Subgroups — D₁₃⋊(Q₈⋊C₁₆) #16787351

Order 3328.

382 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	D₁₃⋊(C₄⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#13129803	C₂	1	2	yes	maximal, complemented
4	Q₈⋊#6092	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂×D₁₃⋊C₁₆	—	2	1	no	—
8	C₂×D₁₃⋊C₁₆	—	2	1	no	—
9	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
10	Q₈×#1055	C₂²	1	1	yes	characteristic
11	D₁₃×C₄⋊C₈	—	2	1	no	—
12	D₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	#11718491	—	2	1	no	—
14	#11718491	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₁₃	—	2	1	no	—
16	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	C₈×Dic₂₆	—	2	1	no	—
18	C₈×Dic₂₆	C₄	1	1	yes	characteristic
19	#11718260	—	2	1	no	—
20	#11718257	—	2	1	no	—
21	C₄⋊#6092	C₂²	1	2	yes	—
22	C₄×#6092	C₂²	1	2	yes	—
23	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
26	C₂×C₈×D₁₃	—	2	1	no	—
27	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
29	C₄⋊C₄×D₁₃	—	2	1	no	—
30	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
31	#88141	—	2	1	no	—
32	#88141	C₈	1	1	yes	characteristic
33	C₄²×D₁₃	—	2	1	no	—
34	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₂₆	—	2	1	no	—
36	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#6092	—	2	2	no	—
38	C₂×#6092	—	2	2	no	—
39	D₁₃⋊C₁₆	—	2	4	no	—
40	D₁₃⋊C₁₆	—	4	2	no	—
41	C₁₃×C₄⋊C₈	—	2	1	no	—
42	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	#88054	—	2	1	no	—
46	#88054	—	2	1	no	—
47	#88054	—	2	1	no	—
48	C₈×Dic₁₃	—	2	1	no	—
49	C₈×Dic₁₃	—	2	1	no	—
50	C₈×Dic₁₃	D₄	1	1	yes	characteristic
51	C₈×Dic₁₃	D₄	1	1	yes	characteristic
52	C₄⋊#1055	—	2	1	no	—
53	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
54	C₄×#1055	—	2	1	no	—
55	C₄×#1055	C₂³	1	1	yes	characteristic
56	C₂×Q₈⋊C₁₆	—	13	1	no	maximal, Sylow
57	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
58	Q₈×D₁₃	—	2	2	no	—
59	Q₈×D₁₃	C₁₆	1	4	yes	complemented
60	C₂×C₄×D₁₃	—	2	1	no	—
61	C₂×C₄×D₁₃	—	2	1	no	—
62	C₂×C₄×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₁₃	M₁₆	1	1	yes	characteristic
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	1	no	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₃	—	2	1	no	—
75	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
76	#6101	—	2	1	no	—
77	#6101	—	2	1	no	—
78	#6101	—	2	1	no	—
79	C₄×Dic₁₃	—	2	1	no	—
80	C₄×Dic₁₃	—	2	1	no	—
81	C₄×Dic₁₃	—	2	1	no	—
82	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₁₃	M₁₆	1	1	yes	characteristic
85	C₂×#1055	—	2	1	no	—
86	C₂×#1055	C₂×D₄	1	1	yes	characteristic
87	C₂×#1055	C₂×D₄	1	1	yes	characteristic
88	C₈×D₁₃	—	2	2	no	—
89	C₈×D₁₃	—	4	2	no	—
90	C₈×D₁₃	SD₁₆	1	2	yes	—
91	C₈×D₁₃	Q₁₆	1	2	yes	—
92	#6092	—	2	4	no	—
93	#6092	—	4	2	no	—
94	C₂×C₈×Q₈	—	13	1	no	—
95	C₂×C₄⋊C₁₆	—	13	1	no	—
96	C₂×C₄×C₁₆	—	13	1	no	—
97	Q₈⋊C₁₆	—	13	2	no	—
