Subgroups — C₂×#19127779 #24916150

Order 10624.

552 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13050574	C₂	1	2	yes	maximal, complemented
2	#19127779	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊(C₈₃⋊C₈)	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13050574	—	2	8	no	—
8	C₂×#13050574	C₂²	1	8	yes	complemented
9	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₈₃⋊C₈)	C₂²	1	4	yes	complemented
13	C₂²⋊(C₈₃⋊C₈)	C₂²	1	8	yes	complemented
14	C₄×C₈₃⋊C₈	C₂²	1	4	yes	complemented
15	C₂²×D₄×C₈₃	C₄	1	1	yes	characteristic
16	C₂³×C₄×C₈₃	C₂²	1	2	yes	—
17	C₄×D₄×C₈₃	C₄	1	4	yes	—
18	C₄×D₄×C₈₃	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₈₃	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₈₃	C₄	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
22	#13050574	—	2	16	no	—
23	C₂×C₈₃⋊C₈	—	2	4	no	—
24	C₂×C₈₃⋊C₈	—	2	2	no	—
25	C₂×C₈₃⋊C₈	D₄	1	4	yes	complemented
26	C₂×C₈₃⋊C₈	C₂³	1	4	yes	—
27	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
28	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
29	C₂⁴×C₈₃	C₂×C₄	1	2	yes	—
30	C₂×D₄×C₈₃	—	2	4	no	—
31	C₂×D₄×C₈₃	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₈₃	—	2	4	no	—
33	C₂²×C₄×C₈₃	—	2	4	no	—
34	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₈₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₃	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₈₃	C₂×C₄	1	8	yes	—
42	C₄²×C₈₃	C₂×C₄	1	2	yes	—
43	C₄²×C₈₃	C₂³	1	2	yes	—
44	C₂³×C₈₃	—	2	4	no	—
45	C₂³×C₈₃	—	2	4	no	—
46	C₂³×C₈₃	—	2	4	no	—
47	C₂³×C₈₃	C₂²×C₄	1	4	yes	—
48	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	D₄×C₈₃	—	2	8	no	—
50	C₂×C₄×C₈₃	—	2	4	no	—
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	—	2	8	no	—
53	C₂×C₄×C₈₃	—	2	4	no	—
54	C₂×C₄×C₈₃	—	2	4	no	—
55	C₂×C₄×C₈₃	M₁₆	1	4	yes	complemented
56	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₈₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₈₃	#41	1	2	yes	—
62	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
63	C₈₃⋊C₈	—	2	4	no	—
64	C₈₃⋊C₈	—	2	2	no	—
65	C₈₃⋊C₈	C₂×D₄	1	4	yes	complemented
66	C₂²×C₈₃	—	2	4	no	—
67	C₂²×C₈₃	—	2	4	no	—
68	C₂²×C₈₃	—	2	4	no	—
69	C₂²×C₈₃	—	4	8	no	—
70	C₂²×C₈₃	C₄×D₄	1	2	yes	—
71	C₂²×C₈₃	C₂×M₁₆	1	2	yes	—
72	C₂²×C₈₃	#131	1	2	yes	—
73	C₂²×C₈₃	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₈₃	—	2	4	no	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	C₄×D₄	1	2	yes	—
77	C₄×C₈₃	C₂×M₁₆	1	4	yes	complemented
78	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
80	C₂×C₈₃	—	4	4	no	—
81	C₂×C₈₃	#436	1	4	yes	complemented
82	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
84	C₂×C₈₃	C₂×#131	1	1	yes	characteristic
85	C₂×#436	—	83	1	no	maximal, Sylow
86	C₈₃	C₂×#436	1	1	yes	characteristic, complemented, Sylow
87	C₂²×M₁₆	—	83	2	no	—
88	C₂×C₄×D₄	D₈₃	1	1	yes	characteristic
89	#436	—	83	8	no	—
90	C₂×C₄⋊C₈	—	83	1	no	—
91	C₂×C₂²⋊C₈	—	83	2	no	—
92	C₂×C₄×C₈	—	83	1	no	—
93	C₂²×D₄	Dic₈₃	1	1	yes	characteristic
94	C₂³×C₄	C₂×D₈₃	1	2	yes	—
95	C₂×M₁₆	—	83	8	no	—
96	C₂×M₁₆	—	166	8	no	—
97	C₂²×C₈	—	83	1	no	—
98	C₂²×C₈	—	83	2	no	—
