Subgroups — C₂×Q₈×D₁₃⋊C₈ #16808406

Order 3328.

2478 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₃⋊C₈	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11719074	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×#1057	C₂	1	2	yes	maximal, complemented
4	C₂×D₁₃⋊(C₄⋊C₈)	C₂	1	3	yes	maximal
5	C₂×C₄×D₁₃⋊C₈	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×D₁₃⋊C₈	C₂²	1	3	yes	—
9	C₂²×D₁₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×Q₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₄×Q₈×D₁₃	C₄	1	8	yes	—
13	C₄×Q₈×D₁₃	C₂²	1	8	yes	—
14	C₂×C₄⋊C₄×D₁₃	C₄	1	3	yes	—
15	C₂×#88141	C₂²	1	3	yes	—
16	C₂×C₄²×D₁₃	C₂²	1	3	yes	—
17	C₂×C₄×Dic₂₆	C₄	1	3	yes	—
18	Q₈×#1057	C₂²	1	16	yes	complemented
19	C₂×#88112	C₂²	1	6	yes	—
20	C₂×C₄⋊#1057	C₂²	1	6	yes	—
21	C₂×C₄×#1057	C₂²	1	6	yes	—
22	C₂×C₄×#1057	C₂²	1	6	yes	—
23	#11719074	C₂²	1	48	yes	complemented
24	D₁₃⋊(C₄⋊C₈)	C₂²	1	24	yes	—
25	C₄×D₁₃⋊C₈	C₂²	1	24	yes	—
26	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×Q₈×D₁₃	C₈	1	16	yes	complemented
28	C₂×Q₈×D₁₃	C₂×C₄	1	8	yes	—
29	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
30	C₂²×C₄×D₁₃	C₂×C₄	1	3	yes	—
31	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
32	C₂²×Dic₂₆	C₂×C₄	1	3	yes	—
33	C₂²×#1057	C₂³	1	6	yes	—
34	C₂²×#1057	C₂³	1	2	yes	—
35	C₂×D₁₃⋊C₈	—	2	12	no	—
36	C₂×D₁₃⋊C₈	Q₈	1	8	yes	direct product factor
37	C₂×D₁₃⋊C₈	C₂³	1	12	yes	—
38	C₂×D₁₃⋊C₈	C₂³	1	4	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
42	Q₈×Dic₁₃	C₂×C₄	1	4	yes	—
43	Q₈×Dic₁₃	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₁₃	C₂×C₄	1	3	yes	—
45	C₂×#6101	C₂×C₄	1	6	yes	—
46	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
47	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
48	C₂×C₄×Dic₁₃	C₂³	1	3	yes	—
49	C₄⋊C₄×D₁₃	C₂×C₄	1	24	yes	—
50	#88141	C₂×C₄	1	12	yes	—
51	#88141	C₂³	1	12	yes	—
52	C₄²×D₁₃	C₂×C₄	1	12	yes	—
53	C₄²×D₁₃	C₂³	1	12	yes	—
54	C₄×Dic₂₆	C₂×C₄	1	24	yes	—
55	#88112	C₂³	1	24	yes	—
56	C₄⋊#1057	C₂³	1	24	yes	—
57	C₄×#1057	C₂³	1	24	yes	—
58	C₄×#1057	C₂³	1	24	yes	—
59	C₂²×C₈×Q₈	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂×C₈	1	8	yes	complemented
62	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
68	C₂²×Dic₁₃	C₂⁴	1	1	yes	characteristic
69	Q₈×D₁₃	C₂×C₈	1	64	yes	complemented
70	C₂×C₄×D₁₃	—	2	12	no	—
71	C₂×C₄×D₁₃	C₂×C₈	1	24	yes	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
73	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
