Subgroups — C₂²×D₄×Q₈×C₁₃ #15749035

Order 3328.

11798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₁₃	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×Q₈×C₁₃	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₁₃×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
4	C₂²×C₁₃×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄×C₁₃	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₁₃	—	2	32	no	—
8	C₂³×Q₈×C₁₃	C₂²	1	6	yes	—
9	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₁₃	C₂²	1	48	yes	complemented
11	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	6	yes	—
13	D₄×Q₈×C₁₃	C₂²	1	256	yes	direct product factor
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	72	yes	complemented
15	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	144	yes	complemented
16	C₂×C₄×Q₈×C₁₃	C₂²	1	24	yes	complemented
17	C₂×C₄×D₄×C₁₃	C₂²	1	72	yes	—
18	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
19	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
20	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
21	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
22	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
23	C₂⁵×C₁₃	C₂³	1	2	yes	—
24	C₂²×Q₈×C₁₃	—	2	384	no	—
25	C₂²×Q₈×C₁₃	—	2	48	no	—
26	C₂²×Q₈×C₁₃	D₄	1	16	yes	direct product factor
27	C₂²×Q₈×C₁₃	C₂³	1	72	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	12	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	128	yes	complemented
30	C₂²×D₄×C₁₃	Q₈	1	16	yes	direct product factor
31	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
32	C₂³×C₄×C₁₃	—	2	48	no	—
33	C₂³×C₄×C₁₃	C₂³	1	72	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	C₂³	1	192	yes	complemented
39	C₁₃×C₂²⋊Q₈	C₂³	1	384	yes	complemented
40	C₄×Q₈×C₁₃	C₂³	1	64	yes	complemented
41	C₄×D₄×C₁₃	C₂³	1	192	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	72	yes	—
45	C₂×C₁₃×C₂²⋊C₄	C₂³	1	72	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	36	yes	—
47	C₂²×D₄×Q₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	2	8	no	—
49	C₂⁴×C₁₃	—	2	8	no	—
50	C₂⁴×C₁₃	C₂×Q₈	1	16	yes	complemented
51	C₂⁴×C₁₃	C₂⁴	1	12	yes	—
52	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₂×Q₈×C₁₃	—	2	512	no	—
54	C₂×Q₈×C₁₃	—	2	288	no	—
55	C₂×Q₈×C₁₃	C₂×D₄	1	96	yes	direct product factor
56	C₂×Q₈×C₁₃	C₂⁴	1	96	yes	—
57	C₂×Q₈×C₁₃	C₂⁴	1	16	yes	—
58	C₂×D₄×C₁₃	C₂×Q₈	1	96	yes	direct product factor
59	C₂×D₄×C₁₃	C₂⁴	1	16	yes	—
60	C₂²×C₄×C₁₃	—	2	24	no	—
61	C₂²×C₄×C₁₃	—	2	12	no	—
62	C₂²×C₄×C₁₃	—	2	288	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
64	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	complemented
65	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	36	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	96	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
