Subgroups — C₈×C₁₃×#117 #15706810

Order 3328.

1174 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₃×C₄⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₈×C₁₃×C₂²⋊C₄	C₂	1	4	yes	maximal
6	C₁₃×#1744	C₂	1	2	yes	maximal, complemented
7	C₄²×C₈×C₁₃	C₂	1	2	yes	maximal, complemented
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₁₃×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#431	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₄	1	8	yes	complemented
15	C₂×C₄×C₈×C₁₃	C₂²	1	8	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
23	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₁₃×#376	C₄	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#336	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	—	2	8	no	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	8	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₁₃×#117	C₈	1	16	yes	direct product factor
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₃	—	2	16	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
54	C₁₃×#95	C₂×C₄	1	4	yes	—
55	C₁₃×#95	C₂×C₄	1	4	yes	—
56	C₈×#117	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₁₃	—	2	8	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	16	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₃	#41	1	8	yes	—
79	C₁₃×C₄⋊C₄	C₂×C₈	1	16	yes	complemented
80	C₁₃×C₂²⋊C₄	C₂×C₈	1	16	yes	—
81	C₄²×C₁₃	—	2	8	no	—
82	C₄²×C₁₃	—	2	8	no	—
83	C₄²×C₁₃	C₄²	1	4	yes	—
84	C₄²×C₁₃	C₄²	1	4	yes	—
85	C₄²×C₁₃	C₂×C₈	1	16	yes	complemented
86	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×#431	C₂×C₁₃	1	1	yes	characteristic
91	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
92	C₄×#117	C₂×C₁₃	1	1	yes	characteristic
93	C₈×C₄⋊C₄	C₂×C₁₃	1	4	yes	complemented
94	C₈×C₂²⋊C₄	C₂×C₁₃	1	4	yes	—
95	#1744	C₂×C₁₃	1	2	yes	complemented
96	C₄²×C₈	C₂×C₁₃	1	2	yes	complemented
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
100	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	8	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	8	no	—
109	C₂×C₄×C₁₃	C₄×C₈	1	16	yes	complemented
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	16	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
118	C₂×C₄×C₁₃	#131	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
122	C₈×C₁₃	—	2	4	no	—
123	C₈×C₁₃	—	2	8	no	—
124	C₈×C₁₃	#117	1	8	yes	direct product factor
125	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
126	C₂×#117	C₄×C₁₃	1	2	yes	—
127	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
128	#431	C₄×C₁₃	1	8	yes	complemented
129	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
130	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄×C₈	C₄×C₁₃	1	8	yes	complemented
132	C₂×C₄×C₈	C₂²×C₁₃	1	8	yes	—
133	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
135	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₄×C₁₃	1	2	yes	—
138	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₄×C₂²⋊C₄	C₄×C₁₃	1	2	yes	—
140	C₄×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
141	#376	C₄×C₁₃	1	2	yes	—
142	C₄³	C₂²×C₁₃	1	2	yes	—
143	#336	C₂²×C₁₃	1	8	yes	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	8	no	—
146	C₂²×C₁₃	—	2	2	no	—
147	C₂²×C₁₃	C₂×C₄×C₈	1	4	yes	—
148	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₁₃	C₄×#41	1	2	yes	—
150	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
151	C₂²×C₁₃	C₂×#131	1	2	yes	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	2	16	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
156	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
157	C₄×C₁₃	#405	1	4	yes	—
158	C₄×C₁₃	C₂×#117	1	2	yes	—
159	C₄×C₁₃	C₄×#41	1	2	yes	—
160	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
163	C₂²×C₈	—	2	8	no	—
164	C₂²×C₈	C₂×C₄×C₁₃	1	8	yes	—
165	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
167	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
168	#117	C₈×C₁₃	1	16	yes	direct product factor
169	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
