Subgroups — C₄×#19104768 #35223359

Order 20864.

484 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	C₂×#19104768	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579699	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₃₂₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#23781773	C₂	1	2	yes	maximal, complemented
5	#23781767	C₂	1	2	yes	maximal, complemented
6	C₄×Dic₆₅₂	C₂	1	2	yes	maximal, complemented
7	C₄×#14579688	C₂	1	2	yes	maximal, complemented
8	#23781707	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781675	C₂	1	1	yes	maximal, characteristic, complemented
10	#23781667	C₂	1	1	yes	maximal, characteristic
11	C₄×M₁₆×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₂₆	C₂²	1	1	yes	characteristic
14	#19104768	C₄	1	8	yes	direct product factor
15	C₂×Dic₆₅₂	C₂²	1	2	yes	—
16	C₂×#14579688	C₂²	1	2	yes	—
17	C₄×Dic₃₂₆	—	2	1	no	—
18	C₄×Dic₃₂₆	—	2	1	no	—
19	C₄×Dic₃₂₆	C₂²	1	2	yes	—
20	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₂₆	C₂²	1	2	yes	—
22	C₄×Dic₃₂₆	C₂²	1	1	yes	characteristic
23	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
24	C₄×D₆₅₂	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
26	C₄×C₂²⋊D₁₆₃	—	2	1	no	—
27	#19104694	C₂²	1	1	yes	characteristic
28	#19104692	—	2	1	no	—
29	C₂×C₄⋊Dic₁₆₃	C₂²	1	1	yes	characteristic
30	C₂×Dic₁₆₃⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₁₆₃	—	2	1	no	—
32	C₄²×D₁₆₃	—	2	1	no	—
33	#19104669	C₂²	1	2	yes	—
34	#19104667	C₂²	1	2	yes	—
35	#19104660	C₂²	1	2	yes	—
36	#19104660	C₂²	1	2	yes	—
37	#19104660	C₂²	1	2	yes	—
38	#19104657	C₂²	1	2	yes	—
39	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₆₃	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
42	#97×C₁₆₃	C₂²	1	2	yes	—
43	C₄×C₈×C₁₆₃	C₂²	1	2	yes	—
44	#14579699	—	2	2	no	—
45	#14579699	C₂×C₄	1	4	yes	—
46	C₂×Dic₃₂₆	—	2	1	no	—
47	C₂×Dic₃₂₆	—	2	2	no	—
48	C₂×Dic₃₂₆	C₂×C₄	1	4	yes	complemented
49	C₂×Dic₃₂₆	C₂³	1	2	yes	—
50	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
51	C₂×D₆₅₂	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
53	C₂²×Dic₁₆₃	—	2	1	no	—
54	C₂²×Dic₁₆₃	—	2	1	no	—
55	C₂×C₄×D₁₆₃	—	2	1	no	—
56	C₂×C₄×D₁₆₃	—	2	1	no	—
57	C₂×C₄×D₁₆₃	—	2	1	no	—
58	Dic₆₅₂	C₂×C₄	1	8	yes	complemented
59	#14579688	C₂×C₄	1	8	yes	complemented
60	C₄⋊Dic₁₆₃	—	2	1	no	—
61	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
62	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₁₆₃	C₂³	1	1	yes	characteristic
64	Dic₁₆₃⋊C₄	—	2	2	no	—
65	Dic₁₆₃⋊C₄	—	2	2	no	—
66	Dic₁₆₃⋊C₄	—	2	1	no	—
67	Dic₁₆₃⋊C₄	—	2	1	no	—
68	C₂²⋊Dic₁₆₃	—	2	1	no	—
69	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
70	(C₂×D₁₆₃)⋊C₄	—	2	1	no	—
71	C₄×Dic₁₆₃	—	2	1	no	—
72	C₄×Dic₁₆₃	—	2	1	no	—
73	C₄×Dic₁₆₃	—	2	1	no	—
74	C₄×Dic₁₆₃	—	2	2	no	—
75	C₄×Dic₁₆₃	—	2	2	no	—
76	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₆₃	D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₆₃	—	2	2	no	—
80	M₁₆×C₁₆₃	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
82	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
83	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
84	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
85	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
86	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
87	C₂²×D₁₆₃	—	2	1	no	—
88	Dic₃₂₆	—	2	4	no	—
89	Dic₃₂₆	—	2	2	no	—
90	Dic₃₂₆	—	2	1	no	—
91	Dic₃₂₆	C₂²×C₄	1	4	yes	—