70	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	—
71	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	—
72	C₁₃×C₄⋊C₄	C₂⁴	1	96	yes	—
73	C₁₃×C₂²⋊C₄	C₂⁴	1	96	yes	—
74	C₄²×C₁₃	C₂⁴	1	48	yes	—
75	C₂⁴×Q₈	C₂×C₁₃	1	2	yes	complemented
76	C₂×D₄×Q₈	C₂×C₁₃	1	48	yes	direct product factor
77	C₂²×C₄⋊Q₈	C₂×C₁₃	1	3	yes	complemented
78	C₂²×C₂²⋊Q₈	C₂×C₁₃	1	6	yes	complemented
79	C₂²×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
80	C₂²×C₄×D₄	C₂×C₁₃	1	3	yes	—
81	C₂³×C₁₃	—	2	64	no	—
82	C₂³×C₁₃	—	2	24	no	—
83	C₂³×C₁₃	—	2	24	no	—
84	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
85	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
86	C₂³×C₁₃	C₂²×Q₈	1	48	yes	complemented
87	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
88	C₂³×C₁₃	C₂⁵	1	8	yes	—
89	C₂³×C₁₃	C₂⁵	1	3	yes	—
90	Q₈×C₁₃	—	2	192	no	—
91	Q₈×C₁₃	C₂²×D₄	1	64	yes	direct product factor
92	D₄×C₁₃	C₂²×Q₈	1	64	yes	direct product factor
93	C₂×C₄×C₁₃	—	2	72	no	—
94	C₂×C₄×C₁₃	—	2	36	no	—
95	C₂×C₄×C₁₃	—	2	192	no	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	72	yes	—
97	C₂×C₄×C₁₃	C₂²×Q₈	1	24	yes	complemented
98	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
99	C₂×C₄×C₁₃	C₂⁵	1	24	yes	—
100	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
101	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
102	C₂³×Q₈	—	2	32	no	—
103	C₂³×Q₈	C₂²×C₁₃	1	6	yes	—
104	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂³×Q₈	C₂²×C₁₃	1	48	yes	complemented
106	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
107	C₂⁴×C₄	C₂²×C₁₃	1	6	yes	—
108	D₄×Q₈	C₂²×C₁₃	1	256	yes	direct product factor
109	C₂×C₄⋊Q₈	C₂²×C₁₃	1	72	yes	complemented
110	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	144	yes	complemented
111	C₂×C₄×Q₈	C₂²×C₁₃	1	24	yes	complemented
112	C₂×C₄×D₄	C₂²×C₁₃	1	72	yes	—
113	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
114	C₂²×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
115	C₂²×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
116	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
117	C₂²×C₄²	C₂²×C₁₃	1	3	yes	—
118	C₂²×C₁₃	—	2	8	no	—
119	C₂²×C₁₃	—	2	8	no	—
120	C₂²×C₁₃	—	2	96	no	—
121	C₂²×C₁₃	D₄×Q₈	1	16	yes	direct product factor
122	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
123	C₂²×C₁₃	C₂³×Q₈	1	6	yes	—
124	C₂²×C₁₃	C₂³×Q₈	1	16	yes	complemented
125	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	6	yes	—
126	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
127	C₄×C₁₃	—	2	24	no	—
128	C₄×C₁₃	—	2	12	no	—
129	C₄×C₁₃	C₂³×D₄	1	24	yes	—
130	C₄×C₁₃	C₂³×Q₈	1	8	yes	complemented
131	C₂⁵	C₂³×C₁₃	1	2	yes	—
132	C₂²×Q₈	—	2	384	no	—
133	C₂²×Q₈	—	2	48	no	—
134	C₂²×Q₈	D₄×C₁₃	1	16	yes	direct product factor
135	C₂²×Q₈	C₂³×C₁₃	1	72	yes	—
136	C₂²×Q₈	C₂³×C₁₃	1	12	yes	—