98	Q₈⋊C₁₆	—	13	2	no	—
99	C₂²×D₁₃	C₂²⋊C₈	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	1	no	—
101	Q₈×C₁₃	C₂×C₁₆	1	2	yes	complemented
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₁₃	—	2	1	no	—
108	C₂×Dic₁₃	—	2	1	no	—
109	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₁₃	C₂²⋊C₈	1	1	yes	characteristic
112	C₄×D₁₃	—	2	4	no	—
113	C₄×D₁₃	—	2	2	no	—
114	C₄×D₁₃	—	4	2	no	—
115	C₄×D₁₃	#103	1	2	yes	—
116	C₄×D₁₃	#104	1	2	yes	—
117	C₄×D₁₃	C₂×C₁₆	1	2	yes	—
118	C₄×D₁₃	M₃₂	1	2	yes	—
119	Dic₂₆	—	2	1	no	—
120	Dic₂₆	—	2	4	no	—
121	Dic₂₆	C₂×C₁₆	1	2	yes	complemented
122	C₈×C₁₃	—	2	1	no	—
123	C₈×C₁₃	—	4	1	no	—
124	C₈×C₁₃	#103	1	2	yes	—
125	#1055	—	2	1	no	—
126	#1055	—	4	1	no	—
127	#1055	C₂×SD₁₆	1	1	yes	characteristic
128	#1055	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×C₄×Q₈	—	13	1	no	—
130	C₂²×C₁₆	—	26	1	no	—
131	C₂²×C₁₆	—	26	1	no	—
132	C₈×Q₈	—	13	1	no	—
133	C₈×Q₈	—	13	1	no	—
134	C₈×Q₈	—	13	1	no	—
135	C₈×Q₈	—	26	1	no	—
136	C₈×Q₈	—	26	1	no	—
137	C₈×Q₈	#261	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	13	1	no	—
139	C₂×C₄⋊C₈	—	26	1	no	—
140	C₂×C₄×C₈	—	13	1	no	—
141	C₂×C₄×C₈	—	26	1	no	—
142	C₄⋊C₁₆	—	13	2	no	—
143	C₄⋊C₁₆	—	26	1	no	—
144	C₄×C₁₆	—	13	2	no	—
145	C₄×C₁₆	—	26	1	no	—
146	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×D₁₃	Q₈⋊C₈	1	2	yes	—
148	C₂×D₁₃	C₂²⋊C₁₆	1	2	yes	—
149	C₂×D₁₃	#350	1	2	yes	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	1	no	—
152	C₄×C₁₃	—	4	1	no	—
153	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
154	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
155	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
156	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
157	Dic₁₃	—	2	2	no	—
158	Dic₁₃	—	2	1	no	—
159	Dic₁₃	—	4	1	no	—
160	Dic₁₃	Q₈⋊C₈	1	2	yes	—
161	Dic₁₃	C₂²⋊C₁₆	1	2	yes	—
162	C₂²×Q₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	26	1	no	—
166	C₄×Q₈	—	13	1	no	—
167	C₄×Q₈	—	13	1	no	—
168	C₄×Q₈	—	13	1	no	—
169	C₄×Q₈	—	26	1	no	—
170	C₄×Q₈	—	26	1	no	—
171	C₄×Q₈	C₂×#261	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	13	1	no	—
173	C₂×C₄⋊C₄	—	26	1	no	—
174	C₂×C₄²	—	13	1	no	—
175	C₂×C₄²	—	26	1	no	—
176	C₂×C₁₆	—	26	2	no	—
177	C₂×C₁₆	—	26	4	no	—
178	C₂×C₁₆	—	26	2	no	—
179	C₂×C₁₆	—	52	2	no	—
180	C₄⋊C₈	—	2	1	no	—
181	C₄⋊C₈	—	13	1	no	—
182	C₄⋊C₈	—	26	1	no	—
183	C₄⋊C₈	—	26	1	no	—
184	C₄⋊C₈	—	26	1	no	—
185	C₄⋊C₈	—	26	1	no	—
186	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	13	1	no	—
189	C₄×C₈	—	13	1	no	—
190	C₄×C₈	—	13	1	no	—
191	C₄×C₈	—	26	1	no	—
192	C₄×C₈	—	26	1	no	—
193	C₄×C₈	—	26	1	no	—
194	C₄×C₈	C₂×#261	1	1	yes	characteristic
195	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
196	C₂×C₁₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
197	C₂×C₁₃	C₂×#350	1	1	yes	characteristic
198	D₁₃	Q₈⋊C₁₆	1	4	yes	complemented
199	C₂×Q₈	—	13	4	no	—
