Subgroups — C₂×Q₈×C₁₃×D₈ #15747690

Order 3328.

2286 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₁₃	C₂	1	2	yes	maximal, complemented
2	Q₈×C₁₃×D₈	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#500	C₂	1	3	yes	maximal, complemented
4	C₂×C₁₃×#474	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₃×D₈	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₁₃	—	2	2	no	—
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	D₄×Q₈×C₁₃	—	2	8	no	—
10	D₄×Q₈×C₁₃	C₂²	1	16	yes	complemented
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	6	yes	—
12	C₂×C₁₃×C₂²⋊Q₈	—	2	6	no	—
13	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₃	C₂²	1	6	yes	—
15	C₁₃×#500	C₂²	1	24	yes	complemented
16	C₁₃×#474	C₂²	1	48	yes	complemented
17	C₈×Q₈×C₁₃	C₂²	1	8	yes	complemented
18	C₄×C₁₃×D₈	C₂²	1	24	yes	—
19	C₂×C₁₃×#107	C₂²	1	6	yes	—
20	C₂×C₁₃×#107	C₂²	1	3	yes	—
21	C₂×C₁₃×C₄⋊C₈	C₂²	1	3	yes	—
22	C₂×C₁₃×#102	C₂²	1	6	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
24	C₂²×Q₈×C₁₃	—	2	16	no	—
25	C₂²×Q₈×C₁₃	—	2	6	no	—
26	C₂²×Q₈×C₁₃	—	4	16	no	—
27	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃	—	2	6	no	—
30	C₂×C₁₃×D₈	Q₈	1	8	yes	direct product factor
31	C₂×C₁₃×D₈	C₂³	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×C₄⋊Q₈	—	2	12	no	—
35	C₁₃×C₄⋊Q₈	C₂³	1	24	yes	—
36	C₁₃×C₂²⋊Q₈	—	2	48	no	—
37	C₄×Q₈×C₁₃	D₄	1	4	yes	—
38	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
39	C₄×D₄×C₁₃	—	2	12	no	—
40	C₄×D₄×C₁₃	C₂³	1	24	yes	—
41	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	6	no	—
45	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
46	C₁₃×#107	C₂³	1	12	yes	—
47	C₁₃×#107	C₂³	1	24	yes	—
48	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
49	C₁₃×#102	C₂³	1	24	yes	—
50	C₄×C₈×C₁₃	C₂³	1	12	yes	—
51	C₂×Q₈×D₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	2	24	no	—
54	C₂×Q₈×C₁₃	—	4	64	no	—
55	C₂×Q₈×C₁₃	—	4	24	no	—
56	C₂×Q₈×C₁₃	D₈	1	8	yes	direct product factor
57	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	C₂×Q₈	1	8	yes	complemented
61	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
62	C₂²×C₄×C₁₃	—	2	6	no	—
63	C₂²×C₄×C₁₃	—	2	24	no	—
64	C₂²×C₄×C₁₃	—	4	24	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×D₈	C₂×Q₈	1	16	yes	direct product factor
69	C₂×C₈×C₁₃	—	2	6	no	—
70	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	complemented
71	C₂×C₈×C₁₃	C₂⁴	1	6	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
73	C₁₃×C₄⋊C₄	—	2	24	no	—
74	C₁₃×C₄⋊C₄	—	2	6	no	—
75	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
76	C₁₃×C₄⋊C₄	C₂⁴	1	12	yes	—
77	C₁₃×C₂²⋊C₄	—	2	24	no	—
78	C₄²×C₁₃	C₂×D₄	1	6	yes	—
79	C₄²×C₁₃	C₂⁴	1	6	yes	—
80	C₂×D₄×Q₈	C₂×C₁₃	1	2	yes	complemented
81	Q₈×D₈	C₂×C₁₃	1	16	yes	direct product factor
82	C₂×#500	C₂×C₁₃	1	3	yes	complemented
83	C₂×#474	C₂×C₁₃	1	6	yes	complemented
84	C₂×C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂×C₄×D₈	C₂×C₁₃	1	3	yes	—
86	C₂³×C₁₃	—	2	8	no	—
87	C₂³×C₁₃	—	2	4	no	—
88	C₂³×C₁₃	—	4	4	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₁₃	—	4	48	no	—
92	Q₈×C₁₃	C₂×D₈	1	16	yes	direct product factor
93	D₄×C₁₃	—	2	8	no	—
94	D₄×C₁₃	—	2	4	no	—
95	D₄×C₁₃	C₂²×Q₈	1	8	yes	complemented
