Subgroups — C₄×#19104755 #35223306

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19104755	C₂	1	1	yes	maximal, characteristic
2	C₄×#14579699	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14579689	C₂	1	2	yes	maximal, complemented
4	C₄×#14579686	C₂	1	2	yes	maximal, complemented
5	C₄×#14579687	C₂	1	2	yes	maximal, complemented
6	C₄×#14579684	C₂	1	2	yes	maximal, complemented
7	#23781701	C₂	1	1	yes	maximal, characteristic, complemented
8	#23781661	C₂	1	1	yes	maximal, characteristic, complemented
9	#23781659	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₁₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	#19104755	C₄	1	8	yes	direct product factor
14	C₂×#14579689	C₂²	1	1	yes	characteristic
15	C₂×#14579687	C₂²	1	1	yes	characteristic
16	C₂×#14579686	C₂²	1	1	yes	characteristic
17	C₂×#14579684	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₃₂₆	C₂²	1	2	yes	—
21	C₄×D₆₅₂	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₁₆₃	—	2	2	no	—
23	#19104694	C₂²	1	1	yes	characteristic
24	#19104692	—	2	1	no	—
25	C₄²×D₁₆₃	—	2	1	no	—
26	#19104668	C₂²	1	2	yes	—
27	#19104666	C₂²	1	2	yes	—
28	#19104659	C₂²	1	2	yes	—
29	#19104658	C₂²	1	2	yes	—
30	#19104656	C₂²	1	2	yes	—
31	#19104655	C₂²	1	2	yes	—
32	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
33	C₄×C₁₆₃⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₆₃	C₂²	1	2	yes	—
36	C₄×D₄×C₁₆₃	—	2	2	no	—
37	C₄×D₄×C₁₆₃	C₂²	1	2	yes	—
38	#117×C₁₆₃	—	2	1	no	—
39	#117×C₁₆₃	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₁₆₃	—	2	1	no	—
41	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
42	#14579699	—	2	2	no	—
43	#14579699	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₃₂₆	C₂³	1	1	yes	characteristic
45	C₂×D₆₅₂	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₆₃	—	2	1	no	—
47	C₂×C₄×D₁₆₃	—	2	1	no	—
48	C₂×C₄×D₁₆₃	—	2	2	no	—
49	#14579689	C₂×C₄	1	4	yes	complemented
50	#14579687	C₂×C₄	1	4	yes	complemented
51	#14579686	C₂×C₄	1	4	yes	complemented
52	#14579684	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆₃⋊C₈	—	2	2	no	—
54	C₂×C₁₆₃⋊C₈	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
56	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₆₃⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₆₃	C₂³	1	2	yes	—
59	Dic₁₆₃⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₁₆₃	—	2	1	no	—
61	(C₂×D₁₆₃)⋊C₄	—	2	2	no	—
62	C₄×Dic₁₆₃	—	2	1	no	—
63	C₄×Dic₁₆₃	—	2	2	no	—
64	#41×C₁₆₃	—	2	2	no	—
65	#41×C₁₆₃	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₆₃	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₆₃	—	2	1	no	—
68	C₂×D₄×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₆₃	—	2	1	no	—
70	C₂²×C₄×C₁₆₃	—	2	2	no	—
71	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
72	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₆₃	—	2	2	no	—
74	C₄⋊C₄×C₁₆₃	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₁₆₃	—	2	1	no	—
76	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
77	C₄²×C₁₆₃	—	2	1	no	—
78	C₄²×C₁₆₃	—	2	2	no	—
79	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
80	C₄²×C₁₆₃	D₄	1	1	yes	characteristic
81	C₄²×C₁₆₃	C₂³	1	2	yes	—
82	C₂²×D₁₆₃	—	2	1	no	—
83	Dic₃₂₆	—	2	1	no	—
