Subgroups — C₂×Q₃₂×#261 #16807189

Order 3328.

678 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×#261	C₂	1	2	yes	maximal
2	Q₃₂×#261	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718221	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718280	C₂	1	2	yes	maximal
5	C₂×C₁₆×#261	C₂	1	1	yes	maximal, characteristic
6	C₂×D₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×#261	—	2	2	no	—
8	C₂×Q₁₆×D₁₃	C₂²	1	2	yes	—
9	C₂×#88117	—	2	2	no	—
10	Q₁₆×#261	—	2	4	no	—
11	Q₁₆×#261	C₂²	1	8	yes	—
12	C₂×#88102	C₂²	1	2	yes	—
13	C₂×C₈×#261	C₂²	1	1	yes	characteristic
14	C₂×D₁₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
16	C₂×#88069	C₄	1	2	yes	complemented
17	D₁₃×Q₃₂	C₄	1	4	yes	complemented
18	D₁₃×Q₃₂	C₂²	1	4	yes	—
19	C₂×Dic₁₀₄	C₄	1	1	yes	characteristic, complemented
20	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
21	#11718280	C₂²	1	8	yes	—
22	#11718221	C₂²	1	4	yes	—
23	C₁₆×#261	C₂²	1	4	yes	—
24	C₂×Q₈×D₁₃	—	2	2	no	—
25	Q₈×#261	—	2	8	no	—
26	Q₈×#261	—	4	4	no	—
27	C₂×C₄⋊#261	—	2	2	no	—
28	C₂×C₄⋊#261	—	4	2	no	—
29	C₂×C₄×#261	—	4	2	no	—
30	C₂×C₄×#261	D₄	1	1	yes	characteristic
31	C₂×D₁₃⋊C₈	D₄	1	1	yes	characteristic
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
33	C₂×#6107	—	2	2	no	—
34	Q₁₆×D₁₃	—	2	2	no	—
35	Q₁₆×D₁₃	—	2	2	no	—
36	Q₁₆×D₁₃	C₂×C₄	1	4	yes	—
37	Q₁₆×D₁₃	C₂³	1	4	yes	—
38	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
40	#88117	—	2	8	no	—
41	C₂×#6092	—	2	1	no	—
42	#88102	—	2	2	no	—
43	#88102	C₂³	1	4	yes	—
44	C₈×#261	D₄	1	2	yes	—
45	C₈×#261	C₂³	1	2	yes	—
46	D₁₃⋊C₁₆	—	2	1	no	—
47	D₁₃⋊C₁₆	C₂³	1	2	yes	—
48	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
50	#88069	C₂×C₄	1	8	yes	complemented
51	C₂×#6090	C₂×C₄	1	1	yes	characteristic
52	Dic₁₀₄	C₂×C₄	1	4	yes	complemented
53	C₁₆×D₁₃	C₂×C₄	1	2	yes	—
54	C₁₆×D₁₃	C₂³	1	2	yes	—
55	C₂×C₄×Q₃₂	—	13	1	no	maximal, Sylow
56	C₂²×#261	—	4	2	no	—
57	C₂²×#261	D₈	1	1	yes	characteristic
58	C₂²×#261	D₈	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	Q₈×D₁₃	—	2	4	no	—
61	Q₈×D₁₃	—	2	4	no	—
62	Q₈×D₁₃	—	4	2	no	—
63	Q₈×D₁₃	—	4	2	no	—
64	C₂×C₄×D₁₃	—	4	2	no	—
65	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂×Dic₂₆	—	2	2	no	—
67	C₂×Dic₂₆	—	4	2	no	—
68	C₂×#1057	—	2	1	no	—
69	C₄⋊#261	—	2	4	no	—
70	C₄⋊#261	—	4	8	no	—
71	C₄⋊#261	—	4	2	no	—
72	C₄×#261	—	4	8	no	—
73	C₄×#261	D₈	1	2	yes	—
74	C₄×#261	C₂×D₄	1	2	yes	—
75	D₁₃⋊C₈	—	2	1	no	—
76	D₁₃⋊C₈	C₂×D₄	1	2	yes	—
77	C₁₃×Q₁₆	—	2	2	no	—
78	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	#6107	—	2	4	no	—
81	#6107	—	2	4	no	—
82	C₂×#1055	#41	1	1	yes	characteristic
83	Dic₅₂	—	2	2	no	—
84	Dic₅₂	C₂²×C₄	1	4	yes	—
85	C₈×D₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₈×D₁₃	C₂×D₄	1	1	yes	characteristic
87	C₈×D₁₃	#41	1	1	yes	characteristic
88	C₈×D₁₃	C₂⁴	1	1	yes	characteristic
89	#6092	—	2	2	no	—
90	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
91	#6090	C₂²×C₄	1	2	yes	—