96	C₂×C₄×C₁₃	—	2	6	no	—
97	C₂×C₄×C₁₃	—	2	12	no	—
98	C₂×C₄×C₁₃	—	4	12	no	—
99	C₂×C₄×C₁₃	—	4	48	no	—
100	C₂×C₄×C₁₃	C₂×D₈	1	12	yes	—
101	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
102	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
104	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
106	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
107	C₈×C₁₃	—	2	6	no	—
108	C₈×C₁₃	C₂²×Q₈	1	4	yes	complemented
109	C₂³×Q₈	—	2	2	no	—
110	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
111	D₄×Q₈	—	2	8	no	—
112	D₄×Q₈	C₂²×C₁₃	1	16	yes	complemented
113	C₂×C₄⋊Q₈	C₂²×C₁₃	1	6	yes	—
114	C₂×C₂²⋊Q₈	—	2	6	no	—
115	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
116	C₂×C₄×D₄	C₂²×C₁₃	1	6	yes	—
117	#500	C₂²×C₁₃	1	24	yes	complemented
118	#474	C₂²×C₁₃	1	48	yes	complemented
119	C₈×Q₈	C₂²×C₁₃	1	8	yes	complemented
120	C₄×D₈	C₂²×C₁₃	1	24	yes	—
121	C₂×#107	C₂²×C₁₃	1	3	yes	—
122	C₂×#107	C₂²×C₁₃	1	6	yes	—
123	C₂×C₄⋊C₈	C₂²×C₁₃	1	3	yes	—
124	C₂×#102	C₂²×C₁₃	1	6	yes	—
125	C₂×C₄×C₈	C₂²×C₁₃	1	3	yes	—
126	C₂²×C₁₃	—	2	8	no	—
127	C₂²×C₁₃	—	4	16	no	—
128	C₂²×C₁₃	—	4	4	no	—
129	C₂²×C₁₃	—	4	4	no	—
130	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
131	C₂²×C₁₃	C₂²×D₈	1	2	yes	—
132	C₂²×C₁₃	#578	1	2	yes	—
133	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₁₃	—	2	6	no	—
135	C₄×C₁₃	—	4	12	no	—
136	C₄×C₁₃	D₄×Q₈	1	2	yes	—
137	C₄×C₁₃	C₂²×D₈	1	12	yes	—
138	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
139	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
140	C₂²×Q₈	—	2	16	no	—
141	C₂²×Q₈	—	2	6	no	—
142	C₂²×Q₈	—	4	16	no	—
143	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
144	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
145	C₂³×C₄	—	2	6	no	—
146	C₂×D₈	Q₈×C₁₃	1	8	yes	direct product factor
147	C₂×D₈	C₂³×C₁₃	1	4	yes	—
148	C₂²×C₈	C₂³×C₁₃	1	3	yes	—
149	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	12	no	—
151	C₄⋊Q₈	C₂³×C₁₃	1	24	yes	—
152	C₂²⋊Q₈	—	2	48	no	—
153	C₄×Q₈	D₄×C₁₃	1	4	yes	—
154	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
155	C₄×D₄	—	2	12	no	—
156	C₄×D₄	C₂³×C₁₃	1	24	yes	—
157	C₂×C₄⋊C₄	—	2	6	no	—
158	C₂×C₄⋊C₄	D₄×C₁₃	1	3	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
160	C₂×C₂²⋊C₄	—	2	6	no	—
161	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
162	#107	C₂³×C₁₃	1	12	yes	—
163	#107	C₂³×C₁₃	1	24	yes	—
164	C₄⋊C₈	C₂³×C₁₃	1	12	yes	—
165	#102	C₂³×C₁₃	1	24	yes	—
166	C₄×C₈	C₂³×C₁₃	1	12	yes	—
167	C₂×C₁₃	—	4	8	no	—
168	C₂×C₁₃	Q₈×D₈	1	4	yes	direct product factor
169	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
170	C₂×C₁₃	C₂³×D₈	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#578	1	1	yes	characteristic
172	C₂⁴	—	2	2	no	—
173	C₂×Q₈	—	2	24	no	—
174	C₂×Q₈	—	4	64	no	—
175	C₂×Q₈	—	4	24	no	—
176	C₂×Q₈	C₁₃×D₈	1	8	yes	direct product factor
177	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	C₂×Q₈×C₁₃	1	8	yes	complemented
181	C₂×D₄	C₂⁴×C₁₃	1	4	yes	—
182	C₂²×C₄	—	2	6	no	—
183	C₂²×C₄	—	2	24	no	—
184	C₂²×C₄	—	4	24	no	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
186	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
187	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
