Subgroups — #24916157

Order 10624.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581798	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128212	C₂	1	1	yes	maximal, characteristic
3	#19127772	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₈₃	C₂	1	4	yes	maximal, complemented
5	#19127727	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127667	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127665	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127656	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13050590	C₂²	1	1	yes	characteristic
11	#14581798	—	2	4	no	—
12	C₂×#13050574	C₂²	1	1	yes	characteristic
13	C₂×#13050574	C₂²	1	1	yes	characteristic
14	C₂×#13050573	C₂²	1	2	yes	complemented
15	C₂×C₈×D₈₃	C₂²	1	2	yes	complemented
16	#14581775	C₄	1	2	yes	—
17	#14581771	C₄	1	2	yes	—
18	#14581766	C₄	1	2	yes	—
19	#14581761	C₄	1	2	yes	—
20	C₄×C₂²⋊D₈₃	C₂²	1	4	yes	—
21	#14581751	C₂²	1	1	yes	characteristic
22	Dic₈₃⋊C₈	C₂²	1	4	yes	complemented
23	C₂²⋊(C₈₃⋊C₈)	C₂²	1	4	yes	—
24	(C₂×D₈₃)⋊C₈	C₂²	1	4	yes	complemented
25	#14581702	C₂²	1	2	yes	complemented
26	C₈×Dic₈₃	C₂²	1	2	yes	complemented
27	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	—	2	4	no	—
29	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
31	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
32	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
33	#13050590	—	2	2	no	—
34	#13050590	—	2	2	no	—
35	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
36	C₂×D₃₃₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊D₈₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊D₈₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₈₃	C₂³	1	2	yes	—
40	#13050574	—	2	2	no	—
41	#13050574	—	2	2	no	—
42	#13050573	—	2	4	no	—
43	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
44	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
45	C₈×D₈₃	—	2	4	no	—
46	C₄⋊Dic₈₃	C₂×C₄	1	2	yes	—
47	Dic₈₃⋊C₄	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
49	C₂²⋊Dic₈₃	C₂×C₄	1	2	yes	—
50	(C₂×D₈₃)⋊C₄	C₂×C₄	1	4	yes	—
51	C₄×Dic₈₃	C₂³	1	2	yes	—
52	C₂⁴×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	—	2	2	no	—
54	C₂²×C₄×C₈₃	—	2	2	no	—
55	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
56	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₈₃	—	2	2	no	—
60	C₂×C₈×C₈₃	—	2	4	no	—
61	C₂×C₈×C₈₃	—	2	4	no	—
62	C₂×C₈×C₈₃	D₄	1	4	yes	complemented
63	C₂×C₈×C₈₃	C₂³	1	2	yes	—
64	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
66	C₂³×C₈₃	—	2	2	no	—
67	C₂³×C₈₃	—	2	4	no	—
68	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂³×C₈₃	C₂²×C₄	1	2	yes	—
70	C₂²×D₈₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₈₃	—	2	2	no	—
72	C₂×C₄×C₈₃	—	2	4	no	—
73	C₂×C₄×C₈₃	—	2	2	no	—
74	C₂×C₄×C₈₃	—	2	2	no	—
75	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₈₃	#41	1	2	yes	—
79	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₈₃	—	2	4	no	—
81	C₂²⋊D₈₃	—	2	8	no	—
82	C₂×Dic₈₃	C₂²×C₄	1	4	yes	—
83	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
84	D₃₃₂	—	2	2	no	—
85	C₄×D₈₃	—	2	2	no	—
86	C₄×D₈₃	—	2	2	no	—
87	Dic₁₆₆	—	2	2	no	—
88	C₈×C₈₃	—	2	2	no	—
89	C₈×C₈₃	C₂×D₄	1	2	yes	—
90	C₈×C₈₃	C₂×D₄	1	2	yes	—
91	C₈₃⋊C₈	—	2	2	no	—
92	C₈₃⋊C₈	—	2	2	no	—
93	C₂²×C₈₃	—	2	8	no	—
94	C₂²×C₈₃	—	2	2	no	—
95	C₂²×C₈₃	—	2	2	no	—
96	C₂²×C₈₃	—	2	2	no	—
97	C₂²×C₈₃	C₄×D₄	1	2	yes	—
98	C₂²×C₈₃	#131	1	4	yes	complemented
99	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
100	C₂×D₈₃	—	2	2	no	—
101	C₂×D₈₃	—	2	4	no	—
102	C₄×C₈₃	—	2	2	no	—
103	C₄×C₈₃	C₄×D₄	1	2	yes	—
104	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
106	Dic₈₃	—	2	2	no	—
107	Dic₈₃	—	2	4	no	—
108	C₂×C₈₃	—	2	2	no	—
109	C₂×C₈₃	—	2	4	no	—
110	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₈₃	C₂×#131	1	2	yes	—
112	D₈₃	—	4	2	no	—
113	#2929	—	83	1	no	maximal, Sylow