75	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
76	C₂×C₄×D₁₃	#41	1	4	yes	—
77	C₂×C₄×D₁₃	C₂⁴	1	2	yes	—
78	C₂×Dic₂₆	C₂×C₈	1	24	yes	complemented
79	C₂×Dic₂₆	C₂²×C₄	1	12	yes	—
80	C₂×#1057	—	2	12	no	—
81	C₂×#1057	C₂×Q₈	1	8	yes	complemented
82	C₂×#1057	C₂⁴	1	12	yes	—
83	C₂×#1057	C₂⁴	1	4	yes	—
84	D₁₃⋊C₈	—	2	24	no	—
85	D₁₃⋊C₈	C₂×Q₈	1	16	yes	direct product factor
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	12	yes	—
87	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
88	C₄⋊Dic₁₃	C₂²×C₄	1	12	yes	—
89	#6101	C₂²×C₄	1	24	yes	—
90	C₄×Dic₁₃	C₂²×C₄	1	12	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
92	C₄×Dic₁₃	C₂²×C₄	1	6	yes	—
93	C₄×Dic₁₃	C₂⁴	1	6	yes	—
94	C₄×Dic₁₃	C₂⁴	1	6	yes	—
95	C₂²×C₄×Q₈	—	13	1	no	—
96	C₂×C₈×Q₈	—	13	2	no	—
97	C₂×C₈×Q₈	—	13	6	no	—
98	C₂×C₈×Q₈	—	13	16	no	—
99	C₂²×C₄⋊C₈	—	13	3	no	—
100	C₂²×C₄×C₈	—	13	3	no	—
101	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂²×D₁₃	C₄×Q₈	1	4	yes	—
103	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
104	C₂²×D₁₃	#131	1	4	yes	—
105	C₂²×D₁₃	C₂³×C₄	1	2	yes	—
106	Q₈×C₁₃	C₂²×C₈	1	16	yes	complemented
107	C₂×C₄×C₁₃	—	2	6	no	—
108	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₁₃	C₂²×C₈	1	12	yes	—
110	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	6	yes	—
113	C₂×Dic₁₃	—	2	6	no	—
114	C₂×Dic₁₃	C₂²×C₈	1	12	yes	—
115	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
116	C₂×Dic₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂³×C₄	1	6	yes	—
118	C₂×Dic₁₃	C₂²×Q₈	1	2	yes	—
119	C₂×Dic₁₃	C₂×#41	1	2	yes	—
120	C₂×Dic₁₃	C₂⁵	1	1	yes	characteristic
121	C₄×D₁₃	—	2	24	no	—
122	C₄×D₁₃	C₄×Q₈	1	8	yes	—
123	C₄×D₁₃	C₂²×C₈	1	48	yes	—
124	C₄×D₁₃	C₂²×Q₈	1	4	yes	—
125	C₄×D₁₃	C₂×#41	1	4	yes	—
126	Dic₂₆	C₂²×C₈	1	48	yes	complemented
127	#1057	—	2	12	no	—
128	#1057	C₂²×Q₈	1	8	yes	complemented
129	C₂³×Q₈	—	13	1	no	—
130	C₂³×C₈	—	13	1	no	—
131	C₂³×C₈	—	13	3	no	—
132	C₂×C₄×Q₈	—	13	1	no	—
133	C₂×C₄×Q₈	—	13	8	no	—
134	C₂×C₄×Q₈	—	13	8	no	—
135	C₂×C₄×Q₈	—	13	3	no	—
136	C₂×C₄×Q₈	—	13	3	no	—
137	C₂×C₄×Q₈	#261	1	1	yes	characteristic
138	C₂²×C₄⋊C₄	—	13	3	no	—
139	C₂²×C₄²	—	13	3	no	—
140	C₈×Q₈	—	13	48	no	—
141	C₈×Q₈	—	13	16	no	—
142	C₂×C₄⋊C₈	—	13	24	no	—
143	C₂×C₄⋊C₈	—	13	6	no	—
144	C₂×C₄⋊C₈	—	13	6	no	—
145	C₂×C₄×C₈	—	13	24	no	—
146	C₂×C₄×C₈	—	13	6	no	—
147	C₂×C₄×C₈	—	13	6	no	—
148	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
149	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
