Subgroups — C₄×#14581840 #24920675

Order 10624.

1492 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581840	C₂	1	1	yes	maximal, characteristic
2	#19128183	C₂	1	4	yes	maximal, complemented
3	#19128182	C₂	1	6	yes	maximal, complemented
4	#19128179	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128177	C₂	1	4	yes	maximal, complemented
6	C₄×#13050591	C₂	1	4	yes	maximal, complemented
7	C₄×#13050590	C₂	1	6	yes	maximal, complemented
8	C₄×Q₈×D₈₃	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#14581840	C₄	1	16	yes	direct product factor
11	C₂×#13050591	C₂²	1	4	yes	—
12	C₂×#13050590	C₂²	1	6	yes	—
13	C₂×Q₈×D₈₃	C₂²	1	4	yes	—
14	Q₈×Dic₈₃	C₂²	1	4	yes	—
15	Q₈×Dic₈₃	C₂²	1	4	yes	—
16	C₄×Dic₁₆₆	C₂²	1	6	yes	—
17	C₄×Dic₁₆₆	C₂²	1	6	yes	—
18	#14581757	C₂²	1	12	yes	—
19	C₄×D₃₃₂	C₂²	1	6	yes	—
20	C₄×D₃₃₂	C₂²	1	6	yes	—
21	#14581754	C₂²	1	12	yes	—
22	C₄×C₂²⋊D₈₃	C₂²	1	4	yes	—
23	C₄×C₂²⋊D₈₃	C₂²	1	12	yes	—
24	#14581751	C₂²	1	6	yes	—
25	#14581750	C₂²	1	12	yes	—
26	#14581749	C₂²	1	12	yes	—
27	C₄⋊C₄×D₈₃	C₂²	1	12	yes	—
28	C₄²×D₈₃	C₂²	1	12	yes	—
29	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₈₃	C₂²	1	4	yes	—
31	C₄×Q₈×C₈₃	C₂²	1	4	yes	—
32	C₂×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
33	C₂×C₄²×C₈₃	C₂²	1	3	yes	—
34	#13050591	C₂×C₄	1	32	yes	complemented
35	#13050590	C₂×C₄	1	48	yes	complemented
36	C₂×Dic₁₆₆	C₂³	1	6	yes	—
37	Q₈×D₈₃	C₂×C₄	1	32	yes	complemented
38	C₂×D₃₃₂	C₂³	1	6	yes	—
39	C₂×C₂²⋊D₈₃	C₂³	1	4	yes	—
40	C₂×C₄×D₈₃	C₂³	1	4	yes	—
41	C₂×C₄×D₈₃	C₂³	1	12	yes	—
42	C₂×C₄×D₈₃	C₂³	1	12	yes	—
43	C₄⋊Dic₈₃	C₂³	1	6	yes	—
44	C₄⋊Dic₈₃	C₂³	1	6	yes	—
45	Dic₈₃⋊C₄	C₂³	1	12	yes	—
46	Dic₈₃⋊C₄	C₂³	1	4	yes	—
47	C₂²⋊Dic₈₃	C₂³	1	4	yes	—
48	(C₂×D₈₃)⋊C₄	C₂³	1	4	yes	—
49	(C₂×D₈₃)⋊C₄	C₂³	1	12	yes	—
50	C₄×Dic₈₃	C₂³	1	4	yes	—
51	C₄×Dic₈₃	C₂³	1	12	yes	—
52	C₄×Dic₈₃	C₂³	1	12	yes	—
53	C₂×Q₈×C₈₃	C₂×C₄	1	8	yes	complemented
54	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
55	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
57	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
58	C₄⋊C₄×C₈₃	C₂³	1	12	yes	—
59	C₄²×C₈₃	C₂³	1	6	yes	—
60	C₄²×C₈₃	C₂³	1	6	yes	—
61	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
62	C₂²×D₈₃	C₂⁴	1	4	yes	—
63	Q₈×C₈₃	C₂²×C₄	1	16	yes	—
64	C₂×C₄×C₈₃	—	2	2	no	—
65	C₂×C₄×C₈₃	—	4	3	no	—
66	C₂×C₄×C₈₃	C₂²×C₄	1	12	yes	—
67	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
