Subgroups — C₂×Q₈×C₄⋊Dic₁₃ #16022222

Order 3328.

3246 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₁₃	C₂	1	2	yes	maximal
2	Q₈×C₄⋊Dic₁₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718646	C₂	1	6	yes	maximal
4	C₂×#11718596	C₂	1	3	yes	maximal
5	C₂×C₄×C₄⋊Dic₁₃	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	16	yes	complemented
9	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
12	C₂×Q₈×Dic₁₃	—	2	16	no	—
13	C₂×Q₈×Dic₁₃	C₂²	1	16	yes	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	3	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
16	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
17	C₂²×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×Dic₁₃	C₂²	1	6	yes	—
19	C₂×#88071	C₂²	1	6	yes	—
20	#11718646	C₂²	1	48	yes	—
21	#11718596	C₂²	1	24	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	24	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₃	D₄	1	8	yes	—
25	C₂²×Q₈×C₁₃	Q₈	1	8	yes	—
26	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
30	C₂³×Dic₁₃	C₂³	1	6	yes	—
31	C₂³×Dic₁₃	C₂³	1	2	yes	—
32	C₄×Q₈×C₁₃	C₂×C₄	1	64	yes	complemented
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	24	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
37	Q₈×Dic₁₃	—	2	64	no	—
38	C₂×C₄⋊Dic₁₃	—	2	24	no	—
39	C₂×C₄⋊Dic₁₃	—	2	12	no	—
40	C₂×C₄⋊Dic₁₃	—	4	12	no	—
41	C₂×C₄⋊Dic₁₃	Q₈	1	8	yes	direct product factor
42	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	12	yes	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
46	C₂×C₄×Dic₁₃	—	2	24	no	—
47	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
48	#88071	C₂³	1	24	yes	—
49	C₂×Q₈×C₄⋊C₄	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×Q₈×C₁₃	C₄⋊C₄	1	64	yes	complemented
52	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
53	C₂×Q₈×C₁₃	C₂×D₄	1	16	yes	—
54	C₂×Q₈×C₁₃	C₂×Q₈	1	16	yes	—
55	C₂²×C₄×C₁₃	—	2	6	no	—
56	C₂²×C₄×C₁₃	—	2	6	no	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₁₃	#41	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
67	C₂²×Dic₁₃	—	2	4	no	—
68	C₂²×Dic₁₃	—	2	4	no	—
69	C₂²×Dic₁₃	—	2	12	no	—
70	C₂²×Dic₁₃	—	2	12	no	—
71	C₂²×Dic₁₃	—	2	12	no	—
72	C₂²×Dic₁₃	C₂×Q₈	1	8	yes	—
73	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
74	C₂²×Dic₁₃	C₂⁴	1	4	yes	—
75	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
76	C₄²×C₁₃	C₂²×C₄	1	48	yes	—
77	C₄⋊Dic₁₃	—	2	48	no	—
78	C₄⋊Dic₁₃	—	2	24	no	—
79	C₄⋊Dic₁₃	—	4	24	no	—
80	C₄⋊Dic₁₃	C₂×Q₈	1	16	yes	direct product factor
81	C₄×Dic₁₃	—	2	48	no	—
82	C₂²×C₄×Q₈	—	13	2	no	—
83	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
84	Q₈×C₄⋊C₄	—	13	16	no	—
85	C₂×#389	—	13	6	no	—
86	C₂×#384	—	13	3	no	—
87	C₂×C₄×C₄⋊C₄	—	13	3	no	—
88	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
89	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
90	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
91	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
92	C₂³×C₁₃	C₂×#41	1	2	yes	—
93	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
94	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
95	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
96	Q₈×C₁₃	C₂×C₄⋊C₄	1	64	yes	complemented
97	C₂×C₄×C₁₃	—	2	24	no	—
98	C₂×C₄×C₁₃	—	2	6	no	—
99	C₂×C₄×C₁₃	—	2	6	no	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	48	yes	—
101	C₂×C₄×C₁₃	C₄×Q₈	1	16	yes	complemented
102	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