92	C₂²×Q₃₂	—	13	1	no	—
93	C₂×C₄×Q₁₆	—	13	2	no	—
94	C₄×Q₃₂	—	13	8	no	—
95	C₂×#366	—	13	1	no	—
96	C₂×#358	—	13	2	no	—
97	C₂×C₄×C₁₆	—	13	1	no	—
98	C₂²×D₁₃	C₂×D₈	1	1	yes	characteristic
99	C₂×#261	—	2	2	no	—
100	C₂×#261	—	4	4	no	—
101	C₂×#261	—	8	4	no	—
102	C₂×#261	Q₃₂	1	4	yes	direct product factor
103	C₂×#261	C₂×D₈	1	2	yes	—
104	C₂×#261	C₂×D₈	1	2	yes	—
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×Dic₁₃	—	4	2	no	—
110	C₂×Dic₁₃	#135	1	1	yes	characteristic
111	C₄×D₁₃	—	4	4	no	—
112	C₄×D₁₃	—	4	4	no	—
113	C₄×D₁₃	C₄×D₄	1	1	yes	characteristic
114	C₄×D₁₃	C₂×D₈	1	1	yes	characteristic
115	C₄×D₁₃	#135	1	1	yes	characteristic
116	C₄×D₁₃	C₂²×D₄	1	1	yes	characteristic
117	Dic₂₆	—	2	4	no	—
118	Dic₂₆	—	4	2	no	—
119	Dic₂₆	—	4	4	no	—
120	Dic₂₆	—	4	4	no	—
121	#1057	—	2	2	no	—
122	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	#1055	C₄×D₄	1	1	yes	characteristic
125	#1055	C₂×#41	1	1	yes	characteristic
126	C₂²×Q₁₆	—	13	2	no	—
127	C₂×C₄×Q₈	—	26	2	no	—
128	C₂×Q₃₂	—	13	2	no	—
129	C₂×Q₃₂	—	13	4	no	—
130	C₂×Q₃₂	—	13	4	no	—
131	C₂×Q₃₂	—	13	1	no	—
132	C₂×Q₃₂	#261	1	1	yes	characteristic, direct product factor
133	C₂²×C₁₆	—	13	1	no	—
134	C₂²×C₁₆	—	13	1	no	—
135	C₄×Q₁₆	—	13	8	no	—
136	C₄×Q₁₆	—	26	4	no	—
137	C₂×#107	—	13	2	no	—
138	C₂×#103	—	26	2	no	—
139	C₂×C₄×C₈	—	13	1	no	—
140	#366	—	13	4	no	—
141	#358	—	13	8	no	—
142	C₄×C₁₆	—	13	4	no	—
143	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
144	C₂×D₁₃	C₄×D₈	1	1	yes	characteristic
145	C₂×D₁₃	C₂×Q₃₂	1	2	yes	—
146	C₂×D₁₃	#508	1	2	yes	—
147	C₂×D₁₃	C₂²×D₈	1	1	yes	characteristic
148	#261	—	2	2	no	—
149	#261	—	8	4	no	—
150	#261	C₂×Q₃₂	1	4	yes	direct product factor
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
153	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
154	Dic₁₃	—	4	4	no	—
155	Dic₁₃	C₄×D₈	1	1	yes	characteristic
156	Dic₁₃	C₂×#135	1	1	yes	characteristic
157	C₂²×Q₈	—	26	2	no	—
158	C₂×Q₁₆	—	13	4	no	—
159	C₂×Q₁₆	—	13	4	no	—
160	C₂×Q₁₆	—	13	2	no	—
161	C₂×Q₁₆	—	26	2	no	—
162	C₂×Q₁₆	—	26	2	no	—
163	C₂×Q₁₆	—	26	2	no	—
164	C₂×Q₁₆	C₂×#261	1	2	yes	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	1	no	—
167	C₄×Q₈	—	26	8	no	—
168	C₄×Q₈	—	52	4	no	—
169	C₂×C₄⋊C₄	—	26	2	no	—
170	C₂×C₄⋊C₄	—	52	2	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	52	2	no	—
173	Q₃₂	—	13	8	no	—
174	Q₃₂	—	13	4	no	—
175	Q₃₂	C₂×#261	1	4	yes	direct product factor
176	C₂×C₁₆	—	13	1	no	—
177	C₂×C₁₆	—	13	2	no	—
178	C₂×C₁₆	—	13	2	no	—
179	C₂×C₁₆	—	13	2	no	—
180	C₂×C₁₆	—	26	1	no	—
181	C₂×C₁₆	—	26	1	no	—
182	C₂×C₁₆	C₂×#261	1	1	yes	characteristic
183	#107	—	13	4	no	—
184	#107	—	26	2	no	—
185	#103	—	26	8	no	—
186	C₄×C₈	—	13	2	no	—
187	C₄×C₈	—	13	2	no	—
188	C₂×C₁₃	C₄×Q₃₂	1	2	yes	complemented
189	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
190	D₁₃	C₄×Q₃₂	1	2	yes	complemented
191	D₁₃	C₂²×Q₃₂	1	1	yes	characteristic
192	D₁₃	C₂×#508	1	1	yes	characteristic