69	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
70	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₈₃	C₂²×C₄	1	16	yes	—
72	C₂×Dic₈₃	C₂⁴	1	4	yes	—
73	C₂×Dic₈₃	C₂⁴	1	4	yes	—
74	C₂×Dic₈₃	C₂⁴	1	4	yes	—
75	D₃₃₂	C₂²×C₄	1	24	yes	—
76	C₄×D₈₃	—	2	8	no	—
77	C₄×D₈₃	—	4	12	no	—
78	C₄×D₈₃	C₂²×C₄	1	48	yes	—
79	Dic₁₆₆	C₂²×C₄	1	24	yes	—
80	C₂²×C₈₃	—	2	1	no	—
81	C₂²×C₈₃	—	2	1	no	—
82	C₂²×C₈₃	C₂³×C₄	1	2	yes	—
83	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
84	C₂×D₈₃	—	2	4	no	—
85	C₂×D₈₃	—	2	4	no	—
86	C₂×D₈₃	C₂³×C₄	1	8	yes	—
87	C₄×C₈₃	—	2	1	no	—
88	C₄×C₈₃	—	2	6	no	—
89	C₄×C₈₃	C₂³×C₄	1	12	yes	—
90	C₄×C₈₃	2₋¹⁺⁴	1	2	yes	complemented
91	Dic₈₃	—	2	4	no	—
92	Dic₈₃	—	2	4	no	—
93	Dic₈₃	C₂³×C₄	1	8	yes	—
94	C₂×C₈₃	—	2	2	no	—
95	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
96	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×C₈₃	#585	1	1	yes	characteristic
98	D₈₃	—	2	8	no	—
99	C₄×2₋¹⁺⁴	—	83	1	no	maximal, Sylow
100	C₈₃	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
101	C₂×2₋¹⁺⁴	—	83	1	no	—
102	#520	—	83	4	no	—
103	#520	—	83	6	no	—
104	#519	—	83	1	no	—
105	#519	—	83	4	no	—
106	C₄×#41	—	83	4	no	—
107	C₄×#41	—	83	6	no	—
108	C₂×C₄×Q₈	—	83	4	no	—
109	C₂×C₄×Q₈	D₈₃	1	1	yes	characteristic, complemented
110	2₋¹⁺⁴	—	83	16	no	—
111	C₂×#41	—	83	4	no	—
112	C₂×#41	—	83	6	no	—
113	C₂²×Q₈	—	83	4	no	—
114	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
115	C₄×Q₈	—	83	4	no	—
116	C₄×Q₈	—	83	4	no	—
117	C₄×Q₈	—	83	12	no	—
118	C₄×Q₈	—	83	6	no	—
119	C₄×Q₈	—	83	6	no	—
120	C₄×Q₈	C₂×D₈₃	1	4	yes	—
121	C₄×Q₈	C₂×D₈₃	1	4	yes	—
122	C₄×D₄	—	83	4	no	—
123	C₄×D₄	—	83	12	no	—
124	C₄×D₄	—	83	12	no	—
125	C₄×D₄	—	83	6	no	—
126	C₄×D₄	—	83	6	no	—
127	#117	—	83	6	no	—
128	#117	—	83	12	no	—
129	#117	—	83	12	no	—
130	C₂×C₄⋊C₄	—	83	12	no	—
131	C₂×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
132	C₂×C₄²	—	83	12	no	—
133	C₂×C₄²	C₂×D₈₃	1	3	yes	—
134	#41	—	83	32	no	—
135	#41	—	83	48	no	—
136	C₂×Q₈	—	83	32	no	—
137	C₂×Q₈	—	83	6	no	—
138	C₂×Q₈	C₄×D₈₃	1	8	yes	complemented
139	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
140	C₂×D₄	—	83	4	no	—
141	C₂×D₄	—	83	6	no	—
142	C₂²×C₄	—	83	12	no	—
143	C₂²×C₄	—	83	12	no	—
144	C₂²×C₄	—	83	4	no	—
145	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
147	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
148	C₄⋊C₄	—	83	12	no	—
149	C₄⋊C₄	—	83	6	no	—
150	C₄⋊C₄	—	83	4	no	—
