	yes	characteristic
328	C₂	C₁₃×#3208	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₁₃×C₄⋊Q₈	Q₈	semidirect
59. C₁₃×C₄⋊C₈	Q₈	semidirect
63. #29206	C₁₃	direct (N × H)
193. C₄⋊Q₈	Q₈×C₁₃	semidirect
216. C₄⋊C₈	Q₈×C₁₃	semidirect
268. C₁₃	#29206	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2868	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2058	2
C₁₃×#1990	2
C₁₃×#1948	2
C₁₃×#1844	2
C₄×C₁₃×C₄⋊C₈	2
#29206	13

Subgroups — C13×#29206 #15715545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29206 #15715545