Subgroups — C₈×C₁₃×#125 #15708011

Order 3328.

742 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1939	C₂	1	2	yes	maximal
3	C₁₃×#1916	C₂	1	4	yes	maximal
4	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₃×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₈×C₁₃×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#389	C₄	1	2	yes	—
17	C₁₃×#387	C₄	1	4	yes	—
18	C₁₃×#382	C₄	1	1	yes	characteristic
19	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄³×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#336	C₂²	1	4	yes	—
23	C₁₃×#336	C₂²	1	4	yes	—
24	C₁₃×#336	C₂²	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₁₃×#125	C₈	1	8	yes	direct product factor
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	—	2	8	no	—
41	C₁₃×C₄⋊C₈	—	2	4	no	—
42	C₄×C₈×C₁₃	—	2	4	no	—
43	C₄×C₈×C₁₃	—	2	4	no	—
44	C₄×C₈×C₁₃	—	2	8	no	—
45	C₄×C₈×C₁₃	—	2	2	no	—
46	C₄×C₈×C₁₃	Q₈	1	4	yes	—
47	C₁₃×#95	C₂×C₄	1	4	yes	—
48	C₁₃×#95	C₂×C₄	1	4	yes	—
49	C₁₃×#95	C₂×C₄	1	4	yes	—
50	C₈×#125	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₁₃	#41	1	4	yes	—
67	C₂×C₈×C₁₃	#41	1	4	yes	—
68	C₁₃×C₄⋊C₄	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	C₂×C₈	1	16	yes	—
71	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
72	C₄²×C₁₃	—	2	4	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	4	no	—
78	C₄²×C₁₃	C₂×C₈	1	4	yes	—
79	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₃	#41	1	2	yes	—
81	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
82	#1939	C₂×C₁₃	1	2	yes	—
83	#1916	C₂×C₁₃	1	4	yes	—
84	#1831	C₂×C₁₃	1	1	yes	characteristic
85	C₈×C₄⋊C₄	C₂×C₁₃	1	4	yes	—
86	C₈×C₄⋊C₄	C₂×C₁₃	1	2	yes	—
87	C₄²×C₈	C₂×C₁₃	1	1	yes	characteristic
88	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	4	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₁₃	#131	1	4	yes	—
105	C₂×C₄×C₁₃	#131	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
110	C₈×C₁₃	—	2	4	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	—	4	4	no	—
113	C₈×C₁₃	#125	1	4	yes	direct product factor
114	C₂×#125	C₄×C₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
116	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
117	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
120	C₂×C₄×C₈	C₂²×C₁₃	1	4	yes	—
121	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
122	#389	C₄×C₁₃	1	2	yes	—
123	#387	C₄×C₁₃	1	4	yes	—
124	#382	C₄×C₁₃	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
126	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
127	C₄³	C₂²×C₁₃	1	1	yes	characteristic
128	#336	C₂²×C₁₃	1	4	yes	—
129	#336	C₂²×C₁₃	1	4	yes	—
130	#336	C₂²×C₁₃	1	4	yes	—
131	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₁₃	C₄×#41	1	2	yes	—
133	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#131	1	2	yes	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	8	no	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	4	4	no	—
141	C₄×C₁₃	C₈×Q₈	1	4	yes	—
142	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
143	C₄×C₁₃	#529	1	2	yes	—
144	C₄×C₁₃	#533	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
146	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
148	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
149	#125	C₈×C₁₃	1	8	yes	direct product factor
150	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	4	yes	—
153	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
155	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
158	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
159	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
160	C₄⋊C₈	—	2	8	no	—
161	C₄⋊C₈	—	2	4	no	—
162	C₄×C₈	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	C₄×C₈	—	2	8	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	Q₈×C₁₃	1	4	yes	—
167	#95	C₂×C₄×C₁₃	1	4	yes	—
168	#95	C₂×C₄×C₁₃	1	4	yes	—
169	#95	C₂×C₄×C₁₃	1	4	yes	—
170	C₂×C₁₃	C₄×#125	1	1	yes	characteristic
171	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
172	C₂×C₁₃	#2960	1	1	yes	characteristic
173	C₂×C₁₃	C₈×#41	1	2	yes	—
174	C₂×C₁₃	#2966	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
179	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
189	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
190	C₂×C₈	C₁₃×#41	1	4	yes	—
191	C₂×C₈	C₁₃×#41	1	4	yes	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	C₂×C₈×C₁₃	1	16	yes	—
195	C₄⋊C₄	C₂×C₈×C₁₃	1	8	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	C₂×C₈×C₁₃	1	4	yes	—
203	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
204	C₄²	C₁₃×#41	1	2	yes	—
205	C₁₃	C₈×#125	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
218	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
219	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
220	C₂×C₄	C₂²×C₈×C₁₃	1	8	yes	—
221	C₂×C₄	C₂²×C₈×C₁₃	1	2	yes	—
222	C₂×C₄	C₁₃×#131	1	4	yes	—
223	C₂×C₄	C₁₃×#131	1	2	yes	—
224	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
226	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
227	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
228	C₈	—	2	4	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	4	no	—
231	C₈	C₁₃×#125	1	4	yes	direct product factor
232	C₂²	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
233	C₂²	C₄×C₁₃×#41	1	2	yes	—
234	C₂²	C₂³×C₈×C₁₃	1	1	yes	characteristic
235	C₂²	C₂×C₁₃×#131	1	1	yes	characteristic
236	C₂²	C₂×C₁₃×#131	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	8	no	—
240	C₄	—	2	4	no	—
241	C₄	—	4	4	no	—
242	C₄	C₈×Q₈×C₁₃	1	4	yes	—
243	C₄	C₂×C₁₃×#125	1	1	yes	characteristic
244	C₄	C₁₃×#529	1	2	yes	—
245	C₄	C₁₃×#533	1	1	yes	characteristic
246	C₂	C₄×C₁₃×#125	1	1	yes	characteristic
247	C₂	C₂×C₈×Q₈×C₁₃	1	1	yes	characteristic
248	C₂	C₁₃×#2960	1	1	yes	characteristic
249	C₂	C₈×C₁₃×#41	1	2	yes	—
250	C₂	C₁₃×#2966	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
29. C₁₃×#125	C₈	direct (N × H)
50. C₈×#125	C₁₃	direct (N × H)
113. C₈×C₁₃	#125	direct (N × H)
149. #125	C₈×C₁₃	direct (N × H)
205. C₁₃	C₈×#125	direct (N × H)
231. C₈	C₁₃×#125	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#125	2
C₁₃×#1939	2
C₁₃×#1916	2
C₁₃×#1831	2
C₈×C₁₃×C₄⋊C₄	2
C₄²×C₈×C₁₃	2
C₈×#125	13

Subgroups — C8×C13×#125 #15708011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₁₃×#125 #15708011