84	Dic₃₂₆	C₂²×C₄	1	2	yes	—
85	D₆₅₂	—	2	1	no	—
86	D₆₅₂	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₆₃	—	2	4	no	—
88	C₂×Dic₁₆₃	—	2	1	no	—
89	C₂×Dic₁₆₃	—	2	2	no	—
90	C₄×D₁₆₃	—	2	4	no	—
91	C₄×D₁₆₃	—	4	4	no	—
92	C₁₆₃⋊C₈	—	2	2	no	—
93	C₁₆₃⋊C₈	C₂²×C₄	1	2	yes	—
94	C₁₆₃⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂³×C₁₆₃	—	2	1	no	—
96	C₂³×C₁₆₃	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₁₆₃	—	2	1	no	—
98	Q₈×C₁₆₃	C₂²×C₄	1	2	yes	—
99	D₄×C₁₆₃	—	2	4	no	—
100	D₄×C₁₆₃	—	2	1	no	—
101	D₄×C₁₆₃	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₁₆₃	—	2	1	no	—
103	C₂×C₄×C₁₆₃	—	2	4	no	—
104	C₂×C₄×C₁₆₃	—	2	4	no	—
105	C₂×C₄×C₁₆₃	—	2	2	no	—
106	C₂×C₄×C₁₆₃	—	4	4	no	—
107	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₆₃	#41	1	2	yes	—
112	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
113	C₂²×C₁₆₃	—	2	2	no	—
114	C₂²×C₁₆₃	—	2	2	no	—
115	C₂²×C₁₆₃	—	4	2	no	—
116	C₂²×C₁₆₃	C₄×D₄	1	2	yes	—
117	C₂²×C₁₆₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₆₃	—	2	2	no	—
119	C₂×D₁₆₃	—	4	2	no	—
120	C₄×C₁₆₃	—	2	2	no	—
121	C₄×C₁₆₃	—	2	2	no	—
122	C₄×C₁₆₃	—	4	2	no	—
123	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
124	C₄×C₁₆₃	#135	1	4	yes	complemented
125	C₄×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₁₆₃	C₂×#41	1	1	yes	characteristic
127	Dic₁₆₃	—	2	2	no	—
128	Dic₁₆₃	—	4	2	no	—
129	C₂×C₁₆₃	—	2	2	no	—
130	C₂×C₁₆₃	—	4	2	no	—
131	C₂×C₁₆₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₆₃	C₂×#135	1	2	yes	—
133	D₁₆₃	—	4	2	no	—
134	C₁₆₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
135	C₄×#135	—	163	1	no	maximal, Sylow
136	C₂×#135	—	163	1	no	—
137	C₄×#41	—	163	1	no	—
138	C₄×#41	D₁₆₃	1	1	yes	characteristic, complemented
139	C₄×Q₁₆	—	163	2	no	—
140	C₄×SD₁₆	—	163	2	no	—
141	C₄×SD₁₆	—	163	2	no	—
142	C₄×D₈	—	163	2	no	—
143	#427	—	163	1	no	—
144	#416	—	163	1	no	—
145	#416	—	163	1	no	—
146	C₂×C₄×C₈	—	163	1	no	—
147	C₂×#41	—	163	1	no	—
148	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
149	#135	—	163	8	no	—
150	C₂×Q₁₆	—	163	1	no	—
151	C₂×SD₁₆	—	163	1	no	—
152	C₂×SD₁₆	—	163	1	no	—
153	C₂×D₈	—	163	1	no	—
154	C₂²×C₈	—	163	1	no	—
155	C₂²×C₈	—	163	1	no	—
156	C₄×Q₈	—	163	2	no	—
157	C₄×Q₈	C₂×D₁₆₃	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	326	2	no	—
160	C₄×D₄	—	163	2	no	—
161	C₄×D₄	C₂×D₁₆₃	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	326	1	no	—
164	#117	—	163	1	no	—
165	#117	C₂×D₁₆₃	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	326	1	no	—
168	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
169	#107	—	163	2	no	—
170	#106	—	163	2	no	—
171	#103	—	163	2	no	—
172	#103	—	163	2	no	—
173	#102	—	163	2	no	—
174	#102	—	163	2	no	—
175	C₄×C₈	—	163	2	no	—
176	C₄×C₈	—	163	2	no	—
177	#41	—	2	2	no	—
178	#41	—	326	2	no	—
179	#41	—	163	4	no	—
180	#41	C₄×D₁₆₃	1	4	yes	complemented
181	C₂×Q₈	—	163	1	no	—
182	C₂×Q₈	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	326	1	no	—
185	C₂×D₄	—	163	1	no	—
186	C₂×D₄	C₂²×D₁₆₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	326	1	no	—
190	C₂²×C₄	—	326	2	no	—
191	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
192	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