137	C₂²×Q₈	C₂³×C₁₃	1	128	yes	complemented
138	C₂²×D₄	Q₈×C₁₃	1	16	yes	direct product factor
139	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
140	C₂³×C₄	—	2	48	no	—
141	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
142	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
143	C₂³×C₄	C₂³×C₁₃	1	72	yes	—
144	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
145	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
146	C₄⋊Q₈	C₂³×C₁₃	1	192	yes	complemented
147	C₂²⋊Q₈	C₂³×C₁₃	1	384	yes	complemented
148	C₄×Q₈	C₂³×C₁₃	1	64	yes	complemented
149	C₄×D₄	C₂³×C₁₃	1	192	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₃	1	36	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₃	1	36	yes	—
152	C₂×C₄⋊C₄	C₂³×C₁₃	1	72	yes	—
153	C₂×C₂²⋊C₄	C₂³×C₁₃	1	72	yes	—
154	C₂×C₄²	C₂³×C₁₃	1	36	yes	—
155	C₂×C₁₃	—	2	16	no	—
156	C₂×C₁₃	C₂×D₄×Q₈	1	12	yes	direct product factor
157	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
158	C₂×C₁₃	C₂⁴×Q₈	1	1	yes	characteristic
159	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
160	C₂⁴	—	2	8	no	—
161	C₂⁴	—	2	8	no	—
162	C₂⁴	C₂×Q₈×C₁₃	1	16	yes	complemented
163	C₂⁴	C₂⁴×C₁₃	1	12	yes	—
164	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
165	C₂×Q₈	—	2	512	no	—
166	C₂×Q₈	—	2	288	no	—
167	C₂×Q₈	C₂×D₄×C₁₃	1	96	yes	direct product factor
168	C₂×Q₈	C₂⁴×C₁₃	1	96	yes	—
169	C₂×Q₈	C₂⁴×C₁₃	1	16	yes	—
170	C₂×D₄	C₂×Q₈×C₁₃	1	96	yes	direct product factor
171	C₂×D₄	C₂⁴×C₁₃	1	16	yes	—
172	C₂²×C₄	—	2	24	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	2	288	no	—
175	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
176	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	complemented
177	C₂²×C₄	C₂⁴×C₁₃	1	18	yes	—
178	C₂²×C₄	C₂⁴×C₁₃	1	36	yes	—
179	C₂²×C₄	C₂⁴×C₁₃	1	96	yes	—
180	C₂²×C₄	C₂⁴×C₁₃	1	18	yes	—
181	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
182	C₄⋊C₄	C₂⁴×C₁₃	1	48	yes	—
183	C₄⋊C₄	C₂⁴×C₁₃	1	48	yes	—
184	C₄⋊C₄	C₂⁴×C₁₃	1	96	yes	—
185	C₂²⋊C₄	C₂⁴×C₁₃	1	96	yes	—
186	C₄²	C₂⁴×C₁₃	1	48	yes	—
187	C₁₃	C₂²×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³	—	2	64	no	—
189	C₂³	—	2	24	no	—
190	C₂³	—	2	24	no	—
191	C₂³	C₂²×D₄×C₁₃	1	4	yes	—
192	C₂³	C₂²×Q₈×C₁₃	1	4	yes	—
193	C₂³	C₂²×Q₈×C₁₃	1	48	yes	complemented
194	C₂³	C₁₃×2₋¹⁺⁴	1	4	yes	—
195	C₂³	C₂⁵×C₁₃	1	8	yes	—
196	C₂³	C₂⁵×C₁₃	1	3	yes	—
197	Q₈	—	2	192	no	—
198	Q₈	C₂²×D₄×C₁₃	1	64	yes	direct product factor
199	D₄	C₂²×Q₈×C₁₃	1	64	yes	direct product factor
200	C₂×C₄	—	2	72	no	—
201	C₂×C₄	—	2	36	no	—
202	C₂×C₄	—	2	192	no	—
203	C₂×C₄	C₂²×D₄×C₁₃	1	72	yes	—
204	C₂×C₄	C₂²×Q₈×C₁₃	1	24	yes	complemented
205	C₂×C₄	C₂⁵×C₁₃	1	12	yes	—
206	C₂×C₄	C₂⁵×C₁₃	1	24	yes	—
207	C₂×C₄	C₂⁵×C₁₃	1	12	yes	—
208	C₂×C₄	C₂⁵×C₁₃	1	4	yes	—
209	C₂²	—	2	8	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	96	no	—