114	C₈₃	#2929	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	83	1	no	—
116	C₂³×C₈	D₈₃	1	1	yes	characteristic, complemented
117	#525	—	83	1	no	—
118	#435	—	83	4	no	—
119	C₈×D₄	—	83	4	no	—
120	#424	—	83	1	no	—
121	#408	—	83	1	no	—
122	#407	—	83	1	no	—
123	#405	—	83	1	no	—
124	C₂×#41	—	83	1	no	—
125	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
126	#131	—	166	4	no	—
127	C₂×M₁₆	—	83	1	no	—
128	C₂×M₁₆	—	83	1	no	—
129	C₂×M₁₆	—	83	2	no	—
130	C₂²×C₈	—	2	4	no	—
131	C₂²×C₈	—	83	2	no	—
132	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
133	C₂²×C₈	C₂×D₈₃	1	2	yes	—
134	C₂²×C₈	C₂×D₈₃	1	2	yes	—
135	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
136	#123	—	83	2	no	—
137	C₂²⋊Q₈	—	83	2	no	—
138	#121	—	83	2	no	—
139	#120	—	83	2	no	—
140	C₄×D₄	—	83	4	no	—
141	#117	—	83	1	no	—
142	C₄⋊C₈	—	83	4	no	—
143	C₂²⋊C₈	—	83	4	no	—
144	C₂²⋊C₈	—	83	4	no	—
145	#97	—	83	2	no	—
146	C₄×C₈	—	83	2	no	—
147	C₂⁴	C₄×D₈₃	1	1	yes	characteristic
148	#41	—	166	2	no	—
149	#41	—	166	2	no	—
150	C₂×Q₈	—	83	1	no	—
151	C₂×D₄	—	83	4	no	—
152	C₂×D₄	—	83	2	no	—
153	C₂×D₄	—	83	1	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	83	2	no	—
157	C₂²×C₄	C₄×D₈₃	1	2	yes	—
158	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
161	M₁₆	—	166	2	no	—
162	M₁₆	—	166	2	no	—
163	M₁₆	—	166	4	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	83	2	no	—
168	C₂×C₈	—	83	2	no	—
169	C₂×C₈	—	166	4	no	—
170	C₂×C₈	C₂²⋊D₈₃	1	4	yes	complemented
171	C₂×C₈	C₂²×D₈₃	1	2	yes	—
172	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
174	C₄⋊C₄	—	83	2	no	—
175	C₄⋊C₄	—	83	4	no	—
176	C₂²⋊C₄	—	83	2	no	—
177	C₂²⋊C₄	—	83	4	no	—
178	C₂²⋊C₄	—	83	4	no	—
179	C₄²	—	83	2	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	83	2	no	—
183	C₂³	C₂×C₄×D₈₃	1	2	yes	—
184	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
185	Q₈	—	166	2	no	—
186	D₄	—	166	8	no	—
187	D₄	—	166	4	no	—
188	D₄	—	166	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	83	4	no	—
194	C₂×C₄	—	83	2	no	—
195	C₂×C₄	—	166	2	no	—
196	C₂×C₄	—	166	2	no	—
197	C₂×C₄	C₂×C₄×D₈₃	1	2	yes	—
198	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
200	C₂×C₄	#13050590	1	2	yes	—
201	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	166	2	no	—
204	C₈	—	166	2	no	—
205	C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
206	C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	166	2	no	—
212	C₂²	—	166	4	no	—
213	C₂²	C₄×C₂²⋊D₈₃	1	2	yes	—
214	C₂²	#14581798	1	4	yes	complemented
215	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	166	2	no	—
218	C₄	—	166	4	no	—
219	C₄	C₄×C₂²⋊D₈₃	1	2	yes	—
220	C₄	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
221	C₄	C₂×#13050590	1	1	yes	characteristic
222	C₂	—	2	4	no	—
223	C₂	—	2	2	no	—
224	C₂	—	332	2	no	—
225	C₂	C₂×C₄×C₂²⋊D₈₃	1	1	yes	characteristic
226	C₂	C₂×#14581798	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂³×C₈×C₈₃	C₂
prime radical	#19128212	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂³×C₈	D₈₃
hypocenter	C₈₃	#2929
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14581798	C₂	semidirect
3. #19127772	C₂	semidirect
4. C₈×C₂²⋊D₈₃	C₂	semidirect
5. #19127727	C₂	semidirect
6. #19127667	C₂	semidirect
7. #19127665	C₂	semidirect
8. #19127656	C₂	semidirect
9. C₂³×C₈×C₈₃	C₂	semidirect
14. C₂×#13050573	C₂²	semidirect
15. C₂×C₈×D₈₃	C₂²	semidirect
22. Dic₈₃⋊C₈	C₂²	semidirect
24. (C₂×D₈₃)⋊C₈	C₂²	semidirect
25. #14581702	C₂²	semidirect
26. C₈×Dic₈₃	C₂²	semidirect
62. C₂×C₈×C₈₃	D₄	semidirect
98. C₂²×C₈₃	#131	semidirect
114. C₈₃	#2929	semidirect
116. C₂³×C₈	D₈₃	semidirect
170. C₂×C₈	C₂²⋊D₈₃	semidirect
214. C₂²	#14581798	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581798	2
#19128212	2
#19127772	2
C₈×C₂²⋊D₈₃	2
#19127727	2
#19127667	2
#19127665	2
#19127656	2
C₂³×C₈×C₈₃	2
#2929	83