92	Dic₃₂₆	C₂²×C₄	1	2	yes	—
93	D₆₅₂	—	2	1	no	—
94	D₆₅₂	C₂²×C₄	1	2	yes	—
95	C₂²⋊D₁₆₃	—	2	4	no	—
96	C₂×Dic₁₆₃	—	2	4	no	—
97	C₂×Dic₁₆₃	—	2	1	no	—
98	C₂×Dic₁₆₃	—	2	2	no	—
99	C₂×Dic₁₆₃	—	2	1	no	—
100	C₂×Dic₁₆₃	—	2	1	no	—
101	C₂×Dic₁₆₃	—	2	2	no	—
102	C₂×Dic₁₆₃	—	4	2	no	—
103	C₄×D₁₆₃	—	2	4	no	—
104	C₄×D₁₆₃	—	4	2	no	—
105	C₄×D₁₆₃	—	4	2	no	—
106	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₆₃	—	2	2	no	—
108	C₂×C₄×C₁₆₃	—	4	1	no	—
109	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
110	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₆₃	#41	1	2	yes	—
116	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
117	C₈×C₁₆₃	—	2	2	no	—
118	C₈×C₁₆₃	C₂²×C₄	1	4	yes	—
119	C₂²×C₁₆₃	—	2	1	no	—
120	C₂²×C₁₆₃	—	2	1	no	—
121	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
122	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₆₃	—	2	2	no	—
124	C₂×D₁₆₃	—	4	1	no	—
125	C₂×D₁₆₃	—	4	1	no	—
126	C₄×C₁₆₃	—	2	1	no	—
127	C₄×C₁₆₃	—	2	1	no	—
128	C₄×C₁₆₃	—	2	1	no	—
129	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
130	C₄×C₁₆₃	#137	1	2	yes	complemented
131	C₄×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
133	Dic₁₆₃	—	2	2	no	—
134	Dic₁₆₃	—	2	4	no	—
135	Dic₁₆₃	—	4	1	no	—
136	Dic₁₆₃	—	4	1	no	—
137	Dic₁₆₃	—	4	2	no	—
138	C₂×C₁₆₃	—	2	2	no	—
139	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₆₃	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₆₃	#575	1	1	yes	characteristic
142	D₁₆₃	—	4	2	no	—
143	C₁₆₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	163	1	no	maximal, Sylow
145	C₂×#137	—	163	1	no	—
146	C₄×#41	—	163	1	no	—
147	C₂×C₄×Q₈	—	163	1	no	—
148	#441	—	163	2	no	—
149	#440	—	163	2	no	—
150	C₄×Q₁₆	—	163	2	no	—
151	C₄×SD₁₆	—	163	2	no	—
152	#428	—	163	1	no	—
153	#419	—	163	1	no	—
154	#417	—	163	1	no	—
155	C₄×M₁₆	D₁₆₃	1	1	yes	characteristic, complemented
156	C₂×#41	—	163	1	no	—
157	C₂²×Q₈	—	163	1	no	—
158	#137	—	163	8	no	—
159	C₂×Q₁₆	—	163	2	no	—
160	C₂×SD₁₆	—	163	2	no	—
161	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
162	C₂×M₁₆	C₂×D₁₆₃	1	1	yes	characteristic
163	C₄×Q₈	—	326	1	no	—
164	C₄×Q₈	—	326	1	no	—
165	C₄×Q₈	—	163	2	no	—
166	C₄×Q₈	—	163	2	no	—
167	C₄×Q₈	—	163	1	no	—
168	C₄×Q₈	—	163	1	no	—
169	C₄×D₄	—	326	1	no	—
170	C₄×D₄	—	326	1	no	—
171	C₄×D₄	—	163	1	no	—
172	C₄×D₄	—	163	1	no	—
173	#117	—	326	1	no	—
174	#117	—	163	1	no	—
175	C₂×C₄⋊C₄	—	326	1	no	—
176	C₂×C₄⋊C₄	—	163	1	no	—
177	C₂×C₄²	—	326	1	no	—
178	C₂×C₄²	—	326	1	no	—
179	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
180	#107	—	163	2	no	—
181	#106	—	163	2	no	—
182	#103	—	163	2	no	—
183	#103	—	163	2	no	—
184	#103	—	163	2	no	—
185	#102	—	163	2	no	—
186	#97	C₂×D₁₆₃	1	2	yes	—
187	C₄×C₈	C₂×D₁₆₃	1	2	yes	—
188	#41	—	326	2	no	—
189	#41	—	163	4	no	—
190	C₂×Q₈	—	326	1	no	—
191	C₂×Q₈	—	326	2	no	—
192	C₂×Q₈	—	163	4	no	—
193	C₂×Q₈	—	163	2	no	—
194	C₂×Q₈	—	163	1	no	—
195	C₂×D₄	—	326	1	no	—
196	C₂×D₄	—	163	1	no	—
197	C₂²×C₄	—	326	1	no	—
198	C₂²×C₄	—	326	1	no	—
199	C₂²×C₄	—	326	1	no	—
200	C₂²×C₄	—	326	1	no	—
201	C₂²×C₄	—	326	1	no	—
202	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
203	C₂²×C₄	D₆₅₂	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
205	Q₁₆	—	163	8	no	—
206	SD₁₆	—	163	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	C₄×D₁₆₃	1	4	yes	complemented
209	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