151	C₄⋊C₄	—	83	6	no	—
152	C₄⋊C₄	C₂²×D₈₃	1	12	yes	—
153	C₂²⋊C₄	—	83	4	no	—
154	C₂²⋊C₄	—	83	12	no	—
155	C₂²⋊C₄	—	83	4	no	—
156	C₄²	—	83	12	no	—
157	C₄²	—	83	12	no	—
158	C₄²	—	83	4	no	—
159	C₄²	C₂²×D₈₃	1	6	yes	—
160	C₄²	C₂²×D₈₃	1	6	yes	—
161	C₂³	—	83	4	no	—
162	C₂³	C₂³×D₈₃	1	1	yes	characteristic
163	Q₈	—	83	24	no	—
164	Q₈	C₂×C₄×D₈₃	1	16	yes	—
165	D₄	—	83	16	no	—
166	D₄	—	83	24	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	3	no	—
169	C₂×C₄	—	332	12	no	—
170	C₂×C₄	—	83	4	no	—
171	C₂×C₄	—	83	48	no	—
172	C₂×C₄	—	83	4	no	—
173	C₂×C₄	—	83	4	no	—
174	C₂×C₄	—	166	8	no	—
175	C₂×C₄	C₂×C₄×D₈₃	1	12	yes	—
176	C₂×C₄	C₂³×D₈₃	1	6	yes	—
177	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
178	C₂×C₄	C₂³×D₈₃	1	6	yes	—
179	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	83	8	no	—
183	C₂²	—	166	4	no	—
184	C₂²	—	166	4	no	—
185	C₂²	C₂²×C₄×D₈₃	1	2	yes	—
186	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	6	no	—
189	C₄	—	83	8	no	—
190	C₄	—	166	4	no	—
191	C₄	—	166	4	no	—
192	C₄	C₂²×C₄×D₈₃	1	12	yes	—
193	C₄	#14581840	1	2	yes	direct product factor
194	C₂	—	2	2	no	—
195	C₂	—	166	8	no	—
196	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic
197	C₂	C₂×#14581840	1	1	yes	characteristic
198	C₂	#19128537	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₈₃
commutator	C₂×C₈₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₈₃
Fitting	C₂×C₄×Q₈×C₈₃	C₂
socle, cosocle	C₂²×C₈₃	C₂⁵
prime radical	C₂×#14581840	C₂
hypercenter	C₂×C₄×Q₈	D₈₃
hypocenter	C₈₃	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #19128183	C₂	semidirect
3. #19128182	C₂	semidirect
4. #19128179	C₂	semidirect
5. #19128177	C₂	semidirect
6. C₄×#13050591	C₂	semidirect
7. C₄×#13050590	C₂	semidirect
8. C₄×Q₈×D₈₃	C₂	semidirect
9. C₂×C₄×Q₈×C₈₃	C₂	semidirect
10. #14581840	C₄	direct (N × H)
34. #13050591	C₂×C₄	semidirect
35. #13050590	C₂×C₄	semidirect
37. Q₈×D₈₃	C₂×C₄	semidirect
53. C₂×Q₈×C₈₃	C₂×C₄	semidirect
90. C₄×C₈₃	2₋¹⁺⁴	semidirect
100. C₈₃	C₄×2₋¹⁺⁴	semidirect
109. C₂×C₄×Q₈	D₈₃	semidirect
138. C₂×Q₈	C₄×D₈₃	semidirect
193. C₄	#14581840	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×#14581840	2
#19128183	2
#19128182	2
#19128179	2
#19128177	2
C₄×#13050591	2
C₄×#13050590	2
C₄×Q₈×D₈₃	2
C₂×C₄×Q₈×C₈₃	2
C₄×2₋¹⁺⁴	83

Subgroups — C4×#14581840 #24920675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14581840 #24920675