150	C₂²×C₁₃	C₂×#131	1	2	yes	—
151	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
152	C₂×D₁₃	C₈×Q₈	1	16	yes	complemented
153	C₂×D₁₃	C₂×C₄×Q₈	1	4	yes	—
154	C₂×D₁₃	C₂³×C₈	1	4	yes	—
155	C₂×D₁₃	C₂×#131	1	4	yes	—
156	C₄×C₁₃	—	2	6	no	—
157	C₄×C₁₃	C₂×C₄×Q₈	1	4	yes	—
158	C₄×C₁₃	C₂³×C₈	1	12	yes	—
159	Dic₁₃	—	2	6	no	—
160	Dic₁₃	C₂×C₄×Q₈	1	2	yes	—
161	Dic₁₃	C₂³×C₈	1	12	yes	—
162	Dic₁₃	C₂³×Q₈	1	1	yes	characteristic
163	Dic₁₃	C₂²×#41	1	1	yes	characteristic
164	C₂²×Q₈	—	13	16	no	—
165	C₂²×Q₈	—	13	8	no	—
166	C₂²×Q₈	—	13	3	no	—
167	C₂²×Q₈	C₂×#261	1	1	yes	characteristic
168	C₂³×C₄	—	13	3	no	—
169	C₂³×C₄	—	13	1	no	—
170	C₂³×C₄	—	13	3	no	—
171	C₂²×C₈	—	13	4	no	—
172	C₂²×C₈	—	13	2	no	—
173	C₂²×C₈	—	13	8	no	—
174	C₂²×C₈	—	13	12	no	—
175	C₂²×C₈	—	13	6	no	—
176	C₂²×C₈	—	26	12	no	—
177	C₄×Q₈	—	13	4	no	—
178	C₄×Q₈	—	13	4	no	—
179	C₄×Q₈	—	13	12	no	—
180	C₄×Q₈	—	13	12	no	—
181	C₄×Q₈	—	13	24	no	—
182	C₄×Q₈	C₂×#261	1	8	yes	—
183	C₂×C₄⋊C₄	—	13	6	no	—
184	C₂×C₄⋊C₄	—	13	3	no	—
185	C₂×C₄⋊C₄	—	13	24	no	—
186	C₂×C₄⋊C₄	C₂×#261	1	3	yes	—
187	C₂×C₄²	—	13	3	no	—
188	C₂×C₄²	—	13	3	no	—
189	C₂×C₄²	—	13	3	no	—
190	C₂×C₄²	—	13	12	no	—
191	C₂×C₄²	—	13	12	no	—
192	C₂×C₄²	C₂×#261	1	3	yes	—
193	C₄⋊C₈	—	13	24	no	—
194	C₄⋊C₈	—	13	24	no	—
195	C₄×C₈	—	13	24	no	—
196	C₄×C₈	—	13	24	no	—
197	C₂×C₁₃	C₂×C₈×Q₈	1	4	yes	complemented
198	C₂×C₁₃	C₂²×C₄×Q₈	1	1	yes	characteristic
199	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
200	C₂×C₁₃	C₂²×#131	1	1	yes	characteristic
201	D₁₃	C₂×C₈×Q₈	1	8	yes	complemented
202	C₂⁴	—	13	1	no	—
203	C₂×Q₈	—	13	64	no	—
204	C₂×Q₈	—	13	24	no	—
205	C₂×Q₈	—	13	12	no	—
206	C₂×Q₈	D₁₃⋊C₈	1	8	yes	direct product factor
207	C₂×Q₈	C₂²×#261	1	4	yes	—
208	C₂²×C₄	—	13	1	no	—
209	C₂²×C₄	—	13	3	no	—
210	C₂²×C₄	—	13	24	no	—
211	C₂²×C₄	—	13	4	no	—
212	C₂²×C₄	—	13	4	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	13	12	no	—
216	C₂²×C₄	—	13	3	no	—
217	C₂²×C₄	—	13	12	no	—
218	C₂²×C₄	—	26	12	no	—
219	C₂²×C₄	C₂²×#261	1	3	yes	—
220	C₂²×C₄	C₂²×#261	1	3	yes	—
221	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	—	13	16	no	—
224	C₂×C₈	—	13	8	no	—
225	C₂×C₈	—	13	12	no	—
226	C₂×C₈	—	26	24	no	—
227	C₂×C₈	—	26	12	no	—
228	C₄⋊C₄	—	13	24	no	—
229	C₄⋊C₄	—	13	12	no	—
230	C₄⋊C₄	C₂²×#261	1	12	yes	—
231	C₄²	—	13	6	no	—
232	C₄²	—	13	6	no	—
233	C₄²	—	13	6	no	—
234	C₄²	—	13	6	no	—
235	C₄²	—	13	12	no	—
236	C₄²	C₂²×#261	1	12	yes	—
237	C₁₃	C₂²×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	13	4	no	—
