Subgroups — C₄×C₁₃×#508 #15719786

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#135	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×Q₃₂	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×SD₃₂	C₂	1	4	yes	maximal, complemented
5	C₄×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₄×C₁₃×#41	—	2	2	no	—
11	C₁₃×#508	C₄	1	8	yes	direct product factor
12	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
18	C₄×C₁₃×SD₁₆	—	2	4	no	—
19	C₄×C₁₃×D₈	C₂²	1	4	yes	—
20	C₁₃×#427	C₂²	1	2	yes	—
21	C₁₃×#416	—	2	2	no	—
22	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#366	C₂²	1	2	yes	—
25	C₁₃×#358	C₂²	1	4	yes	—
26	C₁₃×#357	C₂²	1	4	yes	—
27	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₁₃×#135	—	2	4	no	—
31	C₁₃×#135	C₂×C₄	1	8	yes	complemented
32	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
33	C₂×C₁₃×SD₁₆	—	2	2	no	—
34	C₂×C₁₃×D₈	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	—	2	4	no	—
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	—	4	4	no	—
40	C₁₃×#117	—	2	2	no	—
41	C₁₃×#117	—	4	2	no	—
42	C₂×C₄²×C₁₃	—	4	2	no	—
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₃×C₁₆	—	2	2	no	—
48	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#106	—	2	2	no	—
54	C₁₃×#103	—	2	4	no	—
55	C₁₃×#102	—	2	4	no	—
56	C₄×C₈×C₁₃	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	C₄×#508	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	2	8	no	—
60	C₁₃×#41	—	4	4	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	4	4	no	—
66	C₂²×C₄×C₁₃	D₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
70	C₁₃×SD₁₆	—	2	8	no	—
71	C₁₃×D₈	—	2	2	no	—
72	C₁₃×D₈	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	4	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₄²×C₁₃	—	4	4	no	—
85	C₄²×C₁₃	D₈	1	1	yes	characteristic
86	C₄²×C₁₃	D₈	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
91	C₂×#508	C₂×C₁₃	1	1	yes	characteristic
92	C₄×#135	C₂×C₁₃	1	2	yes	complemented
93	C₄×Q₃₂	C₂×C₁₃	1	2	yes	complemented
94	C₄×SD₃₂	C₂×C₁₃	1	4	yes	complemented
95	C₄×D₁₆	C₂×C₁₃	1	2	yes	complemented
96	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#2138	C₂×C₁₃	1	2	yes	complemented
98	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	4	no	—
102	Q₈×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	2	4	no	—
104	D₄×C₁₃	—	4	8	no	—
105	D₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	8	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	—	8	8	no	—
111	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₁₃	#135	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₄×D₄	1	2	yes	—
119	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
121	C₂×#135	C₂²×C₁₃	1	2	yes	—
122	C₄×#41	—	2	2	no	—
123	#508	C₄×C₁₃	1	8	yes	direct product factor
124	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
126	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
130	C₄×SD₁₆	—	2	4	no	—
131	C₄×D₈	C₂²×C₁₃	1	4	yes	—
132	#427	C₂²×C₁₃	1	2	yes	—
133	#416	—	2	2	no	—
134	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
135	#367	C₂²×C₁₃	1	2	yes	—
136	#366	C₂²×C₁₃	1	2	yes	—
137	#358	C₂²×C₁₃	1	4	yes	—
138	#357	C₂²×C₁₃	1	4	yes	—
139	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
140	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	4	4	no	—
143	C₂²×C₁₃	—	8	4	no	—
144	C₂²×C₁₃	C₄×D₈	1	2	yes	—
145	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	4	4	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	C₄×D₈	1	2	yes	—
150	C₄×C₁₃	#508	1	4	yes	direct product factor
151	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
153	C₂×#41	—	2	2	no	—
154	#135	—	2	4	no	—
155	#135	C₂×C₄×C₁₃	1	8	yes	complemented
156	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×D₈	C₂³×C₁₃	1	2	yes	—
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₄×Q₈	—	2	4	no	—