210	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
211	C₄⋊C₄	—	326	2	no	—
212	C₄⋊C₄	—	326	2	no	—
213	C₄⋊C₄	—	326	1	no	—
214	C₄⋊C₄	—	326	1	no	—
215	C₄⋊C₄	—	326	1	no	—
216	C₄⋊C₄	—	163	2	no	—
217	C₄⋊C₄	—	163	1	no	—
218	C₄⋊C₄	—	163	1	no	—
219	C₂²⋊C₄	—	326	1	no	—
220	C₂²⋊C₄	—	326	1	no	—
221	C₂²⋊C₄	—	326	1	no	—
222	C₄²	—	326	1	no	—
223	C₄²	—	326	1	no	—
224	C₄²	—	326	2	no	—
225	C₄²	—	326	1	no	—
226	C₄²	—	326	2	no	—
227	C₄²	D₆₅₂	1	1	yes	characteristic
228	C₄²	D₆₅₂	1	1	yes	characteristic
229	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
230	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
231	C₂³	—	326	1	no	—
232	C₂³	C₂×D₆₅₂	1	1	yes	characteristic
233	Q₈	—	326	2	no	—
234	Q₈	—	326	1	no	—
235	Q₈	—	326	4	no	—
236	Q₈	—	163	4	no	—
237	Q₈	—	163	2	no	—
238	D₄	—	326	4	no	—
239	D₄	—	326	1	no	—
240	D₄	—	163	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	326	1	no	—
244	C₂×C₄	—	326	2	no	—
245	C₂×C₄	—	326	4	no	—
246	C₂×C₄	—	326	4	no	—
247	C₂×C₄	—	326	1	no	—
248	C₂×C₄	—	326	1	no	—
249	C₂×C₄	—	326	2	no	—
250	C₂×C₄	—	652	2	no	—
251	C₂×C₄	—	652	2	no	—
252	C₂×C₄	—	652	2	no	—
253	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
254	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
255	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
256	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
257	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
258	C₂×C₄	C₂×D₆₅₂	1	1	yes	characteristic
259	C₂×C₄	#14579699	1	2	yes	—
260	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	C₂×C₄×D₁₆₃	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	326	2	no	—
266	C₂²	—	652	1	no	—
267	C₂²	—	652	1	no	—
268	C₂²	C₄×D₆₅₂	1	2	yes	—
269	C₂²	C₂²×D₆₅₂	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	326	2	no	—
274	C₄	—	326	4	no	—
275	C₄	—	652	1	no	—
276	C₄	—	652	1	no	—
277	C₄	—	652	2	no	—
278	C₄	C₄×D₆₅₂	1	2	yes	—
279	C₄	#19104768	1	2	yes	direct product factor
280	C₄	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
281	C₄	C₂×#14579699	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	652	2	no	—
284	C₂	C₂×C₄×D₆₅₂	1	1	yes	characteristic
285	C₂	C₂×#19104768	1	1	yes	characteristic
286	C₂	#23782465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₅₂
commutator	C₄×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₄×M₁₆×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#14579699	C₂²
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₆₃
hypocenter	C₁₆₃	C₄×#137
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Dic₃₂₆	C₂	semidirect
4. #23781773	C₂	semidirect
5. #23781767	C₂	semidirect
6. C₄×Dic₆₅₂	C₂	semidirect
7. C₄×#14579688	C₂	semidirect
8. #23781707	C₂	semidirect
9. #23781675	C₂	semidirect
11. C₄×M₁₆×C₁₆₃	C₂	semidirect
14. #19104768	C₄	direct (N × H)
48. C₂×Dic₃₂₆	C₂×C₄	semidirect
58. Dic₆₅₂	C₂×C₄	semidirect
59. #14579688	C₂×C₄	semidirect
80. M₁₆×C₁₆₃	C₂×C₄	semidirect
130. C₄×C₁₆₃	#137	semidirect
143. C₁₆₃	C₄×#137	semidirect
155. C₄×M₁₆	D₁₆₃	semidirect
208. M₁₆	C₄×D₁₆₃	semidirect
279. C₄	#19104768	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×#19104768	2
C₄×#14579699	2
C₂×C₄×Dic₃₂₆	2
#23781773	2
#23781767	2
C₄×Dic₆₅₂	2
C₄×#14579688	2
#23781707	2
#23781675	2
#23781667	2
C₄×M₁₆×C₁₆₃	2
C₄×#137	163

Subgroups — C4×#19104768 #35223359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19104768 #35223359