193	Q₁₆	—	163	4	no	—
194	SD₁₆	—	163	4	no	—
195	SD₁₆	—	163	4	no	—
196	D₈	—	163	4	no	—
197	C₂×C₈	—	326	2	no	—
198	C₂×C₈	—	163	2	no	—
199	C₂×C₈	—	163	4	no	—
200	C₂×C₈	—	163	1	no	—
201	C₂×C₈	—	163	1	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	326	2	no	—
204	C₄⋊C₄	—	163	2	no	—
205	C₄⋊C₄	C₂²×D₁₆₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	326	1	no	—
209	C₂²⋊C₄	—	326	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	326	1	no	—
213	C₄²	—	326	2	no	—
214	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
215	C₄²	C₂²⋊D₁₆₃	1	1	yes	characteristic
216	C₄²	C₂²×D₁₆₃	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	326	1	no	—
219	C₂³	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	326	1	no	—
222	Q₈	—	163	2	no	—
223	Q₈	C₂×C₄×D₁₆₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	326	4	no	—
227	D₄	—	326	1	no	—
228	D₄	—	163	2	no	—
229	D₄	C₂×C₄×D₁₆₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	326	1	no	—
236	C₂×C₄	—	326	4	no	—
237	C₂×C₄	—	326	2	no	—
238	C₂×C₄	—	652	4	no	—
239	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
243	C₂×C₄	#14579699	1	2	yes	—
244	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
245	C₈	—	326	2	no	—
246	C₈	—	163	2	no	—
247	C₈	—	163	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	326	2	no	—
252	C₂²	—	652	2	no	—
253	C₂²	C₄×C₂²⋊D₁₆₃	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₁₆₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	326	2	no	—
259	C₄	—	652	2	no	—
260	C₄	C₄×C₂²⋊D₁₆₃	1	2	yes	—
261	C₄	#19104755	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
263	C₄	C₂×#14579699	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	652	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₁₆₃	1	1	yes	characteristic
268	C₂	C₂×#19104755	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₁₆₃
commutator	C₄×C₁₆₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₆₃
Fitting	C₄×#41×C₁₆₃	C₂
socle	C₂²×C₁₆₃	C₂²×D₄
prime radical	C₂×#19104755	C₂
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	C₄×#41	D₁₆₃
hypocenter	C₁₆₃	C₄×#135
LCS	G ▷ C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14579699	C₂	semidirect
3. C₄×#14579689	C₂	semidirect
4. C₄×#14579686	C₂	semidirect
5. C₄×#14579687	C₂	semidirect
6. C₄×#14579684	C₂	semidirect
7. #23781701	C₂	semidirect
8. #23781661	C₂	semidirect
9. #23781659	C₂	semidirect
10. C₂×C₄×C₁₆₃⋊C₈	C₂	semidirect
11. C₄×#41×C₁₆₃	C₂	semidirect
13. #19104755	C₄	direct (N × H)
43. #14579699	C₂×C₄	semidirect
49. #14579689	C₂×C₄	semidirect
50. #14579687	C₂×C₄	semidirect
51. #14579686	C₂×C₄	semidirect
52. #14579684	C₂×C₄	semidirect
54. C₂×C₁₆₃⋊C₈	C₂×C₄	semidirect
65. #41×C₁₆₃	C₂×C₄	semidirect
124. C₄×C₁₆₃	#135	semidirect
134. C₁₆₃	C₄×#135	semidirect
138. C₄×#41	D₁₆₃	semidirect
180. #41	C₄×D₁₆₃	semidirect
261. C₄	#19104755	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×#19104755	2
C₄×#14579699	2
C₄×#14579689	2
C₄×#14579686	2
C₄×#14579687	2
C₄×#14579684	2
#23781701	2
#23781661	2
#23781659	2
C₂×C₄×C₁₆₃⋊C₈	2
C₄×#41×C₁₆₃	2
C₄×#135	163

Subgroups — C4×#19104755 #35223306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19104755 #35223306