99	C₂²×C₈	—	83	1	no	—
100	C₄×D₄	Dic₈₃	1	4	yes	—
101	C₄×D₄	C₂×D₈₃	1	4	yes	—
102	C₂×C₄⋊C₄	Dic₈₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	Dic₈₃	1	2	yes	—
104	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
105	C₄⋊C₈	—	83	4	no	—
106	C₂²⋊C₈	—	83	8	no	—
107	C₄×C₈	—	83	4	no	—
108	C₂⁴	C₂×Dic₈₃	1	2	yes	—
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	C₂×Dic₈₃	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
114	C₂²×C₄	C₂×Dic₈₃	1	2	yes	—
115	C₂²×C₄	C₂×Dic₈₃	1	4	yes	—
116	C₂²×C₄	C₂×Dic₈₃	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
118	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
119	M₁₆	—	166	16	no	—
120	C₂×C₈	—	83	4	no	—
121	C₂×C₈	—	83	2	no	—
122	C₂×C₈	—	83	2	no	—
123	C₂×C₈	—	83	4	no	—
124	C₂×C₈	—	166	4	no	—
125	C₂×C₈	—	166	2	no	—
126	C₄⋊C₄	C₂×Dic₈₃	1	4	yes	—
127	C₂²⋊C₄	C₂×Dic₈₃	1	8	yes	—
128	C₄²	C₂×Dic₈₃	1	2	yes	—
129	C₄²	C₂²×D₈₃	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂²×Dic₈₃	1	4	yes	—
134	C₂³	C₂²×Dic₈₃	1	1	yes	characteristic
135	D₄	—	2	8	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	#13050574	1	4	yes	complemented
142	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
143	C₂×C₄	C₂²×Dic₈₃	1	4	yes	—
144	C₂×C₄	C₂²×Dic₈₃	1	2	yes	—
145	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
146	C₂×C₄	D₄×D₈₃	1	2	yes	—
147	C₂×C₄	#13050589	1	2	yes	—
148	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
149	C₈	—	83	4	no	—
150	C₈	—	166	4	no	—
151	C₈	—	166	2	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	4	8	no	—
156	C₂²	D₄×Dic₈₃	1	2	yes	—
157	C₂²	C₂×#13050574	1	2	yes	—
158	C₂²	#14581799	1	2	yes	—
159	C₂²	C₂³×Dic₈₃	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	D₄×Dic₈₃	1	2	yes	—
163	C₄	C₂×#13050574	1	4	yes	complemented
164	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
165	C₄	C₂×#13050589	1	1	yes	characteristic
166	C₂	—	4	4	no	—
167	C₂	#19127779	1	4	yes	direct product factor
168	C₂	C₂×D₄×Dic₈₃	1	1	yes	characteristic
169	C₂	C₂²×#13050574	1	1	yes	characteristic
170	C₂	C₂×#14581799	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂×C₄×D₄×C₈₃	C₂
socle	C₂³×C₈₃	C₂²×C₄
prime radical	C₂²×D₄×C₈₃	C₄
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₈₃
hypocenter	C₈₃	C₂×#436
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13050574	C₂	semidirect
2. #19127779	C₂	direct (N × H)
3. C₂×C₄⋊(C₈₃⋊C₈)	C₂	semidirect
4. C₂×C₂²⋊(C₈₃⋊C₈)	C₂	semidirect
5. C₂×C₄×C₈₃⋊C₈	C₂	semidirect
8. C₂×#13050574	C₂²	semidirect
12. C₄⋊(C₈₃⋊C₈)	C₂²	semidirect
13. C₂²⋊(C₈₃⋊C₈)	C₂²	semidirect
14. C₄×C₈₃⋊C₈	C₂²	semidirect
25. C₂×C₈₃⋊C₈	D₄	semidirect
55. C₂×C₄×C₈₃	M₁₆	semidirect
65. C₈₃⋊C₈	C₂×D₄	semidirect
77. C₄×C₈₃	C₂×M₁₆	semidirect
81. C₂×C₈₃	#436	semidirect
86. C₈₃	C₂×#436	semidirect
141. C₂×C₄	#13050574	semidirect
163. C₄	C₂×#13050574	semidirect
167. C₂	#19127779	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13050574	2
#19127779	2
C₂×C₄⋊(C₈₃⋊C₈)	2
C₂×C₂²⋊(C₈₃⋊C₈)	2
C₂×C₄×C₈₃⋊C₈	2
C₂×C₄×D₄×C₈₃	2
C₂×#436	83

Subgroups — C2×#19127779 #24916150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19127779 #24916150