171	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
175	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
177	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
178	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
181	C₂²⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
182	C₄×C₈	—	2	16	no	—
183	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
184	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
185	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
186	C₄×C₈	C₂×C₄×C₁₃	1	8	yes	—
187	#95	C₂×C₄×C₁₃	1	4	yes	—
188	#95	C₂×C₄×C₁₃	1	4	yes	—
189	C₂×C₁₃	—	2	4	no	—
190	C₂×C₁₃	C₄×#117	1	1	yes	characteristic
191	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
192	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
193	C₂×C₁₃	C₈×#41	1	4	yes	—
194	C₂⁴	C₂²×C₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
198	C₂²×C₄	C₄²×C₁₃	1	4	yes	—
199	C₂²×C₄	C₂×C₈×C₁₃	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
213	C₂×C₈	C₂²×C₄×C₁₃	1	8	yes	—
214	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
215	C₂×C₈	C₁₃×#41	1	8	yes	—
216	C₄⋊C₄	C₂×C₈×C₁₃	1	16	yes	complemented
217	C₂²⋊C₄	C₂×C₈×C₁₃	1	16	yes	—
218	C₄²	—	2	8	no	—
219	C₄²	—	2	8	no	—
220	C₄²	C₄²×C₁₃	1	4	yes	—
221	C₄²	C₄²×C₁₃	1	4	yes	—
222	C₄²	C₂×C₈×C₁₃	1	16	yes	complemented
223	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
224	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
225	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
226	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
227	C₁₃	C₈×#117	1	1	yes	characteristic, direct product factor, Sylow
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
231	C₂³	C₂²×C₈×C₁₃	1	4	yes	—
232	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₄×C₈×C₁₃	1	16	yes	complemented
241	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
243	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
244	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
246	C₂×C₄	C₂²×C₈×C₁₃	1	16	yes	—
247	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
248	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
249	C₂×C₄	C₁₃×#131	1	4	yes	—
250	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₁₃×#117	1	8	yes	direct product factor
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	8	no	—
259	C₂²	C₂×C₄×C₈×C₁₃	1	4	yes	—
260	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
261	C₂²	C₄×C₁₃×#41	1	2	yes	—
262	C₂²	C₂³×C₈×C₁₃	1	2	yes	—
263	C₂²	C₂×C₁₃×#131	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	16	no	—
267	C₄	C₂×C₄×C₈×C₁₃	1	4	yes	—
268	C₄	C₂×C₄×C₈×C₁₃	1	4	yes	—
269	C₄	C₁₃×#405	1	4	yes	—
270	C₄	C₂×C₁₃×#117	1	2	yes	—
271	C₄	C₄×C₁₃×#41	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₄×C₁₃×#117	1	1	yes	characteristic
274	C₂	C₂²×C₄×C₈×C₁₃	1	1	yes	characteristic
275	C₂	C₂×C₁₃×#405	1	1	yes	characteristic
276	C₂	C₈×C₁₃×#41	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₁₃	C₂²
commutator	C₂	C₂²×C₄×C₈×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
6. C₁₃×#1744	C₂	semidirect
7. C₄²×C₈×C₁₃	C₂	semidirect
11. C₁₃×#431	C₄	semidirect
14. C₂×C₄×C₈×C₁₃	C₄	semidirect
35. C₁₃×#117	C₈	direct (N × H)
56. C₈×#117	C₁₃	direct (N × H)
79. C₁₃×C₄⋊C₄	C₂×C₈	semidirect
85. C₄²×C₁₃	C₂×C₈	semidirect
93. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
95. #1744	C₂×C₁₃	semidirect
96. C₄²×C₈	C₂×C₁₃	semidirect
109. C₂×C₄×C₁₃	C₄×C₈	semidirect
124. C₈×C₁₃	#117	direct (N × H)
128. #431	C₄×C₁₃	semidirect
131. C₂×C₄×C₈	C₄×C₁₃	semidirect
168. #117	C₈×C₁₃	direct (N × H)
216. C₄⋊C₄	C₂×C₈×C₁₃	semidirect
222. C₄²	C₂×C₈×C₁₃	semidirect
227. C₁₃	C₈×#117	direct (N × H)
240. C₂×C₄	C₄×C₈×C₁₃	semidirect
255. C₈	C₁₃×#117	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×#431	2
C₂²×C₄×C₈×C₁₃	2
C₄×C₁₃×#117	2
C₈×C₁₃×C₄⋊C₄	2
C₈×C₁₃×C₂²⋊C₄	2
C₁₃×#1744	2
C₄²×C₈×C₁₃	2
C₈×#117	13

Subgroups — C8×C13×#117 #15706810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₁₃×#117 #15706810