162	C₄×D₄	—	2	4	no	—
163	C₄×D₄	—	4	4	no	—
164	#117	—	2	2	no	—
165	#117	—	4	2	no	—
166	C₂×C₄²	—	4	2	no	—
167	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
168	Q₃₂	C₂×C₄×C₁₃	1	4	yes	complemented
169	SD₃₂	C₂×C₄×C₁₃	1	8	yes	complemented
170	D₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
173	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
174	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
176	#107	C₂³×C₁₃	1	4	yes	—
177	#106	—	2	2	no	—
178	#103	—	2	4	no	—
179	#102	—	2	4	no	—
180	C₄×C₈	D₄×C₁₃	1	2	yes	—
181	C₄×C₈	C₂³×C₁₃	1	2	yes	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	8	4	no	—
184	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
185	C₂×C₁₃	C₂×#508	1	2	yes	—
186	#41	—	2	8	no	—
187	#41	—	4	4	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	C₁₃×D₈	1	2	yes	—
194	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
195	Q₁₆	—	2	2	no	—
196	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
197	SD₁₆	—	2	8	no	—
198	D₈	—	2	2	no	—
199	D₈	C₂²×C₄×C₁₃	1	4	yes	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
204	C₂×C₈	C₁₃×#41	1	2	yes	—
205	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	—	4	4	no	—
212	C₄²	C₁₃×D₈	1	1	yes	characteristic
213	C₄²	C₁₃×D₈	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₁₃	1	2	yes	—
215	C₁₆	—	2	2	no	—
216	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
217	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
218	C₁₃	C₄×#508	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	4	2	no	—
220	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
232	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
233	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
234	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
235	C₂×C₄	C₁₃×#135	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₄×D₄×C₁₃	1	2	yes	—
239	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
240	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	C₄×C₁₃×D₈	1	2	yes	—
245	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₄×C₁₃×D₈	1	2	yes	—
250	C₄	C₁₃×#508	1	4	yes	direct product factor
251	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
252	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	8	4	no	—
255	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic
256	C₂	C₂×C₁₃×#508	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃	D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂×C₁₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×#135	C₂	semidirect
3. C₄×C₁₃×Q₃₂	C₂	semidirect
4. C₄×C₁₃×SD₃₂	C₂	semidirect
5. C₄×C₁₃×D₁₆	C₂	semidirect
6. C₁₃×#2158	C₂	semidirect
7. C₁₃×#2138	C₂	semidirect
8. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
11. C₁₃×#508	C₄	direct (N × H)
31. C₁₃×#135	C₂×C₄	semidirect
44. C₁₃×Q₃₂	C₂×C₄	semidirect
45. C₁₃×SD₃₂	C₂×C₄	semidirect
46. C₁₃×D₁₆	C₂×C₄	semidirect
48. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
58. C₄×#508	C₁₃	direct (N × H)
92. C₄×#135	C₂×C₁₃	semidirect
93. C₄×Q₃₂	C₂×C₁₃	semidirect
94. C₄×SD₃₂	C₂×C₁₃	semidirect
95. C₄×D₁₆	C₂×C₁₃	semidirect
96. #2158	C₂×C₁₃	semidirect
97. #2138	C₂×C₁₃	semidirect
98. C₂×C₄×C₁₆	C₂×C₁₃	semidirect
123. #508	C₄×C₁₃	direct (N × H)
150. C₄×C₁₃	#508	direct (N × H)
155. #135	C₂×C₄×C₁₃	semidirect
168. Q₃₂	C₂×C₄×C₁₃	semidirect
169. SD₃₂	C₂×C₄×C₁₃	semidirect
170. D₁₆	C₂×C₄×C₁₃	semidirect
172. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
218. C₁₃	C₄×#508	direct (N × H)
250. C₄	C₁₃×#508	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×#508	2
C₄×C₁₃×#135	2
C₄×C₁₃×Q₃₂	2
C₄×C₁₃×SD₃₂	2
C₄×C₁₃×D₁₆	2
C₁₃×#2158	2
C₁₃×#2138	2
C₂×C₄×C₁₃×C₁₆	2
C₄×#508	13

Subgroups — C4×C13×#508 #15719786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#508 #15719786