200	C₂×Q₈	—	13	1	no	—
201	C₂×Q₈	—	26	1	no	—
202	C₂×Q₈	—	26	2	no	—
203	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
204	C₂²×C₄	—	13	1	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	13	1	no	—
207	C₂²×C₄	—	26	1	no	—
208	C₂²×C₄	—	26	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	26	2	no	—
215	C₂×C₈	—	26	1	no	—
216	C₂×C₈	—	52	2	no	—
217	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
219	C₄⋊C₄	—	2	1	no	—
220	C₄⋊C₄	—	13	1	no	—
221	C₄⋊C₄	—	26	1	no	—
222	C₄⋊C₄	—	26	1	no	—
223	C₄⋊C₄	—	26	1	no	—
224	C₄⋊C₄	—	26	1	no	—
225	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
226	C₄²	—	2	1	no	—
227	C₄²	—	13	1	no	—
228	C₄²	—	13	1	no	—
229	C₄²	—	13	1	no	—
230	C₄²	—	26	1	no	—
231	C₄²	—	26	1	no	—
232	C₄²	—	26	1	no	—
233	C₄²	C₂²×#261	1	1	yes	characteristic
234	C₁₆	—	26	4	no	—
235	C₁₆	—	52	2	no	—
236	C₁₃	C₂×Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	13	1	no	—
238	Q₈	—	2	1	no	—
239	Q₈	—	13	2	no	—
240	Q₈	—	26	4	no	—
241	Q₈	—	26	1	no	—
242	Q₈	D₁₃⋊C₁₆	1	2	yes	complemented
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	2	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	26	4	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	52	2	no	—
257	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
258	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
259	C₂×C₄	C₂×C₂²⋊#261	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	13	1	no	—
263	C₈	—	13	1	no	—
264	C₈	—	26	1	no	—
265	C₈	—	52	1	no	—
266	C₈	#88117	1	2	yes	—
267	C₂²	—	13	2	no	—
268	C₂²	—	13	2	no	—
269	C₂²	—	13	2	no	—
270	C₂²	D₁₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	13	2	no	—
275	C₄	—	13	2	no	—
276	C₄	—	26	2	no	—
277	C₄	—	26	1	no	—
278	C₄	—	52	1	no	—
279	C₄	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
280	C₄	D₁₃⋊M₃₂	1	1	yes	characteristic
281	C₄	C₂×#88117	1	1	yes	characteristic
282	C₄	D₁₃⋊#104	1	1	yes	characteristic
283	C₂	—	13	4	no	—
284	C₂	D₁₃⋊(Q₈⋊C₈)	1	1	yes	characteristic
285	C₂	D₁₃⋊(C₂²⋊C₁₆)	1	1	yes	characteristic
286	C₂	D₁₃⋊#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#261
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#261
Fitting	C₈×Q₈×C₁₃	C₄
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂²×D₁₃	C₂²⋊C₈
cosocle	C₄×#1055	C₂³
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	C₂×Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #13129803	C₂	semidirect
4. Q₈⋊#6092	C₂	semidirect
59. Q₈×D₁₃	C₁₆	semidirect
101. Q₈×C₁₃	C₂×C₁₆	semidirect
121. Dic₂₆	C₂×C₁₆	semidirect
198. D₁₃	Q₈⋊C₁₆	semidirect
236. C₁₃	C₂×Q₈⋊C₁₆	semidirect
242. Q₈	D₁₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₁₃⋊(C₄⋊C₁₆)	2
C₄×D₁₃⋊C₁₆	2
#13129803	2
Q₈⋊#6092	2
C₈×Q₈×D₁₃	2
C₂×Q₈⋊C₁₆	13

Subgroups — D13⋊(Q8⋊C16) #16787351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₃⋊(Q₈⋊C₁₆) #16787351