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	26	4	no	—
195	C₂×Q₈	—	26	4	no	—
196	C₂×Q₈	—	26	2	no	—
197	C₂×Q₈	—	52	2	no	—
198	C₂×Q₈	—	52	2	no	—
199	C₂×Q₈	—	52	2	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	52	2	no	—
204	C₂²×C₄	—	52	2	no	—
205	Q₁₆	—	2	2	no	—
206	Q₁₆	—	13	4	no	—
207	Q₁₆	—	26	4	no	—
208	Q₁₆	—	26	4	no	—
209	Q₁₆	—	26	2	no	—
210	Q₁₆	C₂²×#261	1	4	yes	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	26	1	no	—
218	C₂×C₈	—	26	1	no	—
219	C₂×C₈	C₂²×#261	1	1	yes	characteristic
220	C₄⋊C₄	—	26	4	no	—
221	C₄⋊C₄	—	52	2	no	—
222	C₄⋊C₄	—	52	8	no	—
223	C₄²	—	13	2	no	—
224	C₄²	—	13	2	no	—
225	C₄²	—	52	8	no	—
226	C₁₆	—	13	2	no	—
227	C₁₆	—	26	2	no	—
228	C₁₆	C₂²×#261	1	2	yes	—
229	C₁₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	13	1	no	—
231	Q₈	—	2	4	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	26	4	no	—
234	Q₈	—	52	4	no	—
235	Q₈	—	52	4	no	—
236	Q₈	—	52	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	13	2	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	13	1	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	2	no	—
246	C₂×C₄	—	26	2	no	—
247	C₂×C₄	—	52	4	no	—
248	C₂×C₄	—	52	4	no	—
249	C₂×C₄	—	52	2	no	—
250	C₂×C₄	—	52	4	no	—
251	C₂×C₄	—	104	4	no	—
252	C₂×C₄	D₄×#261	1	1	yes	characteristic
253	C₈	—	13	1	no	—
254	C₈	—	13	1	no	—
255	C₈	—	26	2	no	—
256	C₈	D₄×#261	1	1	yes	characteristic
257	C₈	C₂³×#261	1	1	yes	characteristic
258	C₂²	—	13	1	no	—
259	C₂²	—	13	1	no	—
260	C₂²	—	13	2	no	—
261	C₂²	—	13	2	no	—
262	C₂²	D₈×#261	1	1	yes	characteristic
263	C₄	—	4	4	no	—
264	C₄	—	13	4	no	—
265	C₄	—	13	1	no	—
266	C₄	—	13	1	no	—
267	C₄	—	26	2	no	—
268	C₄	—	52	4	no	—
269	C₄	—	104	4	no	—
270	C₄	D₈×#261	1	1	yes	characteristic
271	C₄	C₂×D₄×#261	1	1	yes	characteristic
272	C₂	—	13	1	no	—
273	C₂	—	13	1	no	—
274	C₂	—	13	2	no	—
275	C₂	Q₃₂×#261	1	2	yes	direct product factor
276	C₂	C₂×D₈×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#261
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₈	C₂³×#261
Fitting	C₂×C₁₃×Q₃₂	C₄
socle	C₂²×C₁₃	C₄×D₈
prime radical	C₂²×D₁₃	C₂×D₈
cosocle	C₈×D₁₃	C₂⁴
hypercenter	C₂×Q₃₂	#261
hypocenter	C₁₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₃₂×#261	C₂	direct (N × H)
15. C₂×C₁₃×Q₃₂	C₄	semidirect
16. C₂×#88069	C₄	semidirect
17. D₁₃×Q₃₂	C₄	semidirect
19. C₂×Dic₁₀₄	C₄	semidirect
48. C₁₃×Q₃₂	C₂×C₄	semidirect
50. #88069	C₂×C₄	semidirect
52. Dic₁₀₄	C₂×C₄	semidirect
102. C₂×#261	Q₃₂	direct (N × H)
132. C₂×Q₃₂	#261	direct (N × H)
150. #261	C₂×Q₃₂	direct (N × H)
175. Q₃₂	C₂×#261	direct (N × H)
188. C₂×C₁₃	C₄×Q₃₂	semidirect
190. D₁₃	C₄×Q₃₂	semidirect
229. C₁₃	C₂×C₄×Q₃₂	semidirect
275. C₂	Q₃₂×#261	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×Q₁₆×#261	2
Q₃₂×#261	2
C₂×#11718221	2
C₂×#11718280	2
C₂×C₁₆×#261	2
C₂×D₁₃×Q₃₂	2
C₂×C₄×Q₃₂	13

Subgroups — C2×Q32×#261 #16807189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×#261 #16807189