Subgroups — #24916475

Order 10624.

406 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13050590	C₂	1	1	yes	maximal, characteristic
2	#19127828	C₂	1	2	yes	maximal, complemented
3	#19127775	C₂	1	4	yes	maximal, complemented
4	#19127766	C₂	1	2	yes	maximal, complemented
5	#19127756	C₂	1	2	yes	maximal, complemented
6	C₄×#13050574	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×D₈₃	C₂	1	2	yes	maximal, complemented
8	#432×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13050590	C₄	1	1	yes	characteristic
10	C₂×#13050574	C₂²	1	2	yes	—
11	C₂×C₈×D₈₃	C₂²	1	4	yes	—
12	C₄×Dic₁₆₆	C₄	1	1	yes	characteristic
13	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
14	C₄×D₃₃₂	C₄	1	1	yes	characteristic
15	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₈₃	C₄	1	2	yes	—
17	C₄×C₂²⋊D₈₃	C₂²	1	2	yes	—
18	#14581751	C₄	1	1	yes	characteristic
19	#14581749	C₄	1	2	yes	—
20	C₄²×D₈₃	C₂²	1	2	yes	—
21	Dic₈₃⋊C₈	C₂²	1	4	yes	—
22	(C₂×D₈₃)⋊C₈	C₂²	1	4	yes	—
23	#14581703	C₂²	1	1	yes	characteristic
24	#14581703	C₂²	1	1	yes	characteristic
25	#14581702	C₂²	1	4	yes	—
26	C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
30	#97×C₈₃	C₂²	1	2	yes	—
31	#13050590	—	2	4	no	—
32	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
33	C₂×D₃₃₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₈₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₈₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₈₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₈₃	C₂³	1	2	yes	—
38	#13050574	—	2	4	no	—
39	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
40	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
41	C₈×D₈₃	—	2	8	no	—
42	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₈₃	C₂×C₄	1	1	yes	characteristic
44	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
45	Dic₈₃⋊C₄	C₂×C₄	1	2	yes	—
46	C₂²⋊Dic₈₃	C₂×C₄	1	2	yes	—
47	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
48	(C₂×D₈₃)⋊C₄	C₂×C₄	1	2	yes	—
49	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
50	C₄×Dic₈₃	C₂×C₄	1	2	yes	—
51	C₄×Dic₈₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂³	1	4	yes	—
56	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₈₃	C₂³	1	1	yes	characteristic
59	C₄²×C₈₃	C₂³	1	1	yes	characteristic
60	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₈₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₈₃	—	2	2	no	—
63	C₂×C₄×C₈₃	—	2	2	no	—
64	C₂×C₄×C₈₃	—	4	1	no	—
65	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₈₃	—	2	4	no	—
72	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₈₃	C₂²×C₄	1	2	yes	—
75	D₃₃₂	M₁₆	1	4	yes	complemented
76	C₄×D₈₃	—	2	4	no	—
77	C₄×D₈₃	—	2	4	no	—
78	C₄×D₈₃	—	4	2	no	—
79	Dic₁₆₆	—	2	2	no	—
80	C₈×C₈₃	—	2	4	no	—
81	C₈₃⋊C₈	—	2	2	no	—
82	C₈₃⋊C₈	#41	1	4	yes	complemented
83	C₂²×C₈₃	—	2	1	no	—
84	C₂²×C₈₃	—	2	1	no	—
85	C₂²×C₈₃	—	2	1	no	—
86	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₈₃	—	2	2	no	—
88	C₂×D₈₃	—	2	2	no	—
89	C₂×D₈₃	C₂×M₁₆	1	4	yes	—
90	C₄×C₈₃	—	2	1	no	—
91	C₄×C₈₃	—	2	1	no	—
92	C₄×C₈₃	—	2	1	no	—
93	C₄×C₈₃	—	2	1	no	—
94	C₄×C₈₃	C₂×M₁₆	1	2	yes	—
95	C₄×C₈₃	C₂×#41	1	2	yes	—
96	Dic₈₃	—	2	2	no	—
97	Dic₈₃	—	2	2	no	—
98	Dic₈₃	—	2	2	no	—
99	C₂×C₈₃	—	4	1	no	—
100	C₂×C₈₃	C₄×#41	1	1	yes	characteristic
