Subgroups — #35223182

Order 20864.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579699	C₂	1	1	yes	maximal, characteristic
2	#23781725	C₂	1	2	yes	maximal, complemented
3	#23781723	C₂	1	2	yes	maximal, complemented
4	#23781722	C₂	1	1	yes	maximal, characteristic, complemented
5	#23781720	C₂	1	1	yes	maximal, characteristic, complemented
6	#23781720	C₂	1	1	yes	maximal, characteristic, complemented
7	#23781690	C₂	1	2	yes	maximal, complemented
8	#23781630	C₂	1	2	yes	maximal
9	#23781617	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579699	C₂²	1	1	yes	characteristic
13	C₂×#14579682	C₂²	1	2	yes	—
14	C₂×C₈×D₁₆₃	C₂²	1	2	yes	—
15	C₄×Dic₃₂₆	C₄	1	2	yes	—
16	C₄×D₆₅₂	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₆₃	C₄	1	2	yes	—
18	C₄×C₂²⋊D₁₆₃	C₄	1	2	yes	—
19	#19104694	C₂²	1	1	yes	characteristic
20	#19104692	C₂²	1	1	yes	characteristic
21	#19104692	C₂²	1	1	yes	characteristic
22	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
23	C₄²×D₁₆₃	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
25	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
26	Dic₁₆₃⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊(C₁₆₃⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₁₆₃)⋊C₈	C₂²	1	2	yes	—
30	#19104646	C₂²	1	1	yes	characteristic
31	#19104646	C₂²	1	1	yes	characteristic
32	#19104645	C₂²	1	2	yes	—
33	C₈×Dic₁₆₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₆₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₆₃	C₂²	1	2	yes	—
37	#97×C₁₆₃	C₂²	1	1	yes	characteristic
38	#97×C₁₆₃	C₂²	1	1	yes	characteristic
39	#14579699	—	2	4	no	—
40	C₂×Dic₃₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂×D₆₅₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₆₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₆₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₆₃	C₂³	1	1	yes	characteristic
48	#14579682	—	2	4	no	—
49	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
50	C₂×C₁₆₃⋊C₈	C₂³	1	2	yes	—
51	C₈×D₁₆₃	—	2	4	no	—
52	C₄⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
53	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₆₃⋊C₄	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₆₃	C₂×C₄	1	2	yes	—
56	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₆₃)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₆₃	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₆₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₆₃	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
65	C₂×C₈×C₁₆₃	C₂³	1	2	yes	—
66	C₄²×C₁₆₃	C₂×C₄	1	2	yes	—
67	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
68	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
69	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₆₃	C₂²×C₄	1	1	yes	characteristic
71	Dic₃₂₆	—	2	2	no	—
72	D₆₅₂	—	2	2	no	—
73	C₂²⋊D₁₆₃	—	2	2	no	—
74	C₂²⋊D₁₆₃	—	2	2	no	—
75	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₆₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₆₃	C₂²×C₄	1	1	yes	characteristic
79	C₄×D₁₆₃	—	2	2	no	—
80	C₄×D₁₆₃	—	4	2	no	—
81	C₄×D₁₆₃	—	4	2	no	—
82	C₄×D₁₆₃	#41	1	4	yes	—
83	C₁₆₃⋊C₈	—	2	2	no	—
84	C₁₆₃⋊C₈	—	2	2	no	—
85	C₂³×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₆₃	—	2	2	no	—
87	C₂×C₄×C₁₆₃	—	4	2	no	—
88	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₆₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₆₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₆₃	—	2	2	no	—
93	C₈×C₁₆₃	—	2	2	no	—
94	C₂²×C₁₆₃	—	2	1	no	—
95	C₂²×C₁₆₃	—	2	2	no	—
96	C₂²×C₁₆₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₆₃	—	2	2	no	—
98	C₂×D₁₆₃	—	2	1	no	—
99	C₂×D₁₆₃	—	2	2	no	—
100	C₂×D₁₆₃	#117	1	2	yes	—
101	C₄×C₁₆₃	—	2	1	no	—
102	C₄×C₁₆₃	—	2	2	no	—
103	C₄×C₁₆₃	—	2	2	no	—
104	C₄×C₁₆₃	C₂×#41	1	2	yes	—
105	Dic₁₆₃	—	2	1	no	—
106	Dic₁₆₃	—	2	2	no	—
107	Dic₁₆₃	—	2	2	no	—