106	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
107	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
108	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
109	C₂×Dic₁₃	—	2	16	no	—
110	C₂×Dic₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	2	4	no	—
112	C₂×Dic₁₃	—	2	12	no	—
113	C₂×Dic₁₃	—	2	12	no	—
114	C₂×Dic₁₃	—	2	12	no	—
115	C₂×Dic₁₃	—	4	24	no	—
116	C₂×Dic₁₃	C₂²×Q₈	1	8	yes	—
117	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	13	16	no	—
119	C₂×C₄×Q₈	—	26	16	no	—
120	C₂×C₄×Q₈	Dic₁₃	1	16	yes	complemented
121	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
122	C₂²×C₄⋊C₄	—	13	1	no	—
123	C₂²×C₄⋊C₄	—	13	6	no	—
124	C₂²×C₄⋊C₄	—	13	3	no	—
125	C₂²×C₄⋊C₄	—	13	6	no	—
126	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
127	C₂²×C₄²	—	13	6	no	—
128	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
129	#389	—	13	48	no	—
130	#384	—	13	24	no	—
131	C₄×C₄⋊C₄	—	13	24	no	—
132	C₂×#95	—	13	6	no	—
133	C₂²×C₁₃	C₂²×C₄⋊C₄	1	4	yes	—
134	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
135	C₂²×C₁₃	#520	1	4	yes	—
136	C₂²×C₁₃	D₄×Q₈	1	4	yes	—
137	C₂²×C₁₃	#550	1	4	yes	—
138	C₂²×C₁₃	#557	1	4	yes	—
139	C₂²×C₁₃	Q₈²	1	4	yes	—
140	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
141	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
142	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
143	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₁₃	—	2	12	no	—
148	C₄×C₁₃	C₂²×C₄⋊C₄	1	24	yes	—
149	C₄×C₁₃	C₂×C₄×Q₈	1	8	yes	complemented
150	Dic₁₃	—	2	8	no	—
151	Dic₁₃	—	4	12	no	—
152	C₂²×Q₈	Dic₂₆	1	8	yes	—
153	C₂²×Q₈	D₅₂	1	8	yes	—
154	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
155	C₂²×Q₈	C₂²×D₁₃	1	4	yes	—
156	C₂³×C₄	—	13	2	no	—
157	C₂³×C₄	—	13	6	no	—
158	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
159	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
160	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
161	C₄×Q₈	—	26	64	no	—
162	C₄×Q₈	C₂×Dic₁₃	1	64	yes	complemented
163	C₂×C₄⋊C₄	—	13	8	no	—
164	C₂×C₄⋊C₄	—	13	4	no	—
165	C₂×C₄⋊C₄	—	13	24	no	—
166	C₂×C₄⋊C₄	—	13	12	no	—
167	C₂×C₄⋊C₄	—	13	24	no	—
168	C₂×C₄⋊C₄	—	26	24	no	—
169	C₂×C₄⋊C₄	—	26	12	no	—
170	C₂×C₄⋊C₄	—	52	12	no	—
171	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
172	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
173	C₂×C₄²	—	13	24	no	—
174	C₂×C₄²	—	26	24	no	—
175	C₂×C₄²	C₂×Dic₁₃	1	24	yes	—
176	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
177	#95	—	13	24	no	—
178	C₂×C₁₃	Q₈×C₄⋊C₄	1	8	yes	complemented
179	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#520	1	1	yes	characteristic
182	C₂×C₁₃	C₂×D₄×Q₈	1	1	yes	characteristic
183	C₂×C₁₃	C₂×#550	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#557	1	1	yes	characteristic
185	C₂×C₁₃	C₂×Q₈²	1	1	yes	characteristic
186	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
187	C₂×Q₈	C₄⋊Dic₁₃	1	64	yes	direct product factor
188	C₂×Q₈	C₂×Dic₂₆	1	16	yes	—
189	C₂×Q₈	C₂×D₅₂	1	16	yes	—
190	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
191	C₂²×C₄	—	2	6	no	—
192	C₂²×C₄	—	2	6	no	—
193	C₂²×C₄	—	13	8	no	—
194	C₂²×C₄	—	13	4	no	—
195	C₂²×C₄	—	13	12	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	—	26	12	no	—
199	C₂²×C₄	—	26	12	no	—
200	C₂²×C₄	—	26	12	no	—
201	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
202	C₂²×C₄	C₂×D₅₂	1	12	yes	—
203	C₂²×C₄	Q₈×D₁₃	1	4	yes	—
204	C₂²×C₄	#6131	1	4	yes	—