239	C₂³	—	13	4	no	—
240	C₂³	—	13	4	no	—
241	C₂³	—	13	2	no	—
242	C₂³	C₂³×#261	1	1	yes	characteristic
243	Q₈	—	13	48	no	—
244	Q₈	C₂×D₁₃⋊C₈	1	16	yes	direct product factor
245	C₂×C₄	—	2	6	no	—
246	C₂×C₄	—	13	12	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	6	no	—
252	C₂×C₄	—	13	48	no	—
253	C₂×C₄	—	13	4	no	—
254	C₂×C₄	—	13	4	no	—
255	C₂×C₄	—	13	8	no	—
256	C₂×C₄	—	13	6	no	—
257	C₂×C₄	—	26	24	no	—
258	C₂×C₄	—	26	6	no	—
259	C₂×C₄	C₂×D₁₃⋊C₈	1	12	yes	—
260	C₂×C₄	Q₈×#261	1	4	yes	—
261	C₂×C₄	C₂³×#261	1	6	yes	—
262	C₂×C₄	C₂³×#261	1	6	yes	—
263	C₂×C₄	C₂³×#261	1	2	yes	—
264	C₈	—	13	8	no	—
265	C₈	—	26	12	no	—
266	C₂²	—	13	4	no	—
267	C₂²	—	13	4	no	—
268	C₂²	—	13	4	no	—
269	C₂²	—	13	16	no	—
270	C₂²	C₂²×D₁₃⋊C₈	1	2	yes	—
271	C₂²	C₂×Q₈×#261	1	2	yes	—
272	C₂²	D₁₃⋊#131	1	2	yes	—
273	C₂²	C₂⁴×#261	1	1	yes	characteristic
274	C₄	—	2	6	no	—
275	C₄	—	13	2	no	—
276	C₄	—	13	1	no	—
277	C₄	—	13	1	no	—
278	C₄	—	13	12	no	—
279	C₄	—	26	6	no	—
280	C₄	C₂²×D₁₃⋊C₈	1	12	yes	—
281	C₄	C₂×Q₈×#261	1	4	yes	—
282	C₂	—	13	8	no	—
283	C₂	Q₈×D₁₃⋊C₈	1	4	yes	direct product factor
284	C₂	C₂²×Q₈×#261	1	1	yes	characteristic
285	C₂	C₂³×D₁₃⋊C₈	1	1	yes	characteristic
286	C₂	C₂×D₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#261
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂²	C₂⁴×#261
Fitting	C₂×C₄×Q₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂³×D₁₃	C₂²×C₄
cosocle	C₂×Dic₁₃	C₂⁵
hypercenter	C₂×C₄×Q₈	#261
hypocenter	C₁₃	C₂²×C₈×Q₈
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₃⋊C₈	C₂	direct (N × H)
2. C₂×#11719074	C₂	semidirect
3. C₂×Q₈×#1057	C₂	semidirect
18. Q₈×#1057	C₂²	semidirect
23. #11719074	C₂²	semidirect
27. C₂×Q₈×D₁₃	C₈	semidirect
36. C₂×D₁₃⋊C₈	Q₈	direct (N × H)
61. C₂×Q₈×C₁₃	C₂×C₈	semidirect
69. Q₈×D₁₃	C₂×C₈	semidirect
78. C₂×Dic₂₆	C₂×C₈	semidirect
81. C₂×#1057	C₂×Q₈	semidirect
85. D₁₃⋊C₈	C₂×Q₈	direct (N × H)
106. Q₈×C₁₃	C₂²×C₈	semidirect
126. Dic₂₆	C₂²×C₈	semidirect
128. #1057	C₂²×Q₈	semidirect
152. C₂×D₁₃	C₈×Q₈	semidirect
197. C₂×C₁₃	C₂×C₈×Q₈	semidirect
201. D₁₃	C₂×C₈×Q₈	semidirect
206. C₂×Q₈	D₁₃⋊C₈	direct (N × H)
237. C₁₃	C₂²×C₈×Q₈	semidirect
244. Q₈	C₂×D₁₃⋊C₈	direct (N × H)
283. C₂	Q₈×D₁₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×D₁₃⋊C₈	2
C₂×#11719074	2
C₂×Q₈×#1057	2
C₂×D₁₃⋊(C₄⋊C₈)	2
C₂×C₄×D₁₃⋊C₈	2
C₂×C₄×Q₈×D₁₃	2
C₂²×C₈×Q₈	13

Subgroups — C2×Q8×D13⋊C8 #16808406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×D₁₃⋊C₈ #16808406