212	C₂²	D₄×Q₈×C₁₃	1	16	yes	direct product factor
213	C₂²	C₂³×D₄×C₁₃	1	6	yes	—
214	C₂²	C₂³×Q₈×C₁₃	1	6	yes	—
215	C₂²	C₂³×Q₈×C₁₃	1	16	yes	complemented
216	C₂²	C₂×C₁₃×2₋¹⁺⁴	1	6	yes	—
217	C₂²	C₂⁶×C₁₃	1	1	yes	characteristic
218	C₄	—	2	24	no	—
219	C₄	—	2	12	no	—
220	C₄	C₂³×D₄×C₁₃	1	24	yes	—
221	C₄	C₂³×Q₈×C₁₃	1	8	yes	complemented
222	C₂	—	2	16	no	—
223	C₂	C₂×D₄×Q₈×C₁₃	1	12	yes	direct product factor
224	C₂	C₂⁴×D₄×C₁₃	1	1	yes	characteristic
225	C₂	C₂⁴×Q₈×C₁₃	1	1	yes	characteristic
226	C₂	C₂²×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₃
prime radical	C₂³×D₄×C₁₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₁₃	C₂	semidirect
2. C₂×D₄×Q₈×C₁₃	C₂	direct (N × H)
3. C₂²×C₁₃×C₄⋊Q₈	C₂	semidirect
4. C₂²×C₁₃×C₂²⋊Q₈	C₂	semidirect
5. C₂²×C₄×Q₈×C₁₃	C₂	semidirect
10. C₂³×Q₈×C₁₃	C₂²	semidirect
13. D₄×Q₈×C₁₃	C₂²	direct (N × H)
14. C₂×C₁₃×C₄⋊Q₈	C₂²	semidirect
15. C₂×C₁₃×C₂²⋊Q₈	C₂²	semidirect
16. C₂×C₄×Q₈×C₁₃	C₂²	semidirect
26. C₂²×Q₈×C₁₃	D₄	direct (N × H)
29. C₂²×Q₈×C₁₃	C₂³	semidirect
30. C₂²×D₄×C₁₃	Q₈	direct (N × H)
38. C₁₃×C₄⋊Q₈	C₂³	semidirect
39. C₁₃×C₂²⋊Q₈	C₂³	semidirect
40. C₄×Q₈×C₁₃	C₂³	semidirect
47. C₂²×D₄×Q₈	C₁₃	direct (N × H)
50. C₂⁴×C₁₃	C₂×Q₈	semidirect
55. C₂×Q₈×C₁₃	C₂×D₄	direct (N × H)
58. C₂×D₄×C₁₃	C₂×Q₈	direct (N × H)
64. C₂²×C₄×C₁₃	C₂×Q₈	semidirect
75. C₂⁴×Q₈	C₂×C₁₃	semidirect
76. C₂×D₄×Q₈	C₂×C₁₃	direct (N × H)
77. C₂²×C₄⋊Q₈	C₂×C₁₃	semidirect
78. C₂²×C₂²⋊Q₈	C₂×C₁₃	semidirect
79. C₂²×C₄×Q₈	C₂×C₁₃	semidirect
86. C₂³×C₁₃	C₂²×Q₈	semidirect
91. Q₈×C₁₃	C₂²×D₄	direct (N × H)
92. D₄×C₁₃	C₂²×Q₈	direct (N × H)
97. C₂×C₄×C₁₃	C₂²×Q₈	semidirect
105. C₂³×Q₈	C₂²×C₁₃	semidirect
108. D₄×Q₈	C₂²×C₁₃	direct (N × H)
109. C₂×C₄⋊Q₈	C₂²×C₁₃	semidirect
110. C₂×C₂²⋊Q₈	C₂²×C₁₃	semidirect
111. C₂×C₄×Q₈	C₂²×C₁₃	semidirect
121. C₂²×C₁₃	D₄×Q₈	direct (N × H)
124. C₂²×C₁₃	C₂³×Q₈	semidirect
130. C₄×C₁₃	C₂³×Q₈	semidirect
134. C₂²×Q₈	D₄×C₁₃	direct (N × H)
137. C₂²×Q₈	C₂³×C₁₃	semidirect
138. C₂²×D₄	Q₈×C₁₃	direct (N × H)
146. C₄⋊Q₈	C₂³×C₁₃	semidirect
147. C₂²⋊Q₈	C₂³×C₁₃	semidirect
148. C₄×Q₈	C₂³×C₁₃	semidirect
156. C₂×C₁₃	C₂×D₄×Q₈	direct (N × H)
162. C₂⁴	C₂×Q₈×C₁₃	semidirect
167. C₂×Q₈	C₂×D₄×C₁₃	direct (N × H)
170. C₂×D₄	C₂×Q₈×C₁₃	direct (N × H)
176. C₂²×C₄	C₂×Q₈×C₁₃	semidirect
187. C₁₃	C₂²×D₄×Q₈	direct (N × H)
193. C₂³	C₂²×Q₈×C₁₃	semidirect
198. Q₈	C₂²×D₄×C₁₃	direct (N × H)
199. D₄	C₂²×Q₈×C₁₃	direct (N × H)
204. C₂×C₄	C₂²×Q₈×C₁₃	semidirect
212. C₂²	D₄×Q₈×C₁₃	direct (N × H)
215. C₂²	C₂³×Q₈×C₁₃	semidirect
221. C₄	C₂³×Q₈×C₁₃	semidirect
223. C₂	C₂×D₄×Q₈×C₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₁₃	2
C₂×D₄×Q₈×C₁₃	2
C₂²×C₁₃×C₄⋊Q₈	2
C₂²×C₁₃×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₁₃	2
C₂²×C₄×D₄×C₁₃	2
C₂²×D₄×Q₈	13

Subgroups — C22×D4×Q8×C13 #15749035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×Q₈×C₁₃ #15749035