188	D₈	C₂×Q₈×C₁₃	1	16	yes	direct product factor
189	C₂×C₈	—	2	6	no	—
190	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	complemented
191	C₂×C₈	C₂⁴×C₁₃	1	6	yes	—
192	C₂×C₈	C₂⁴×C₁₃	1	2	yes	—
193	C₄⋊C₄	—	2	24	no	—
194	C₄⋊C₄	—	2	6	no	—
195	C₄⋊C₄	C₂×D₄×C₁₃	1	12	yes	—
196	C₄⋊C₄	C₂⁴×C₁₃	1	12	yes	—
197	C₂²⋊C₄	—	2	24	no	—
198	C₄²	C₂×D₄×C₁₃	1	6	yes	—
199	C₄²	C₂⁴×C₁₃	1	6	yes	—
200	C₁₃	C₂×Q₈×D₈	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³	—	2	8	no	—
202	C₂³	—	2	4	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
206	Q₈	—	4	48	no	—
207	Q₈	C₂×C₁₃×D₈	1	16	yes	direct product factor
208	D₄	—	2	8	no	—
209	D₄	—	2	4	no	—
210	D₄	C₂²×Q₈×C₁₃	1	8	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	4	12	no	—
214	C₂×C₄	—	4	48	no	—
215	C₂×C₄	C₂×C₁₃×D₈	1	12	yes	—
216	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
217	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
219	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
220	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
221	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
222	C₈	—	2	6	no	—
223	C₈	C₂²×Q₈×C₁₃	1	4	yes	complemented
224	C₂²	—	2	8	no	—
225	C₂²	—	4	4	no	—
226	C₂²	—	4	4	no	—
227	C₂²	—	4	16	no	—
228	C₂²	D₄×Q₈×C₁₃	1	2	yes	—
229	C₂²	C₂²×C₁₃×D₈	1	2	yes	—
230	C₂²	C₁₃×#578	1	2	yes	—
231	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	4	12	no	—
234	C₄	D₄×Q₈×C₁₃	1	2	yes	—
235	C₄	C₂²×C₁₃×D₈	1	12	yes	—
236	C₄	C₂³×Q₈×C₁₃	1	1	yes	characteristic
237	C₄	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂	—	4	8	no	—
239	C₂	Q₈×C₁₃×D₈	1	4	yes	direct product factor
240	C₂	C₂×D₄×Q₈×C₁₃	1	1	yes	characteristic
241	C₂	C₂³×C₁₃×D₈	1	1	yes	characteristic
242	C₂	C₂×C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×Q₈×C₁₃	C₂	semidirect
2. Q₈×C₁₃×D₈	C₂	direct (N × H)
3. C₂×C₁₃×#500	C₂	semidirect
4. C₂×C₁₃×#474	C₂	semidirect
5. C₂×C₈×Q₈×C₁₃	C₂	semidirect
10. D₄×Q₈×C₁₃	C₂²	semidirect
15. C₁₃×#500	C₂²	semidirect
16. C₁₃×#474	C₂²	semidirect
17. C₈×Q₈×C₁₃	C₂²	semidirect
30. C₂×C₁₃×D₈	Q₈	direct (N × H)
51. C₂×Q₈×D₈	C₁₃	direct (N × H)
56. C₂×Q₈×C₁₃	D₈	direct (N × H)
60. C₂×D₄×C₁₃	C₂×Q₈	semidirect
68. C₁₃×D₈	C₂×Q₈	direct (N × H)
70. C₂×C₈×C₁₃	C₂×Q₈	semidirect
80. C₂×D₄×Q₈	C₂×C₁₃	semidirect
81. Q₈×D₈	C₂×C₁₃	direct (N × H)
82. C₂×#500	C₂×C₁₃	semidirect
83. C₂×#474	C₂×C₁₃	semidirect
84. C₂×C₈×Q₈	C₂×C₁₃	semidirect
92. Q₈×C₁₃	C₂×D₈	direct (N × H)
95. D₄×C₁₃	C₂²×Q₈	semidirect
108. C₈×C₁₃	C₂²×Q₈	semidirect
112. D₄×Q₈	C₂²×C₁₃	semidirect
117. #500	C₂²×C₁₃	semidirect
118. #474	C₂²×C₁₃	semidirect
119. C₈×Q₈	C₂²×C₁₃	semidirect
146. C₂×D₈	Q₈×C₁₃	direct (N × H)
168. C₂×C₁₃	Q₈×D₈	direct (N × H)
176. C₂×Q₈	C₁₃×D₈	direct (N × H)
180. C₂×D₄	C₂×Q₈×C₁₃	semidirect
188. D₈	C₂×Q₈×C₁₃	direct (N × H)
190. C₂×C₈	C₂×Q₈×C₁₃	semidirect
200. C₁₃	C₂×Q₈×D₈	direct (N × H)
207. Q₈	C₂×C₁₃×D₈	direct (N × H)
210. D₄	C₂²×Q₈×C₁₃	semidirect
223. C₈	C₂²×Q₈×C₁₃	semidirect
239. C₂	Q₈×C₁₃×D₈	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×D₄×Q₈×C₁₃	2
Q₈×C₁₃×D₈	2
C₂×C₁₃×#500	2
C₂×C₁₃×#474	2
C₂×C₈×Q₈×C₁₃	2
C₂×C₄×C₁₃×D₈	2
C₂×Q₈×D₈	13

Subgroups — C2×Q8×C13×D8 #15747690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₁₃×D₈ #15747690