108	Dic₁₆₃	#117	1	2	yes	—
109	C₂×C₁₆₃	—	4	1	no	—
110	C₂×C₁₆₃	C₂×#117	1	1	yes	characteristic
111	C₂×C₁₆₃	#568	1	2	yes	—
112	D₁₆₃	—	2	2	no	—
113	D₁₆₃	—	4	1	no	—
114	C₁₆₃	#2923	1	1	yes	characteristic, complemented, Sylow
115	#2923	—	163	1	no	maximal, Sylow
116	C₄×#41	—	163	1	no	—
117	#433	—	163	2	no	—
118	#433	—	163	2	no	—
119	#432	—	163	1	no	—
120	#432	—	163	1	no	—
121	#432	—	163	1	no	—
122	#432	D₁₆₃	1	1	yes	characteristic, complemented
123	#424	—	163	2	no	—
124	#408	—	163	2	no	—
125	C₄×M₁₆	—	163	1	no	—
126	C₂×#97	—	163	1	no	—
127	C₂×#41	—	163	1	no	—
128	C₂×M₁₆	—	163	2	no	—
129	C₂²×C₈	—	163	2	no	—
130	C₄×Q₈	—	163	2	no	—
131	C₄×D₄	—	163	2	no	—
132	C₄×D₄	—	163	2	no	—
133	C₄×D₄	—	163	2	no	—
134	#117	—	163	1	no	—
135	#117	—	163	1	no	—
136	#117	—	163	1	no	—
137	C₂×C₄²	—	163	1	no	—
138	C₂×C₄²	—	163	1	no	—
139	C₂×C₄²	C₂×D₁₆₃	1	1	yes	characteristic
140	C₄⋊C₈	—	163	2	no	—
141	C₄⋊C₈	—	163	2	no	—
142	C₄⋊C₈	—	163	2	no	—
143	C₄⋊C₈	C₂×D₁₆₃	1	2	yes	—
144	C₂²⋊C₈	—	163	2	no	—
145	C₂²⋊C₈	—	163	2	no	—
146	C₂²⋊C₈	—	163	2	no	—
147	C₂²⋊C₈	C₂×D₁₆₃	1	2	yes	—
148	#97	—	163	2	no	—
149	#97	—	163	1	no	—
150	#97	—	163	1	no	—
151	#97	C₂×D₁₆₃	1	1	yes	characteristic
152	#97	C₂×D₁₆₃	1	1	yes	characteristic
153	C₄×C₈	—	163	2	no	—
154	#41	—	326	4	no	—
155	C₂×Q₈	—	163	1	no	—
156	C₂×D₄	—	163	1	no	—
157	C₂×D₄	—	163	1	no	—
158	C₂×D₄	—	163	1	no	—
159	C₂²×C₄	—	163	1	no	—
160	C₂²×C₄	—	163	2	no	—
161	C₂²×C₄	—	163	2	no	—
162	C₂²×C₄	—	163	1	no	—
163	C₂²×C₄	C₄×D₁₆₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
165	M₁₆	—	326	4	no	—
166	C₂×C₈	—	326	4	no	—
167	C₂×C₈	—	163	2	no	—
168	C₂×C₈	—	163	2	no	—
169	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
170	C₂×C₈	C₂²×D₁₆₃	1	2	yes	—
171	C₄⋊C₄	—	163	2	no	—
172	C₄⋊C₄	—	163	2	no	—
173	C₄⋊C₄	—	163	2	no	—
174	C₂²⋊C₄	—	163	2	no	—
175	C₂²⋊C₄	—	163	2	no	—
176	C₂²⋊C₄	—	163	2	no	—
177	C₄²	—	163	2	no	—
178	C₄²	—	163	2	no	—
179	C₄²	—	163	1	no	—
180	C₄²	—	163	1	no	—
181	C₄²	C₄×D₁₆₃	1	2	yes	—
182	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
183	C₄²	C₂²×D₁₆₃	1	1	yes	characteristic
184	C₂³	—	163	1	no	—
185	C₂³	—	163	1	no	—
186	C₂³	C₂×C₄×D₁₆₃	1	1	yes	characteristic
187	Q₈	—	326	2	no	—
188	D₄	—	326	2	no	—
189	D₄	—	326	2	no	—
190	D₄	—	326	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	326	2	no	—
194	C₂×C₄	—	652	2	no	—
195	C₂×C₄	—	652	2	no	—
196	C₂×C₄	—	163	2	no	—
197	C₂×C₄	—	163	2	no	—
198	C₂×C₄	—	163	1	no	—
199	C₂×C₄	—	163	1	no	—
200	C₂×C₄	—	163	4	no	—
201	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₁₆₃	1	2	yes	—
204	C₂×C₄	C₂³×D₁₆₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	326	2	no	—
208	C₈	—	326	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	326	2	no	—
212	C₂²	—	326	2	no	—
213	C₂²	—	326	1	no	—
214	C₂²	—	163	2	no	—
215	C₂²	C₂²×C₄×D₁₆₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	326	2	no	—
220	C₄	—	326	2	no	—
221	C₄	—	326	1	no	—
222	C₄	—	163	2	no	—
223	C₄	#41×D₁₆₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	326	2	no	—
226	C₂	—	652	1	no	—
227	C₂	#117×D₁₆₃	1	1	yes	characteristic
228	C₂	#23782431	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₆₃
commutator, socle	C₂²×C₁₆₃	C₂³×C₄
Fitting	#432×C₁₆₃	C₂
prime radical	C₂×#14579699	C₂²
cosocle	C₂×C₄×C₁₆₃	C₂⁴
hypercenter	#432	D₁₆₃
hypocenter	C₁₆₃	#2923
LCS	G ▷ C₂²×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #23781725	C₂	semidirect
3. #23781723	C₂	semidirect
4. #23781722	C₂	semidirect
5. #23781720	C₂	semidirect
6. #23781720	C₂	semidirect
7. #23781690	C₂	semidirect
9. #23781617	C₂	semidirect
10. #97×D₁₆₃	C₂	semidirect
11. #432×C₁₆₃	C₂	semidirect
114. C₁₆₃	#2923	semidirect
122. #432	D₁₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579699	2
#23781725	2
#23781723	2
#23781722	2
#23781720	2
#23781690	2
#23781630	2
#23781617	2
#97×D₁₆₃	2
#432×C₁₆₃	2
#2923	163