101	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₈₃	#568	1	1	yes	characteristic
103	D₈₃	—	2	4	no	—
104	#2970	—	83	1	no	maximal, Sylow
105	C₈₃	#2970	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	—	83	1	no	—
107	#446	—	83	2	no	—
108	#435	—	83	4	no	—
109	C₈×D₄	—	83	2	no	—
110	#432	—	83	2	no	—
111	#432	D₈₃	1	1	yes	characteristic, complemented
112	C₄×M₁₆	—	83	1	no	—
113	C₂×#97	—	83	2	no	—
114	C₂×#41	—	83	1	no	—
115	C₂×M₁₆	—	83	2	no	—
116	C₂²×C₈	—	83	4	no	—
117	C₄×Q₈	—	83	1	no	—
118	C₄×Q₈	—	83	1	no	—
119	C₄×D₄	—	83	2	no	—
120	C₄×D₄	—	83	2	no	—
121	C₄×D₄	—	83	1	no	—
122	C₄×D₄	—	83	1	no	—
123	#117	—	83	1	no	—
124	#117	—	83	2	no	—
125	C₂×C₄²	—	83	2	no	—
126	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
127	C₄⋊C₈	—	83	4	no	—
128	C₄⋊C₈	C₂×D₈₃	1	2	yes	—
129	C₂²⋊C₈	—	83	4	no	—
130	C₂²⋊C₈	C₂×D₈₃	1	2	yes	—
131	#97	—	83	4	no	—
132	#97	—	83	1	no	—
133	#97	—	83	1	no	—
134	#97	C₂×D₈₃	1	2	yes	—
135	C₄×C₈	—	83	2	no	—
136	#41	—	166	4	no	—
137	C₂×Q₈	—	83	1	no	—
138	C₂×D₄	—	83	2	no	—
139	C₂×D₄	—	83	1	no	—
140	C₂²×C₄	—	83	2	no	—
141	C₂²×C₄	—	83	2	no	—
142	C₂²×C₄	—	83	2	no	—
143	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
146	M₁₆	—	166	4	no	—
147	C₂×C₈	—	83	2	no	—
148	C₂×C₈	—	83	2	no	—
149	C₂×C₈	—	166	8	no	—
150	C₂×C₈	C₂²×D₈₃	1	4	yes	—
151	C₄⋊C₄	—	83	2	no	—
152	C₄⋊C₄	—	83	2	no	—
153	C₄⋊C₄	—	83	1	no	—
154	C₄⋊C₄	—	83	1	no	—
155	C₂²⋊C₄	—	83	2	no	—
156	C₂²⋊C₄	—	83	2	no	—
157	C₂²⋊C₄	—	83	2	no	—
158	C₄²	—	83	2	no	—
159	C₄²	—	83	2	no	—
160	C₄²	—	83	2	no	—
161	C₄²	C₄×D₈₃	1	1	yes	characteristic
162	C₄²	C₄×D₈₃	1	1	yes	characteristic
163	C₄²	C₂²×D₈₃	1	1	yes	characteristic
164	C₄²	C₂²×D₈₃	1	1	yes	characteristic
165	C₂³	—	83	2	no	—
166	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
167	Q₈	—	166	2	no	—
168	D₄	—	83	4	no	—
169	D₄	—	166	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	332	2	no	—
174	C₂×C₄	—	83	2	no	—
175	C₂×C₄	—	83	2	no	—
176	C₂×C₄	—	83	2	no	—
177	C₂×C₄	—	166	4	no	—
178	C₂×C₄	—	166	4	no	—
179	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	83	4	no	—
187	C₈	—	166	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	83	4	no	—
192	C₂²	—	166	2	no	—
193	C₂²	—	166	2	no	—
194	C₂²	C₂²×C₄×D₈₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	166	2	no	—
200	C₄	—	166	2	no	—
201	C₄	—	166	2	no	—
202	C₄	M₁₆×D₈₃	1	2	yes	—
203	C₄	#41×D₈₃	1	2	yes	—
204	C₂	—	4	1	no	—
205	C₂	—	166	4	no	—
206	C₂	#19128169	1	1	yes	characteristic
207	C₂	C₂×M₁₆×D₈₃	1	1	yes	characteristic
208	C₂	#19128467	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₃
commutator, socle	C₂²×C₈₃	C₂³×C₄
Fitting	#432×C₈₃	C₂
prime radical	C₂×#13050590	C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	#432	D₈₃
hypocenter	C₈₃	#2970
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #19127828	C₂	semidirect
3. #19127775	C₂	semidirect
4. #19127766	C₂	semidirect
5. #19127756	C₂	semidirect
6. C₄×#13050574	C₂	semidirect
7. #97×D₈₃	C₂	semidirect
8. #432×C₈₃	C₂	semidirect
75. D₃₃₂	M₁₆	semidirect
82. C₈₃⋊C₈	#41	semidirect
105. C₈₃	#2970	semidirect
111. #432	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13050590	2
#19127828	2
#19127775	2
#19127766	2
#19127756	2
C₄×#13050574	2
#97×D₈₃	2
#432×C₈₃	2
#2970	83