205	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
207	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
208	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
209	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
210	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
211	C₄⋊C₄	—	13	16	no	—
212	C₄⋊C₄	—	26	48	no	—
213	C₄⋊C₄	—	26	24	no	—
214	C₄⋊C₄	—	52	24	no	—
215	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
216	C₄²	—	26	48	no	—
217	C₄²	C₂²×Dic₁₃	1	48	yes	—
218	C₁₃	C₂×Q₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
219	C₂³	C₂²×Dic₂₆	1	2	yes	—
220	C₂³	C₂²×D₅₂	1	2	yes	—
221	C₂³	C₂×Q₈×D₁₃	1	2	yes	—
222	C₂³	C₂×#6131	1	2	yes	—
223	C₂³	#88256	1	2	yes	—
224	C₂³	#88257	1	2	yes	—
225	C₂³	C₂³×Dic₁₃	1	2	yes	—
226	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
227	Q₈	C₂×C₄⋊Dic₁₃	1	64	yes	direct product factor
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	2	6	no	—
231	C₂×C₄	—	13	8	no	—
232	C₂×C₄	—	26	16	no	—
233	C₂×C₄	—	26	4	no	—
234	C₂×C₄	—	26	4	no	—
235	C₂×C₄	—	26	12	no	—
236	C₂×C₄	—	26	12	no	—
237	C₂×C₄	—	26	12	no	—
238	C₂×C₄	—	52	24	no	—
239	C₂×C₄	C₂×C₄⋊Dic₁₃	1	48	yes	—
240	C₂×C₄	Q₈×Dic₁₃	1	16	yes	complemented
241	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
242	C₂×C₄	C₂²×D₅₂	1	12	yes	—
243	C₂×C₄	C₂×Q₈×D₁₃	1	4	yes	—
244	C₂×C₄	C₂×#6131	1	4	yes	—
245	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
247	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
248	C₂²	Q₈×Dic₂₆	1	4	yes	—
249	C₂²	#11719275	1	4	yes	—
250	C₂²	Q₈×D₅₂	1	4	yes	—
251	C₂²	#11719282	1	4	yes	—
252	C₂²	C₂²×C₄⋊Dic₁₃	1	4	yes	—
253	C₂²	C₂×Q₈×Dic₁₃	1	4	yes	—
254	C₂²	#11719532	1	4	yes	—
255	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
256	C₂²	C₂³×D₅₂	1	1	yes	characteristic
257	C₂²	C₂²×Q₈×D₁₃	1	1	yes	characteristic
258	C₂²	C₂²×#6131	1	1	yes	characteristic
259	C₂²	C₂×#88256	1	1	yes	characteristic
260	C₂²	C₂×#88257	1	1	yes	characteristic
261	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
262	C₄	—	2	12	no	—
263	C₄	—	26	8	no	—
264	C₄	—	52	12	no	—
265	C₄	C₂²×C₄⋊Dic₁₃	1	24	yes	—
266	C₄	C₂×Q₈×Dic₁₃	1	8	yes	complemented
267	C₂	Q₈×C₄⋊Dic₁₃	1	8	yes	direct product factor
268	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
269	C₂	C₂²×Q₈×Dic₁₃	1	1	yes	characteristic
270	C₂	C₂×#11719532	1	1	yes	characteristic
271	C₂	C₂×Q₈×D₅₂	1	1	yes	characteristic
272	C₂	C₂×#11719282	1	1	yes	characteristic
273	C₂	C₂×#11719275	1	1	yes	characteristic
274	C₂	C₂×Q₈×Dic₂₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×Q₈×C₄⋊C₄
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊Dic₁₃	C₂	direct (N × H)
8. C₂×C₄×Q₈×C₁₃	C₄	semidirect
32. C₄×Q₈×C₁₃	C₂×C₄	semidirect
41. C₂×C₄⋊Dic₁₃	Q₈	direct (N × H)
51. C₂×Q₈×C₁₃	C₄⋊C₄	semidirect
80. C₄⋊Dic₁₃	C₂×Q₈	direct (N × H)
96. Q₈×C₁₃	C₂×C₄⋊C₄	semidirect
101. C₂×C₄×C₁₃	C₄×Q₈	semidirect
120. C₂×C₄×Q₈	Dic₁₃	semidirect
149. C₄×C₁₃	C₂×C₄×Q₈	semidirect
162. C₄×Q₈	C₂×Dic₁₃	semidirect
178. C₂×C₁₃	Q₈×C₄⋊C₄	semidirect
187. C₂×Q₈	C₄⋊Dic₁₃	direct (N × H)
218. C₁₃	C₂×Q₈×C₄⋊C₄	semidirect
227. Q₈	C₂×C₄⋊Dic₁₃	direct (N × H)
240. C₂×C₄	Q₈×Dic₁₃	semidirect
266. C₄	C₂×Q₈×Dic₁₃	semidirect
267. C₂	Q₈×C₄⋊Dic₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈×Dic₁₃	2
Q₈×C₄⋊Dic₁₃	2
C₂×#11718646	2
C₂×#11718596	2
C₂×C₄×C₄⋊Dic₁₃	2
C₂²×C₄×Q₈×C₁₃	2
C₂×Q₈×C₄⋊C₄	13

Subgroups — C2×Q8×C4⋊Dic13 #16022222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₄⋊Dic₁₃ #16022222